# A logical analysis of instrumentality judgments:
means-end relations in the context of experience and expectations

Kees van Berkel Electronic address<EMAIL_ADDRESS>Ruhr Universität Bochum,
Institute for Philosophy II, Universitätsstraße 150, 44801 Bochum, Germany
Tim S. Lyon Electronic address<EMAIL_ADDRESS>Technische
Universität Dresden, Institute of Artificial Intelligence, Nöthnitzer Str. 46,
01069 Dresden, Germany  Matteo Pascucci Electronic address:
<EMAIL_ADDRESS>Institute of Philosophy, Slovak Academy of
Sciences, Klemensova 19, 811 09 Bratislava, Slovak Republic

###### Abstract

This article proposes the use of temporal logic for an analysis of
instrumentality inspired by the work of G.H. von Wright. The first part of the
article contains the philosophical foundations. We discuss von Wright’s
general theory of agency and his account of instrumentality. Moreover, we
propose several refinements to this framework via rigorous definitions of the
core notions involved. In the second part, we develop a logical system called
Temporal Logic of Action and Expectations ($\mathsf{TLAE}$). The logic is
inspired by a fragment of propositional dynamic logic based on indeterministic
time. The system is proven to be weakly complete relative to its given
semantics. We then employ $\mathsf{TLAE}$ to formalise and analyse the
instrumentality relations defined in the first part of the paper. Last, we
point out philosophical implications and possible extensions of our work.

Keywords: Action logic $\cdot$ Instrumentality $\cdot$ Means-end relation
$\cdot$ Philosophical logic $\cdot$ Temporal logic $\cdot$ Von Wright

##  1 Introduction

Agents shape their world by making choices, performing actions, and exercising
abilities. They reason practically about attaining ends, plan short-term and
long-term, and comply with and violate norms. Agents may be right, lucky, and
mistaken in their planning. In all of the above, instrumentality
statements—also referred to as means-end relations—play a vital role. They
provide reasons to act in one way instead of the other. For instance, the
statement ‘Taking the A-train is an excellent means for going to Harlem’ may
influence whether I visit my friend Eduard, who lives in Harlem, by train. My
experience with traffic jams in New York may cause me to refrain from going by
car instead.

The concepts of action, ability, and choice are central to any theory of
agency [2, 15, 17, 18, 32] and have been extensively investigated in
philosophical logic [3, 5, 12, 27, 34]. In contrast, the philosophical and
logical investigation of instrumentality relations has thus far received
comparably little attention in the literature. In particular, the formal study
of how agents acquire and comparatively assess the quality of instrumentality
judgments remains to be conducted. This is especially noteworthy due to the
role of instrumentality statements in the fields of practical reasoning [4,
13, 18, 25, 33, 36], AI planning [23, 24], and linguistics [14, 26, 28].
Although it is common to analyse reasoning with such statements, none of the
above accounts treat how various instrumentality judgments are obtained,
compared, and assessed. To our knowledge, Georg Henrik von Wright is the only
philosopher who provides such a philosophical, yet brief, account of
instrumentality judgments. This article provides a formal account of
instrumentality inspired by von Wright’s philosophy.

Consider the following practical problem:

> I would like to have this small parcel opened. What should I do? Which
> action is the best choice for securing my desired end? For instance, would
> ripping the parcel’s cardboard, cutting the tape with a knife, or using a
> pair of scissors be most suitable for my purpose?

To satisfactorily address the above problem, I must find out which actions are
(most) suitable for attaining my goal. Various challenges arise: I need to
somehow collect candidate instruments that will satisfactorily resolve my
practical problem. Subsequently, none of the instruments at my disposal may be
necessary, that is, there are only sufficient means available. How should
candidate means be compared? Which must be preferred and chosen? Is there a
difference in the quality of these available instruments?

In the aforementioned fields, instrumentality relations are commonly taken as
given. For instance, most accounts of practical reasoning limit illustrations
of practical inference to cases of necessary means, yielding oversimplified
representations [18]. Exceptions are [13, 21, 30] which discuss versions of
comparitivism as a way to resolve cases in which various sufficient
instruments are available. Nonetheless, in all these accounts instrumentality
relations are assumed to be present from the outset, bypassing the following
questions:

  * (q1)

How are instrumentality judgments acquired by an agent?

  * (q2)

How can the comparative quality be assessed for such judgments?

This article addresses questions q1 and q2. Concerning the above practical
problem, one may respond to these questions in various ways. For instance, one
may invoke experience, apply theoretical knowledge, or follow the advice of a
trustful adviser. Given my desire to open the parcel, I may recall from
personal experience that using a knife or a pair of scissors always sufficed.
Then again, I may recall a piece of advice that one should avoid using knives
to open parcels since they may damage the contents. Alternatively, I may
search the internet for possible ways to open my parcel. In this article, we
propose several ways to compare actions as instruments for a given purpose. In
particular, we assign a pivotal role to the agent’s past in yielding
instrumentality judgments.

##### Three Criteria.

Guided by questions q1 and q2, any account of instrumentality should be able
to address the following three points: First (I), the account must clarify
what it means to say that for an agent $\alpha$, action $\Delta$ is an
instrument serving purpose $\varphi$. The fact that instrumentality is a
relative notion becomes clear when we see that different actions may serve
different ends for different agents with different abilities. Considering the
previous example, I might not be comfortable with knives which means that
using a knife to open a parcel would not be a suitable instrument for me
(although it might be for someone else).

Second (II), the account must not only provide procedures to determine which
instruments are suitable for the purpose at hand, but it must also allow for a
comparison of the different instruments collected. The assessment of
instrumentality judgments is axiological (from the Greek ‘axía’, for ‘value’),
i.e., it provides a label expressing a specific value of the instruments at
hand; e.g., scissors are good instruments for opening parcels, although using
your hands would be better. Eventually, such axiological judgments concerning
instruments may serve as a guide in practical decision-making.

As a third point (III), we observe that one of the typical features of
instrumentality relations is that they are not given once and for all. In many
cases, such judgments are established via inductive arguments, relying on past
experience witnessing the connection between the terms involved. As famously
noted by David Hume [20], inductive arguments are affected by a fundamental
problem: how are we justified in making inferences from an observed connection
in the past to instances of that connection of which we have no experience?
For example, in forming instrumentality judgments based on experience, an
individual agent can often not collect all relevant past cases that would
settle the issue. Even then, future cases may still be different. For this
reason, judgments of instrumentality are essentially defeasible. For instance,
such judgments may need to be revised due to new personal experience,
additionally secured information on the past, and newly received data from
other agents.

##### Contributions.

In this work, we address the above three criteria. We do this by developing a
formal account of instrumentality. Our account is grounded in von Wright’s
theory of agency in general and his analysis of instrumentality in particular.
In the work ‘The Variety of Goodness’ [35], von Wright provides a
philosophical discussion of instrumentality by which actions can be judged as
‘good instruments’ for specific purposes.

In order to address (I)-(III), we take von Wright’s account as a departure
point and extend it where necessary. Those parts rooted in von Wright’s
account are made explicit throughout this work. We define a logical system
called the Temporal Logic of Actions and Expectations ($\mathsf{TLAE}$) that
will formally capture the developed theory of instrumentality. We prove the
weak completeness of $\mathsf{TLAE}$ with respect to a corresponding Kripke-
style semantics using a variation of the Fischer-Ladner construction for
propositional dynamic logic [16]. Employing $\mathsf{TLAE}$, we provide
formalisations of the acquired conceptions of (comparative) instrumentality
and discuss the philosophical implications of our formal setting.

##### Outline.

Section 2 consists of a brief analysis of von Wright’s general theory of
agency and his theory of instrumental goodness. We refine von Wright’s theory
by supplementing criteria for comparing instruments that serve the same
purpose. In particular, in Section 2.2 and 2.3, we address criterion I by
providing various notions of instrumentality deduced from an agent’s past
experience. In Section 2.4, we deal with criterion II by providing different
ways of comparing instrumentality judgments, yielding various value judgments.
Section 2.5 is devoted to the defeasibility of instrumentality judgments as
expressed in criterion III, and additionally contains a refinement of our take
on criterion I. In Section 3, we discuss the requirements of our formal
language and introduce the logical system $\mathsf{TLAE}$, which is a temporal
extension of the logic ‘Logic of Actions and Expectations’ ($\mathsf{LAE}$) in
[6]. In Section 4, we prove that $\mathsf{TLAE}$ is weakly complete. After
that, in Section 5, we discuss the philosophical implications of our formalism
and address criteria I-III formally: we logically formalise different notions
of instrumentality (criterion I), present several semantic notions of
comparative instrumentality (criterion II), and discuss the defeasible nature
of instrumentality statements (criterion III). In Section 6, we address future
work.

##  2 Agency and Instrumentality

The backbone of our philosophical analysis will be von Wright’s general theory
of agency, as laid out in [32, 34, 35]. We start with a brief analysis of the
theory and refer to [3, 6, 29] for a more extensive discussion. We
subsequently provide a philosophical analysis of instrumental goodness and
comparative instrumentality. The former is primarily based on von Wright’s
discussion of instrumental goodness as laid out in [35, pp. 19-40]. In short,
instrumental goodness covers the study of judgments concerning how well
instruments serve their purpose (equivalently, how well means serve their
ends). Since von Wright’s analysis is sparse—i.e., being a sub-topic of his
theory of goodness [35]—we extend his account in two ways: (i) we discuss
several refined instrumentality notions and (ii) consider possible definitions
of comparing instruments. The theory presented in this section will ground the
subsequent formalisation.

###  2.1 Von Wright’s General Theory of Agency

According to von Wright, to act is to interfere with the course of nature
[32]. Such interference manifests itself in bringing something about or
preventing something from happening. What is brought about or prevented is a
particular state of affairs, i.e., a partial description of the world such as
‘the parcel is open’. To bring about a result $\varphi$ means to act “in such
a manner that the state of affairs that $\varphi$ is the result of one’s
action” [32, p. 13]. Likewise, prevention of $\varphi$ indicates that one’s
action has succeeded in ensuring $\lnot\varphi$.

$w_{0}$$\lnot\varphi$$w_{1}$$\varphi$$w_{2}$$\lnot\varphi$(i) produce
$\varphi$$a$$n$

$w_{0}$$\varphi$$w_{1}$$\lnot\varphi$$w_{2}$$\varphi$(ii) destroy
$\varphi$$a$$n$

$w_{0}$$\lnot\varphi$$w_{1}$$\lnot\varphi$$w_{2}$$\varphi$(iii) suppress
$\varphi$$a$$n$

$w_{0}$$\varphi$$w_{1}$$\varphi$$w_{2}$$\lnot\varphi$(iv) preserve
$\varphi$$a$$n$

Figure 1: Von Wright’s four elementary types of action. The transition (arrow)
from $w_{0}$ denoting the agent acting is labelled $a$, and the alternative
transition indicating the agent’s non-interference with nature is labelled
$n$.

The above concept of action is founded on the notion of change. In fact, for
von Wright, any theory of agency and action must presuppose an account of
change, e.g., see [3]. He takes change to define a transition from an initial
state (i.e., the present moment) to an end-state (i.e., a future moment). Such
transitions can be either agent-independent (e.g., a moon eclipse) or agent-
dependent (e.g., me cutting the tape of my parcel). The agent-dependent
setting forces a non-deterministic worldview. That is, to bring something
about forces at least the following three elements: the initial state, in
which the agent finds herself, the actual end-state (which is the state that
emerges after the performed action), and an alternative end-state (which would
result if the agent refrains from performing the action).

Von Wright discusses various relations between these three states. By bringing
together the above account of change with the twofold distinction of bringing
about and prevention, he characterises four types of action: producing,
destroying, suppressing, preserving. The first two bring about something,
whereas the latter two prevent something. (We refer to [3, 6] for a discussion
of these action types in a formal setting.) The four action types are
characterised in Figure 1. For instance, at (iii), the act of suppressing
$\varphi$ indicates that at the initial state $\lnot\varphi$ holds, through
the agent’s acting $\lnot\varphi$ continues to hold, and if the agent would
not have acted $\varphi$ would have come about.

Von Wright’s reading of the four action types is arguably too strong: the
agent’s acting in Figure 1 ultimately decides the fate of $\varphi$. In the
case of producing, by acting, the agent ensures $\varphi$, whereas, by not-
acting, the agent can ensure $\lnot\varphi$. In other words, von Wright’s
account takes agency as causally sufficient in both directions, e.g., see [3].
Furthermore, in Figure 1, there is no distinction made between different kinds
of action an agent can perform at $w_{0}$.

Since a general analysis of agency involves many distinct agents and distinct
agents can simultaneously perform distinct actions, an individual agent’s
action is often not causally sufficient. This is known as the uncertainty of
action [5]. We adopt a generalisation of von Wright’s approach that includes
this uncertainty. A transition involves the following three elements: (i) an
initial state, (ii) a set of actions, and (iii) a set of possible final
states. Henceforth, we also refer to such states as moments in indeterministic
time. A single agent does not entirely control the course of events, but even
the complete set of actions performed by all agents involved does not
necessarily entail a unique end-state (cf. the influence of nature). For this
reason, we say that a set of actions causally contributes to the attainment of
the (actual) end-state of a transition only if the end-state would have been
different without the performance of that set of actions.

$w_{0}$$\lnot\varphi,\psi$$w_{1}$$\varphi,\lnot\psi$$\underline{w_{2}}$$\varphi,\psi$$w_{3}$$\varphi,\psi$$w_{4}$$\lnot\varphi,\psi$$\dots$$w_{n}$$\varphi,\lnot\psi$
$\Delta$ $\overline{\Delta}$ Figure 2: Indeterministic time and uncertainty of
action: producing $\varphi$ through performing action $\Delta$ at $w_{0}$. Let
$\varphi$ stand for ‘the parcel is open’, $\psi$ for ‘the parcel is
undamaged’, and $\Delta$ for ‘using a knife to cut the parcel’s tape’.
Alternatively, one may label arrows to denote $\Delta$-transitions (cf. Figure
3).

For instance, considering Figure 2, we say that the action $\Delta$ brings the
agent from the present moment $w_{0}$ to either one of the future moments
$w_{1}$, $w_{2}$, or $w_{3}$ without strictly determining either of the three.
Still, all three moments satisfy $\varphi$. Thus, we say that the agent can
produce $\varphi$ by performing $\Delta$ at $w_{0}$, even though the agent
cannot secure a unique future moment with action $\Delta$ (e.g., one could
read $\varphi$ as ‘the parcel is open’ and $\Delta$ as ‘using a knife to cut
the parcel’s tape’). Furthermore, performing $\overline{\Delta}$—which is the
complement of $\Delta$—could lead to at least one future moment where
$\lnot\varphi$ holds, namely, $w_{4}$. Suppose that $w_{2}$ is the actual
future moment (underlined in Figure 2). In that case, the agent causally
contributed to the attainment of $w_{2}$ through performing $\Delta$, since if
the agent had refrained from performing $\Delta$, the resulting possible
future moments would have been one of $w_{4},\ldots,w_{n}$. Last, although
$\Delta$ successfully ensures $\varphi$, the action can still fail to secure
other ends such as $\psi$ in Figure 2 (e.g., where $\psi$ reads ‘the parcel is
undamaged’).

As a final remark concerning the nature of the term ‘action’, we will follow
the usual distinction between types (i.e., generic categories, such as
‘writing’) and tokens (i.e., concrete instances in specific circumstances,
such as the action of a particular person writing on a particular blackboard
on a specific date; see [17] for an extensive discussion). Von Wright adopts a
similar demarcation by distinguishing between act-categories, on the one hand,
and act-individuals, on the other hand [32, p. 36]. It suffices to restrict
our analysis to atomic actions, negative actions (e.g., ‘not crossing the
street’) and complex actions (e.g., ‘turning left or turning right’ and
‘turning left and hitting the break’). We do not consider sequences of
actions.

###  2.2 Instrumentality and Instrumental Goodness

We are now in a position to address criterion I of Section 1, and clarify what
it means that an action $\Delta$ is an instrument serving purpose $\varphi$.
Our analysis will yield two provisional definitions at the end of the next
subsection. Central to the study of instrumentality is the relation between an
action and a result. The former is the instrument for the desired outcome
expressed in the latter. The outcome can therefore be seen as the purpose of
performing the action in question. Thus, we refer to the action as an
instrument serving a particular purpose. For example, ‘pulling down the lever
of a door and drawing the door towards you’ is the instrument for the result
‘the door is open’.

Following von Wright [35, p. 21], we can group actions into categories, or
kinds, according to the purposes served. To illustrate, no action will qualify
as a cutting-instrument unless it can serve the purpose of cutting. In this
respect, the ability to serve a ‘cutting purpose’ is a functional
characteristic of members of the kind ‘cutting’(-instruments). Actions do not
necessarily serve a unique purpose. As an instrument, an action can be a
member of several kinds, serving multiple purposes. The action ‘using a knife’
is an instrument for opening parcels and for peeling apples.

The goodness of an instrument is judged in relation to how it serves the
purpose with which it is associated. Consequently, an instrument may serve
certain purposes with excellence while serving others poorly. In this article,
we concentrate on a specific agentive source for judgments of instrumental
goodness: the agent’s (personal) experience with the instruments serving the
purpose. In the case of the agent’s experience, the agent checks her past for
applications of those actions belonging to the same type as the one under
consideration, checking whether these past applications successfully served
the purpose at hand. This temporal component will be central to our definition
of instrumental goodness.

We point out that for instrumentality judgments what matters is an agent’s
_perception_ of the relation between an action and effect. That is, we are
concerned with what the agent believes is the relation between an instrument
and purpose, which may or may not be rooted in any physical causal relation.
For instance, to form a judgement concerning ‘cutting knives’ and ‘opening
parcels’, an agent needs no (prior) knowledge of the physics involved in
cutting the tape of a parcel (the sharpness of the blade, the movement of the
arm, the consistency of the tape, etc.). In our case, what counts is the
agent’s (past) experience of the event (opening a parcel with a knife) and the
beliefs that were formed accordingly. Therefore, when we say that a knife is
an instrument for opening parcels, we do not refer to its set of physical
causal qualities per se. In our opinion, the notion of an instrument is a
practical one, and it would generally be better to keep it distinct from the
notion of a cause, as employed in the analysis of physical connections between
things.111This relates to Hume’s criticism of the notion of causation:
causality judgements depend on the set of beliefs of a specific group of
agents given a specific context and cannot be generalised to constitute
physical laws. They are not intended to have such a universal validity. We see
past experience as a fruitful approach to agentive reasoning since it is a
source accessible to the agent at any given time.222Past experience is not
exhaustive. Additionally, one may consider here-say, advice, education, and
theoretical knowledge as sources alternative to past experience. Such
alternative sources fulfil an important role in accounting for the employment
of instruments that have never been used before. We leave this to future work.

This article further develops the idea of rooting instrumentality judgments in
past experience. Von Wright involves past experience in his analysis of
goodness of skill and instrument, but the expansion of the idea as presented
in this article is our own. We propose that instrumentality judgements of this
type are formed in three steps. In the first step, the agent collects all
relevant evidence available from the past. This is the empirical part of the
procedure. The second step consists in forming context-relative judgments
based on what past experience has taught. For instance, in considering all
cases in which two actions were performed, it may turn out that a particular
outcome was more frequently obtained in association with one of the actions
than with the other. This is the inductive part of the procedure (i.e., the
generalisation of the relevant experience). Third, inferences can be drawn
from these context-relative judgements in combination with considerations of
chance, (unpredictable) interference, and additional circumstances. Such
inferences serve as a guide for the agent’s current behaviour. This is the
deductive part of the argument.

###  2.3 Three Notions of Instrumentality

The ambivalent nature of the philosophical concept of ‘good’ may suggest that
judgments of instrumental goodness are not objective. However, von Wright [35,
p. 25] argues that in dealing with instrumental goodness, the relation between
‘purpose’, ‘instrument’ and ‘good’ is, in fact, objective. Namely, judgments
of instrumental goodness express two types of ‘connexion’. The first is a
causal connexion, expressing a relation between the instrument (as a perceived
cause) and its purpose (the desired effect). Von Wright emphasises that this
connexion can be empirically checked and is thus objective. Extending the
above to cover past experience, we observe that collecting an agent’s relevant
past experience is also an empirical process, having the same objectivity
status. To avoid confusion, we refer to the first connexion as an empirical
connexion instead. The second is a logical connexion that holds between the
purpose in question and the term ‘better’. Namely, given the agent’s desire
for a particular effect (the purpose), one can order the available actions
according to their empirical connexion such that one action can be logically
determined to be a better instrument than another. Accordingly, judgments of
instrumental goodness can be objective since we refer to an observable,
empirical property of the instrument and assign goodness based on an ordering,
which is a judgment of logic.

There are several ways to assess the goodness of an instrument. For von
Wright, ‘goodness’ always refers to particular observable properties of the
instrument that make it suitable for some purpose. This property is called the
good-making property. Good-making properties are the properties of the
instruments that enable the logical ordering of goodness. To illustrate, if
one intends to have some chopped vegetables (the purpose), the good-making
property ‘sharper’ can determine which of the available knives (instruments)
is better. Here, sharpness determines the causal connexion between the knife
and cutting. One can then order all available knives according to their
sharpness to determine which are sharper and, thus, better [35, p. 25].

This article takes past experience as a general property for ordering
instruments. In particular, we focus on the successful applications of the
instrument that guaranteed the effect, as witnessed by the agent. Given this
good-making property, a judgment about one instrument ‘being better’ than
another is a logical consequence of ‘being more successful at guaranteeing the
desired result’.

To qualify a generic instrument—i.e., action type—as good for some purpose, we
need to ground our judgment in the past performance of concrete actions, i.e.,
action tokens. We refer to this temporal component as the historical witness
of an instrument’s suitability. In inquiring about whether to apply a certain
instrument, agents often base decisions on statements such as:

  * (i)

‘it has worked before’;

  * (ii)

‘so far, it has not disappointed me’;

  * (iii)

‘well, it has thus far worked better than any of the alternatives’.

These three remarks illustrate the importance of temporal reference.
Furthermore, we identify three different criteria of instrumental goodness in
(i)-(iii): The first remark exemplifies, what we will call, the minimum
criterion for any appropriate definition of instrumentality: (i) the action
has served the purpose at least once and, for that reason, it can serve the
purpose as an instrument. That is to say, criterion (i) functions as a lower
bound on the instrument’s suitability, thus identifying candidate instruments.
In the second remark, we recognise an upper bound, i.e., a maximum criterion:
(ii) there have been applications of the instrument and these applications
have always served the purpose. Criterion (ii) is referred to as instrumental
excellence. In the last remark, we identify a comparative approach to
instrumental goodness: (iii) the action is suitable in comparison to
alternative actions.

Observe that (i) and (ii) express, respectively, an existential criterion and
a universal (constructive) criterion. These criteria enable us to label
instruments as ‘good’, independent of their relation to other instruments. The
third (iii) interpretation is a comparative notion that labels instruments as
good but only relative to other instruments. If an instrument is not excellent
in itself but is the best among others, we refer to the instrument as
comparatively excellent.

We make our analysis precise in Definition 1. In what follows, we use
$\Delta$, $\Gamma$, $\Sigma,\dots$ to denote action types, $\alpha$,
$\alpha_{1},\alpha_{2}\dots$ to denote agents, $\varphi$, $\psi$, $\chi,\dots$
to denote propositions (i.e., descriptions of states of affairs), and
$w,v,u,\dots$ to denote moments. The terms ‘moment’ and ‘state’ are used
interchangeably and the expression ‘$\varphi$-instrument’ abbreviates
‘instrument to obtain the state of affairs described by $\varphi$’.

######  Definition 1 (Definition of Candidate Instruments).

  * (1)

candidate instruments: An action type $\Delta$ is a candidate
$\varphi$-instrument for agent $\alpha$ at moment $w$ if and only if (i)
$\Delta$ has led to $\varphi$ for $\alpha$ at least once in the past of $w$.

  * (2)

excellent candidate instruments: An action type $\Delta$ is an excellent
candidate $\varphi$-instrument for agent $\alpha$ at moment $w$ if and only if
(i) $\Delta$ is a candidate $\varphi$-instrument for $\alpha$ at $w$ and (ii)
$\Delta$ has always led to $\varphi$ for $\alpha$ in the past of $w$.

  * (3)

better candidate instruments: An action type $\Delta$ is a better candidate
$\varphi$-instrument for agent $\alpha$ at moment $w$ than the candidate
$\varphi$-instruments $\Gamma_{1},\ldots,\Gamma_{n}$ available to $\alpha$ at
$w$ if and only if (i) $\Delta$ is a candidate $\varphi$-instrument for
$\alpha$ at $w$ and (ii) in the past of $w$, action $\Delta$ was more
successful for $\alpha$ in guaranteeing $\varphi$ than
$\Gamma_{1},\ldots,\Gamma_{n}$.

We point out that judgments of the type expressed above vary over agents, and
since the qualification of instruments is defined relative to the past, these
judgments are strictly dependent on a vantage point too. This makes the above
three definitions defeasible: new experience may cause the agent to revise
previous judgments. For example, horses may have been the best option for
private transport before cars. We discuss defeasibility in detail in Section
2.5.

###  2.4 Different Ways of Comparing

In Definition 1, we defined the notion of a ‘better instrument’ in terms of
the relative success of the instrument in question. In what follows, we
address criterion II of Section 1 and discuss various ways of comparing the
success of candidate instruments. The analysis will yield different
definitions of comparative instrumental goodness.

Von Wright does not provide an explicit method for comparing candidate
instruments, but we find some hints in his analysis of technical goodness (the
goodness of ability, capacity, and skill) [35].333Von Wright [35] explicitly
distinguishes technical goodness from instrumental goodness. The former
relates to agents’ skills in obtaining results and applying instruments. The
latter denotes an impersonal analysis of how instruments serve purposes. We do
not differentiate between instrumental and technical goodness, but the logic
developed in this paper does allow for reasoning with agent-dependent and
-independent notions of instrumentality. There are two ways of testing whether
an agent excels at guaranteeing a particular result or at performing some
action, namely, (i) through competition and (ii) through achievement [35, Ch.
2]. Method (i) evaluates personal performance in relation to other agents. In
the context of the Olympic games, competition is a means to determine which of
all candidate athletes excels at, say, the javelin throw. Such competition can
be defined in terms of ‘who comes in first’ in a single game (the absolute
best) or in terms of who ended first in a sequence of games (the overall
best). Method (ii) evaluates an agent’s performance of an action, not in light
of other agents acting, but in relation to a predetermined threshold of
excellence. Achievement may be defined as beating a predetermined time or
record (which may or may not be previously set by other agents) or achieving a
qualifying threshold. Such thresholds function as markers of a specific grade
of excellence.

Following von Wright [35, p. 34], both (i) and (ii) are methods of
establishing goodness by ‘degree of distinction’, thus constituting a
comparison. Method (i) hints at judging goodness on the basis of ordering the
success ratios of participating candidates, whereas method (ii) hints at the
usage of pre-established thresholds that suitable candidates must pass.

Translating the above to instrumental goodness, we find two notions of
comparison. The first compares a candidate instrument to alternative
candidates, which calls for an ordering of (all) suitable instruments serving
the purpose at hand. First, we gather all instruments that potentially deserve
the label ‘good-instrument’, these are candidate instruments (see Definition
1). Then, for each candidate, we collect its successful and unsuccessful
applications in attaining the desired outcome. Thus, we construct a success-
failure ratio for each candidate instrument. An instrument comparatively
excels in serving a particular purpose if it has the highest success ratio
among its alternatives.

However, we point out that this ordering might fail to properly assess whether
an instrumentality relation exists between $\varphi$ and available actions
$\Gamma_{1},\dots,\Gamma_{n}$. This can happen when some of the considered
actions were performed only a small number of times within the considered
time. A very small sample size may have no effective statistical power since
the observed connections between $\varphi$ and $\Gamma,\dots,\Gamma_{n}$ could
have resulted from mere chance. In such cases, it is reasonable to set
thresholds such that one can avoid engaging in a judgment of those actions
that have not been sufficiently tested.

The second method considers a single candidate instrument and checks whether
it has satisfied a certain threshold. We identify two types of threshold. The
first consists in setting a minimum threshold of the success-failure ratio.
For instance, a candidate $\varphi$-instrument is a good $\varphi$-instrument
if it guarantees the effect $\varphi$ at least half of the time. Although such
a threshold is reasonable, there are problems with it. Namely, suppose I throw
a dart at the bullseye and end up hitting the bullseye during my first throw
(without any practice). Naively, for me throwing darts is an excellent
instrument for hitting the bullseye since, thus far, I have always been
successful. Often, it makes sense to require an additional threshold in the
light of which the candidate instrument must be evaluated. This is the second
type: we may require that the instrument has been at least applied $n$-many
times in attempting to attain the end in question (e.g., think of products
that must be tested before they can be sold).

Likewise, time plays a role in setting thresholds: imposing a threshold on the
number of applications entails imposing a minimum length on the time interval
considered. One needs to ensure that the time interval allows for a number of
action performances that is at least equal to the threshold. Setting such a
limit has another function: it excludes cases that lie too far in the past.
For instance, in determining whether I excel at hitting the bullseye it
suffices to consider, say, my last 100 attempts. Without a limit, past cases
when I was still learning how to play darts would be included, thus
misrepresenting my current skills.

Based on the above, we propose the following two definitions of comparison.

######  Definition 2 (Notions of Comparison).

A candidate $\varphi$-instrument $\Delta$ can be evaluated with respect to:

  1. 1.

other candidate $\varphi$-instruments $\Gamma_{1},\ldots,\Gamma_{n}$ based on
a success-failure ratio.

  2. 2.

a threshold:

(i) on a lower bound of past co-occurrences of $\Delta$ and $\varphi$;

(ii) on a lower bound of the ratio of past co-occurrences of $\Delta$ and
$\varphi$ versus past occurrences of $\Delta$ without $\varphi$;

(iii) on both (i) and (ii).

The above distinction gives rise to two types of comparative instrumental
goodness: (1) goodness in relation to other instruments and (2) goodness in
relation to a set threshold. Combinations of (1) and (2) are also possible:
e.g., consider the context of the Olympic games, where an athlete must acquire
a certain amount of credits in order to enter the games initially (threshold).
An advantage of adopting approach (1) is that it completely preserves the
logical status of comparative goodness by ordering available instruments based
on their success ratio, independent of any external measurement such as a
threshold. The downside of (1) is that it allows for cases such as ‘having a
lottery ticket is an excellent instrument for winning the lottery since it is
the only way to win (despite not being sufficient)’. In such cases, a certain
threshold is desirable.444There is another issue concerning the lottery
example. Since a myriad of tickets is available for any single lottery draw,
and tickets are assigned to buyers randomly, the win-loss proportion should be
based on the number of winning tickets vs the number of total tickets. In such
cases, reference to an agent is not a primary parameter. The issue relates to
the low probability of the instrument serving the purpose. This indicates that
some actions may not be suitable as instruments at all. $n$ times and winning
$n$ times will not make it an excellent instrument. There is no sufficient
instrument for winning the lottery (perhaps except by buying all the tickets)
since winning is beyond the agent’s abilities.

Following von Wright, we can label instruments as ‘better’, ‘best’, and
‘poorest’ by ordering the available instruments. The best instrument will be
the instrument which is unsurpassed in its success ratio by any other, and the
poorest will be unsurpassed in its low success ratio. A poor instrument,
however, is not necessarily a bad instrument [35, p. 35]. The former does not
serve the purpose well, whereas the latter may serve the purpose well but have
(legally or socially) undesirable side effects (e.g., think of ‘opening a
parcel by first stealing the knife with which you intend to open the parcel’).

###  2.5 Expectations: The Other Temporal Component

So far, we discussed how the past serves as a fruitful source of information
for qualifying relations of instrumentality. Judgments of instrumentality are
essential for practical reasoning since the latter concerns how states of
affairs can be altered through the intervention of an agent (cf. [4, 13, 18,
25]). Practical reasoning is, thus, essentially directed towards the future.
Through instrumentality judgments, agents may generalise past experience and
project it onto the future. This generalisation and projection lies at the
heart of the problem of induction. We now address criterion III of Section 1.

In [31], von Wright deals with the problem of induction. In particular, von
Wright divides the induction problem into two parts (see [31, p. 50]):

  * (q3)

How can we demonstrate that the generalisations we make about experienced
cases are correct? How do we know that our generalisation is ‘complete’?

  * (q4)

How can we demonstrate that such generalisations are reliable for making
predictions? That is, how can we extend our generalisations to the future?

Interestingly, von Wright’s division is temporal: (q3) deals with the past and
(q4) with the future. Regarding generalisations extending to the past, one can
theoretically acquire universally objective judgments by collecting all past
instances of the object under generalisation. However, when it comes to
predictions, the problem of induction truly shows itself:

> “Scarcely anybody would pretend that predictions, even when based upon the
> safest inductions, might not fail sometimes.” [31, p. 51]

Regarding the future, generalisations are inherently defeasible: future
information may falsify our earlier judgments. Although not explicitly stated,
von Wright’s account of instrumentality judgments (which have an inherently
inductive nature) appear to incorporate the same temporal distinction between
(i) collecting past cases and (ii) extending past generalisations to the
future in terms of predictions. We believe that the discussed theory of
instrumentality can account for the problem of induction through the role of
expectations in instrumentality judgments:

> “Judgments of instrumental goodness, usually, even if not necessarily,
> contain a conjectural element” [35, p. 27].

In relation to instrumentality, an expectation is a projection of the past
onto the nearby future that the action will continue to serve the intended
purpose.

In [31], von Wright deals with the general problem of induction as an
inherently temporal problem that only arises through the involvement of future
cases. Likewise, in the setting of practical reasoning, we find specific roles
assigned to the past and the future. We believe that expectation as a
defeasible cognitive attitude warrants the connection between universal
statements about the past and their projection onto the future. Indeed, one
can say that even if an agent experienced relations between actions and
outcomes in the past, which are not sufficient to form a stable and
universally valid judgment about instrumentality, the experience nevertheless
serves to support context-specific and graded judgments in an agent’s
practical deliberation. By limiting such judgments to (defeasible)
expectations instead of universal claims, the problem of induction is avoided
altogether. Despite being defeasible, such judgments may still guide an agent
in decision-making. For instance:

> According to what I have experienced so far, using a knife to cut the tape
> of a parcel has served the goal of opening a parcel to a sufficiently good
> degree, and I expect it to work out in the same way, at least in the near
> future. Using a knife is a good instrument for realising my goal.

Thus, the limits of inductive arguments do not prevent one from formulating
instrumentality judgments to decide how to act in the immediate future.

Now we know that the tentative definitions (Definition 1) provided in Section
2.2 do not suffice, and reference to the conjectural element must be included.
In judging whether an instrument is suitable for a present purpose, we take
our past experience and project it onto the future, conjecturing that its past
success will sustain in the immediate future. Hence, we recognise two temporal
components in instrumentality judgments: (i) past performance of particular
actions subsumed under a certain type and (ii) the expected continuation of
the performance of actions of this type in the nearby future.555We will see
that in the formal framework, the expectations of an agent $\alpha$ correspond
to those (nearby) future moments that $\alpha$ regards likely to happen. We
stress that expectations are not to be confused with epistemic notions of
(incomplete) knowledge: an agent can have expectations about the future apart
from her knowledge of these expected future moments. The first temporal
component is related to the empirical part of arguments used to establish
instrumentality relations; the second temporal component is related to the
inductive part of such arguments (see Section 1).

$w_{0}$$w_{1}$$\varphi$$w_{2}$$\lnot\varphi$$w_{3}$$\varphi$$w_{4}$$\varphi$$w_{5}$$\quad\varphi$$w_{6}$$\quad\lnot\varphi$$\underline{w_{7}}$$\lnot\varphi$
exp-$\alpha$$\Delta$$\Delta$$\Delta$$\Delta$ Figure 3: Expectations and past
generalizations of “$\Delta$ leading to $\varphi$”: agent $\alpha$ expects
(i.e., ‘exp-$\alpha$’ and the dotted lines) the moments $w_{5}$ and $w_{6}$ as
possible future continuations of $w_{4}$, although $w_{7}$ is the actual
(underlined) continuation of $w_{4}$. The arrows labelled with $\Delta$
represent the transitions designated by the performance of $\Delta$.

As a basic example of past generalisations and expectations, consider the
model presented in Figure 3. Let $\alpha$ be an agent at moment $w_{4}$. The
past (that is, $w_{0}$ up to $w_{4}$) provides the agent with the experience
that ‘so far, $\varphi$ occurred after every transition caused by $\Delta$’.
The agent may generalise this observation to ‘generally, performing $\Delta$
leads to $\varphi$’. Suppose that at $w_{4}$, the agent believes that
‘$\Delta$ leading to $\varphi$’ will continue to hold in the (nearby) future.
The moments $w_{5}$ and $w_{6}$ express those moments the agent expects to be
future continuations of $w_{4}$. At $w_{5}$ and $w_{6}$, it is, in fact, the
case that $\Delta$ leads to $\varphi$. However, the model shows that the
actual future continuation of $w_{4}$, namely $w_{7}$, falsifies the
generalisation by making $\lnot\varphi$ true after the performance of
$\Delta$. This captures the idea that future projections are inherently
defeasible: an agent’s estimations may turn out to be wrong. She may have
expected some other future states to be possible or may have wrongly projected
her past experience onto the future. The inclusion of expectations thus allows
us to adequately address criterion III.

Based on the above, we extend Definition 1 on candidate instruments to include
the conjectural element of instrumentality reasoning. The resulting Definition
3 (below) additionally refines our take on criterion I (the first two items)
as well as criterion II (the last two items).

######  Definition 3 (Four Definitions of Instrument).

  * (1)

instruments: An action type $\Delta$ is a $\varphi$-instrument for agent
$\alpha$ at moment $w$ if and only if (i) $\Delta$ has led to $\varphi$ for
$\alpha$ at least once in the past of $w$ and (ii) $\alpha$ expects at $w$
that $\Delta$ will lead to $\varphi$ in the immediate future.

  * (2)

excellent instruments: An action type $\Delta$ is an excellent
$\varphi$-instrument for agent $\alpha$ at moment $w$ if and only if (i)
$\Delta$ is a $\varphi$-instrument for $\alpha$ at $w$ and (ii) $\Delta$ has
always led to $\varphi$ for $\alpha$ in the past of $w$.

  * (3)

better instruments: An action type $\Delta$ is a better $\varphi$-instrument
for agent $\alpha$ at moment $w$ than the $\varphi$-instruments
$\Gamma_{1},\ldots,\Gamma_{n}$ available to $\alpha$ at $w$ if and only if (i)
$\Delta$ is a $\varphi$-instrument for $\alpha$ at $w$ and (ii) in the past of
$w$, action $\Delta$ was more successful for $\alpha$ in guaranteeing
$\varphi$ than $\Gamma_{1},\ldots,\Gamma_{n}$.

  * (4)

goodn instruments: An action type $\Delta$ is a goodn $\varphi$-instrument for
agent $\alpha$ at moment $w$ if and only if (i) $\Delta$ is a
$\varphi$-instrument for $\alpha$ at $w$ and (ii) $\alpha$’s past performance
of $\Delta$ satisfies threshold $n$ at $w$.

Since items (2)-(4) in the above definition are based on definition (1), all
notions of instrumentality are relative to the agent’s expectations about the
instrument at the moment of evaluation. In Definition 3, item (3) and (4)
refer to judgments of instrumentality involving comparisons, respectively
thresholds, as specified in Definition 2. The logic of actions and
expectations introduced in the subsequent sections allows us to capture these
definitions formally.

##  3  A Temporal Logic of Actions and Expectations: $\mathsf{TLAE}$

On the basis of the analysis provided in Section 2 we will develop a logic
that will enable us to formalise the mentioned notions of instrumentality. We
will refer to the logic as the Temporal Logic of Actions and Expectations,
henceforth $\mathsf{TLAE}$, since these will be the main ingredients of its
syntax and semantics. $\mathsf{TLAE}$ is a linguistic and deductive extension
of $\mathsf{LAE}$, a logic developed in [6]. From the linguistic point of
view, the novelty is the use of operators for temporal reference to the past.
From the deductive point of view, the novelty is the use of axioms that
characterise the behaviour of the new operators, as well as their interaction
with the old ones.

For the sake of a self-contained exposition, we will not assume familiarity
with $\mathsf{LAE}$ and provide a detailed presentation of the new framework,
mentioning differences with the old one when needed. We start by listing all
fundamental concepts that we intend to capture and then introduce the formal
language of $\mathsf{TLAE}$ in a rigorous way.

###  3.1 Fundamental Concepts Expressed

Purposes. These are the desired results of actions, represented by formulas
$\varphi,\psi,\chi,\ldots$ (occasionally annotated). As a matter of fact, a
purpose can be equated with a description of a state of affairs. In principle,
it is possible to have very complex descriptions involving, for instance,
modal concepts (e.g., that it possibly rains or it possibly snows).
Furthermore, descriptions of states of affairs may likewise refer to actions
(e.g., the description that the door has been opened or the proposition that
an agent will open the door).

Actions. These can be (candidate) instruments for achieving a purpose. We use
$\delta_{1},\delta_{2},\delta_{3},\ldots$ to represent atomic action types and
build complex action types $\Delta,\Gamma,\Theta,\ldots$ (possibly annotated)
via complementation ‘$-$’ (overline), union ‘$\cup$’, and intersection
‘$\cap$’ of types (e.g., not-opening the door, opening the door or opening the
window, and both opening the door and closing the window, respectively).

Agents. Actions are performed by agents , which will be here denoted by
$\alpha$, $\beta$, $\gamma$, $\ldots$ (possibly annotated). Different actions
may be available to different agents and the skills of a particular agent
might influence outcomes of actions. We will codify the performance of an
action type $\Delta$ by an agent $\alpha$ via a Boolean formula $\varphi$.
That is to say, such performance is associated with a Boolean formula
$\varphi$ via a function $t$ in the following manner: (i) the performance of
an atomic action type $\delta$ by $\alpha$ is translated via $t$ into a
propositional constant $\mathfrak{d}^{\alpha}$ and (ii) the performance of a
complex action type $\Delta$ by $\alpha$ is translated via $t$ into a formula
$\varphi$ whose Boolean structure matches the algebraic structure of $\Delta$
(as formally defined below). Notice the difference between the font of, e.g.,
$\delta_{1}$, which indicates an action type, and the font of
$\mathfrak{d}_{1}^{\alpha_{1}}$, which indicates a proposition, namely that an
action of type $\delta$ is performed by $\alpha_{1}$.

Temporal reference and the structure of time. Von Wright’s account of agency
takes action as agential change, and change as a transition between moments in
time. Consequently, we need to adopt a temporal perspective towards
instrumentality as well. In particular, we will take each model for our
language to include a set of moments $W$ and evaluate formulae at those
moments.

Judgments of instrumentality refer to the degree in which a certain action, as
an instrument, may lead to a certain state of affairs. This ‘leading to’ is
the temporal component of instrumentality referring to possible future
moments. We therefore use a modal operator $\square$, meaning ‘in all possible
immediate next moments’. This operator is associated with an accessibility
relation between pairs of moments, $R_{\square}$. For instance, let
$\mathfrak{d}^{\alpha_{1}}$ stand for ‘the door has been opened by agent
$\alpha_{1}$’, then the formula $\Box\mathfrak{d}^{\alpha_{1}}$ is interpreted
as ‘in all possible immediate next moments the door has been opened by agent
$\alpha_{1}$’. We take it that outcomes of actions concern primarily the
nearby future, given that an action affects the world as soon as it is
performed by an agent. For this reason, in our framework there is no need of
using a temporal operator for reference to the whole future of a moment of
evaluation (yet, of course, by iterating operator $\square$ it is possible to
make reference to a more distant future).

We also use two modal operators referring to the past, $\blacksquare$ and
$\mathsf{H}$, which extend the language of $\mathsf{LAE}$. The former is the
converse of $\square$ and quantifies over immediate predecessors of a given
moment. The latter is the transitive closure of $\blacksquare$, thus referring
to the whole past, not just the immediate past. Hence, $\mathsf{H}\varphi$
reads ‘everywhere in the past $\varphi$ holds’. These operators are associated
with accessibility relations $R_{\blacksquare}$ and $R_{\mathsf{H}}$,
respectively. The main benefit of using operators for past reference in
$\mathsf{TLAE}$ is that they allow one to count previous performances of an
action, keeping track of this counting at the syntactic level. Moreover,
counting permits us to compute a success ratio with respect to a certain
outcome, giving rise to more refined notions of instrumentality. Such a
syntactic procedure of counting was not possible in $\mathsf{LAE}$.

Finally, we use an operator  $\bm{[}$ A $\bm{]}$ which applies to a formula
$\varphi$ and says that $\varphi$ describes a state of affairs that takes
place in the immediate actual future of the moment of evaluation. This allows
us to provide a comparison between what the reasoning agent expects to be the
case immediately after the moment of evaluation and what will actually be the
case. It is associated with an accessibility relation $R_{\scalebox{0.8}{
\raisebox{0.90417pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip
0.6pt\raisebox{-0.54247pt}{{A}}\hskip 0.6pt}$\bm{]}$}} }}$ (this operator was
originally denoted by $N$ in [6]).

We stress that there is a twofold asymmetry between the past and the future in
our formal framework. First, our indeterministic approach to time allows for
several possible immediate successors of a moment, but only one immediate
predecessor; i.e., the past of a moment is linear, while its future is
possibly branching. Second, the way in which instrumentality judgements are
formed, namely through past experience, requires reference to the entire past
of a moment, but not to its entire future.

Expectations. We employ propositional constants of the form
$\mathfrak{e}^{\alpha_{i}}$ to encode that the most recent expectations of an
agent $\alpha_{i}$ are fulfilled. Such formulae further involve the agent’s
perspective in judging instrumentality relations. Expectations capture the
_conjectural element in judgments of instrumentality._

###  3.2 The Formal Language of $\mathsf{TLAE}$.

Based on the above list, we define two languages: an action language
$\mathcal{L}_{\mathsf{Act}}$, which is an algebra of actions for agent-
independent action types, and the logical language
$\mathcal{L}_{\mathsf{TLAE}}$ into which these actions will be translated. We
let $\mathtt{Action}:=\\{\delta_{1},\ldots,\delta_{n}\\}$ be a finite set of
_atomic action types_ and define the set $\mathcal{L}_{\mathsf{Act}}$ of _all
action types_ to be all strings generated by the following BNF grammar:

$\Delta::=\delta_{i}\ |\ \Delta\cup\Delta\ |\ \overline{\Delta}$

where $\delta_{i}\in\mathtt{Action}$. The $\cup$ operation is used to form a
_disjunction of action types_ (e.g., ‘turning-left or turning-right’) and the
— operation is used to form a _negation of action types_ (e.g., ‘not turning-
right’). We take the intersection of actions, denoted by the operation $\cap$,
as defined, i.e.,
$\Delta\cap\Gamma:=\overline{\overline{\Delta}\cup\overline{\Gamma}}$.

We use $\mathtt{Agent}:=\\{\alpha_{1},\ldots,\alpha_{m}\\}$ to denote the set
of all agent terms and define an _agent-bound action type_ to be an expression
of the form $\Delta^{\alpha_{i}}$, where $\Delta\in\mathcal{L}_{\mathsf{Act}}$
and $\alpha_{i}\in\mathtt{Agent}$. For any $\alpha_{i}\in\mathtt{Agent}$, we
let
$\mathtt{Wit}^{\alpha_{i}}:=\\{\mathfrak{d}^{\alpha_{i}}_{1},\ldots,\mathfrak{d}^{\alpha_{i}}_{n}\\}$
be the set of propositional constants respectively witnessing the performance
of action types $\delta_{1},\ldots,\delta_{n}$ by $\alpha_{i}$. For instance,
suppose $\delta_{1}$ stands for ‘opening the door’, then we read its
corresponding witness $\mathfrak{d}_{1}^{\alpha_{1}}$ as ‘the door has been
opened by agent $\alpha_{1}$’. We make the correspondence between agent-bound
action types and propositional constants formally precise below. Notice that
$|\mathtt{Wit}^{\alpha_{i}}|=|\mathcal{L}_{\mathsf{Act}}|=n$. We use
$\mathtt{Wit}$ to denote the set
$\bigcup_{\alpha_{i}\in\mathtt{Agent}}\mathtt{Wit}^{\alpha_{i}}$.

The language $\mathcal{L}_{\mathsf{TLAE}}$ of $\mathsf{TLAE}$ is defined via
the following BNF grammar:

$\varphi::=p\ |\ \mathfrak{e}^{\alpha_{i}}\ |\ \mathfrak{d}^{\alpha_{i}}_{j}\
|\ \neg\varphi\ |\ \varphi\rightarrow\varphi\ |\ \Box\varphi\ |\
\raisebox{1.29167pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip
0.6pt\raisebox{-0.77496pt}{{A}}\hskip 0.6pt}$\bm{]}$}}\varphi\ |\
\blacksquare\varphi\ |\ \mathsf{H}\varphi$

where $p$ is any _propositional variable_ from the set
$\mathtt{Var}:=\\{p_{k}\ |\ k\in\mathbb{N}\\}$, $i\in\\{1,\ldots,m\\}$, and
$j\in\\{1,\ldots,n\\}$. We use $p$, $q$, $r$, $\ldots$ (possibly annotated) to
denote propositional variables, and $\varphi$, $\psi$, $\chi$, $\ldots$
(possibly annotated) to denote formulae from $\mathcal{L}_{\mathsf{TLAE}}$.
The connectives $\lnot$ and $\rightarrow$ denote ‘negation’ and ‘material
implication’, respectively. The interpretation of each modal operator $\Box$,
$\blacksquare$,  $\bm{[}$ A $\bm{]}$ , and $\mathsf{H}$ is given in Definition
5 below; we take $\Diamond$, $\blacklozenge$,  $\langle$ A $\rangle$ , and
$\mathsf{P}$ to be duals of each respective modal operator. Conjunction
$\land$, disjunction $\lor$, and material equivalence $\leftrightarrow$ are
defined in the usual way. Last, $\mathfrak{e}^{\alpha_{i}}$ is a propositional
constant witnessing the compatibility of a state with $\alpha_{i}$’s
expectations, which can be read as ‘the present state is compatible with
$\alpha_{i}$’s expectations’.

The translation encoding action types from $\mathcal{L}_{\mathsf{Act}}$ into
agent-indexed formulae of $\mathcal{L}_{\mathsf{TLAE}}$ is established
recursively through the following function $t$:

  * –

For all $\delta\in\mathtt{Action}$, and all $\alpha_{i}\in\mathtt{Agent}$,
$t(\delta^{\alpha_{i}})=\mathfrak{d}^{\alpha_{i}}_{i}$

  * –

For all $\Delta\in\mathcal{L}_{\mathsf{Act}}$, and all
$\alpha_{i}\in\mathtt{Agent}$, $t(\overline{\Delta}^{\alpha_{i}})=\lnot
t(\Delta^{\alpha_{i}})$

  * –

For all $\Delta,\Gamma\in\mathcal{L}_{\mathsf{Act}}$, and all
$\alpha_{i}\in\mathtt{Agent}$,
$t(\Delta^{\alpha_{i}}\cup\Gamma^{\alpha_{j}})=t(\Delta^{\alpha_{i}})\lor
t(\Gamma^{\alpha_{j}})$

The advantage of this translation is that it enables us to reason with actions
on the object language level while simultaneously distinguishing such formulae
from other (non-action) formulae in the language. This distinction will prove
beneficial in (i) defining a variety of modal instrumentality operators in
Section 5 and (ii) axiomatising action specific properties.

To give an example of the expressive power of $\mathcal{L}_{\mathsf{TLAE}}$,
we briefly recall the three agentive notions of would, could, and will, as
discussed and defined in [6].

  1. Would
$[\Delta^{\alpha_{i}}]^{would}\varphi:=\square(t(\Delta^{\alpha_{i}})\rightarrow\varphi)$.
($\mathsf{d1}$)

  2. Could
$[\Delta^{\alpha_{i}}]^{could}\varphi:=\square(t(\Delta^{\alpha_{i}})\rightarrow\varphi)\wedge\lozenge
t(\Delta^{\alpha_{i}})$. ($\mathsf{d2}$)

  3. Will
$[\Delta^{\alpha_{i}}]^{will}\varphi:=\square(t(\Delta^{\alpha_{i}})\rightarrow\varphi)\wedge\raisebox{0.55974pt}{\scalebox{0.83}{$\langle$\scalebox{0.9}{{A}}$\rangle$}}t(\Delta^{\alpha_{i}})$.
($\mathsf{d3}$)

The formula $[\Delta^{\alpha_{i}}]^{would}\varphi$ ($\mathsf{d1}$) means that
‘at the current state, by behaving in accordance with $\Delta$, $\alpha_{i}$
would bring about $\varphi$’. The formula
$[\Delta^{\alpha_{i}}]^{could}\varphi$ ($\mathsf{d2}$) means that ‘at the
moment of evaluation, by behaving in accordance with $\Delta$, $\alpha_{i}$
would bring about $\varphi$ and $\alpha_{i}$ could (i.e., is able to) behave
in accordance with $\Delta$’. Finally, the formula
$[\Delta^{\alpha_{i}}]^{will}\varphi$ ($\mathsf{d3}$) means that ‘at the
moment of evaluation, by behaving in accordance with $\Delta$, $\alpha_{i}$
would bring about $\varphi$ and $\alpha_{i}$ will behave in accordance with
$\Delta$’. One can obtain multi-agent variants of the above modalities, such
as $[\Delta^{\alpha}\cap\Gamma^{\beta}]^{could}\varphi$, referring to the
agents $\alpha$ and $\beta$’s ability to jointly secure $\varphi$. As an
example, let $\Delta$ be the generic action ‘push’ and let $\varphi$ stand for
‘the trolley is rolling’. Then, the formula
$[\Delta^{\alpha}\cap\Delta^{\beta}]^{will}\varphi$ reads ‘$\alpha$ and
$\beta$ will both push to ensure the trolley is rolling’.

There are relevant connections between our modalities ‘could’, ‘would’ and
‘will’ and other operators used in the literature on agency logics. For
instance, in the tradition of STIT logics the core ingredients are agent-
relative operators which connect an agent’s (or a group of agents’) behaviour
to a certain outcome. The most basic of these operators is $[\alpha\;stit]$,
which is extensively analysed in [5]. A formula of the form
$[\alpha\;stit]\varphi$ means that agent $\alpha$ behaves in such a way so as
to ensure that $\varphi$ holds. Particularly relevant to our setting is the
variant of this operator denoted by $[\alpha\;xstit]$ and analysed, e.g., in
[10, 11], since it involves a temporal shift towards the immediate future:
$[\alpha\;xstit]\varphi$ means that agent $\alpha$ behaves in such a way so as
to ensure that $\varphi$ holds immediately after. The role played by these
operators in STIT logics is here captured by the ‘will’ modality. As a matter
of fact, the formula $[\Delta^{\alpha_{i}}]^{will}\varphi$ can be regarded as
a $\mathsf{TLAE}$-version of $[\alpha\;xstit]\varphi$ which additionally
includes information about the action chosen by the agent to get the result.

There are also significant connections between our approach and proposals to
integrate reference to actions in the framework of STIT logics. For instance,
the formalism introduced by Xu in [37] includes formulas of the form
$[\alpha,\Delta]\varphi$, meaning that agent $\alpha$ obtains $\varphi$ by
doing $\Delta$. This formalism also includes tense-logical operators for
future reference, although not for reference to the immediate future. In our
framework the situation is reversed: we have operators that make reference to
the immediate future and do not have operators that make reference to the
whole future (as explained in Section 3.1).

Moreover, as pointed out in [6], the ‘would’ modality employed here resembles
operators used in propositional dynamic logic (PDL) [16]. In a broader
perspective, our approach can be seen as a reduction of PDL to alethic modal
logic with constants witnessing the performance of actions, similar to the
reduction of deontic logic to alethic modal logic proposed by Anderson in
[1].666One could also add a counterfactual component to the above definitions,
namely, the formula $\lozenge\neg\varphi$ as a conjunct. This would strengthen
the idea of a causal connection between $\Delta$ and $\varphi$ since it may be
the case that $\varphi$ fails to hold in the immediate future. A
counterfactual component is also taken into account in one of Anderson’s
strategies to reduce deontic logic, as well as in some STIT logics (see, e.g.,
the deliberative STIT operator in [5]).

Finally, we would like to mention that our account of ‘would’, ‘could’ and
‘will’ is also related to proposals that represent concepts of STIT logic
within dynamic logics or logics of temporal computation. For instance, the
logic $\mathcal{DLA}$ proposed by Herzig and Lorini [19, 22] is based on a
language including an agent-indexed operator $[a:\alpha]$ such that the
formula $[a:\alpha]\varphi$ means that agent $a$ ensures that $\varphi$ will
happen by performing an action of type $\alpha$; this interpretation is very
close to our analysis of ‘will’. Furthermore, Boudou and Lorini in [9] propose
to integrate STIT operators in the computational logic $\mathsf{CTL}^{*}$,
whose language is endowed with temporal operators for ‘next-time’
($\mathsf{X}$) and ‘until’ ($\mathsf{U}$). The expressiveness of future
reference in the latter setting goes beyond the one allowed for in ours.

###  3.3 Semantics and Axioms of $\mathsf{TLAE}$

We adopt relational frames for the semantic characterisation of the logic
$\mathsf{TLAE}$. The proposed frame properties will be motivated by the
discussion presented in Section 2. In brief, we will define irreflexive tree-
like structures that are linear with respect to the past, and allow for
branching with respect to the future. Irreflexivity is motivated by the fact
that a moment cannot be its own immediate successor. In other words, the
transitive closure of the immediate successor relation is a strict partial
order. One of the advantages of employing irreflexive structures is that it
allows for counting with respect to the past, that is, $\blacklozenge$ and
$\blacklozenge\blacklozenge$ refer to two distinct moments in the past, the
first immediately preceding the moment of evaluation and the second
immediately preceding the first. Henceforth, we use $\blacklozenge^{i}$
($i\in\mathbb{N}$) to refer to a concatenation of $i$-many $\blacklozenge$
operators, referring to a moment $i$ time units in the past. This feature will
be employed in counting successful applications of an instrument serving a
certain purpose, thus facilitating the comparative notions of instrumentality
from Section 2.

First, we define $\mathcal{L}_{\mathsf{TLAE}}$-frames (Definition 4), which
will subsequently be refined to form the class of envisaged
$\mathsf{TLAE}$-frames (Definition 6). Frames and truth-conditions are defined
as usual, with the exception that the interpretation of constants will not be
varying per model, but will be fixed on the level of frames (Definition 5).
Namely, the valuation of constants is fixed to sets of moments defined on the
frame level which means that the semantic interpretation of such constants is
fixed for every model defined over a frame. One advantage of interpreting
constants on the level of frames is that we can define frame properties (and
corresponding axioms) that will restrict the logical behaviour of certain
constants, thus enhancing the expressiveness of the formal setting.

######  Definition 4 ($\mathcal{L}_{\mathsf{TLAE}}$-frame,
$\mathcal{L}_{\mathsf{TLAE}}$-model).

A _$\mathcal{L}_{\mathsf{TLAE}}$ -frame_ is a tuple

$\mathfrak{F}=\langle W,\\{W_{\mathfrak{d}^{\alpha_{i}}_{j}}\ |\
\mathfrak{d}^{\alpha_{i}}_{j}\in\mathtt{Wit}\\},\\{W_{\mathfrak{e}^{\alpha_{i}}}\
|\ \alpha_{i}\in\mathtt{Agent}\\},R_{\Box},R_{\scalebox{0.8}{
\raisebox{0.90417pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip
0.6pt\raisebox{-0.54247pt}{{A}}\hskip 0.6pt}$\bm{]}$}}
}},R_{\blacksquare},R_{\mathsf{H}}\rangle$

where $W$ is a set of moments $w$, $u$, $v$, $\ldots$ (which are occasionally
annotated),
$W_{\mathfrak{d}^{\alpha_{i}}_{j}},W_{\mathfrak{e}^{\alpha_{i}}}\subseteq W$,
and $R_{\Box}$, $R_{\scalebox{0.8}{
\raisebox{0.90417pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip
0.6pt\raisebox{-0.54247pt}{{A}}\hskip 0.6pt}$\bm{]}$}} }}$,
$R_{\blacksquare}$, and $R_{\mathsf{H}}$ are binary relations over $W$.

A _$\mathcal{L}_{\mathsf{TLAE}}$ -model_
$\mathfrak{M}=\langle\mathfrak{F},V\rangle$ is a tuple where $\mathfrak{F}$ is
a $\mathcal{L}_{\mathsf{TLAE}}$-frame, and $V$ is a valuation function mapping
propositional atoms to sets of moments such that:

  * –

$V(\mathfrak{d}^{\alpha_{i}}_{j}):=W_{\mathfrak{d}^{\alpha_{i}}_{j}}$

  * –

$V(\mathfrak{e}^{\alpha_{i}}):=W_{\mathfrak{e}^{\alpha_{i}}}$

######  Definition 5 (Truth-conditions).

Formulae are evaluated at a state of a model, along the following lines:

  * –

$\mathfrak{M},w\models p$ iff $w\in V(p)$, for any propositional variable
$p\in\mathtt{Var}$

  * –

$\mathfrak{M},w\models\neg\varphi$ iff $\mathfrak{M},w\not\models\varphi$

  * –

$\mathfrak{M},w\models\varphi\rightarrow\psi$ iff
$\mathfrak{M},w\not\models\varphi$ or $\mathfrak{M},w\models\psi$

  * –

$\mathfrak{M},w\models\Box\varphi$ iff for all $v\in W$ s.t. $R_{\Box}wv$, it
follows that $\mathfrak{M},v\models\varphi$

  * –

$\mathfrak{M},w\models\raisebox{1.29167pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip
0.6pt\raisebox{-0.77496pt}{{A}}\hskip 0.6pt}$\bm{]}$}}\varphi$ iff for all
$v\in W$ s.t. $R_{\scalebox{0.8}{
\raisebox{0.90417pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip
0.6pt\raisebox{-0.54247pt}{{A}}\hskip 0.6pt}$\bm{]}$}} }}wv$, it follows that
$\mathfrak{M},v\models\varphi$

  * –

$\mathfrak{M},w\models\blacksquare\varphi$ iff for all $v\in W$ s.t.
$R_{\blacksquare}wv$, it follows that $\mathfrak{M},v\models\varphi$

  * –

$\mathfrak{M},w\models\mathsf{H}\varphi$ iff for all $v\in W$ s.t. $R_{H}wv$,
it follows that $\mathfrak{M},v\models\varphi$

The semantic clauses for $\Diamond$,  $\langle$ A $\rangle$ , $\blacklozenge$,
$\mathsf{P}$, $\land$, $\lor$, and $\leftrightarrow$ are defined as usual.

For an arbitrary $\Delta^{\alpha_{i}}$ such that
$\Delta\in\mathcal{L}_{\mathsf{Act}}$ and $\alpha_{i}\in\mathtt{Agent}$, we
define $W_{t(\Delta^{\alpha_{i}})}$ using the following recursive clauses:

  * –

$W_{t(\delta_{j}^{\alpha_{i}})}:=W_{\mathfrak{d}^{\alpha_{i}}_{j}}$

  * –

$W_{t(\overline{\Delta}^{\alpha_{i}})}:=W{-}W_{t(\Delta^{\alpha_{i}})}$

  * –

$W_{t(\Delta^{\alpha_{i}}\cup\Gamma^{\alpha_{j}})}:=W_{t(\Delta^{\alpha_{i}})}\cup
W_{t(\Gamma^{\alpha_{j}})}$

It can be easily shown (through induction on the complexity of
$\Delta^{\alpha_{i}}$) that for each $\Delta\in\mathcal{L}_{\mathsf{Act}}$,
each $\alpha_{i}\in\mathtt{Agent}$, each $\mathcal{L}_{\mathsf{TLAE}}$-model
$\mathfrak{M}$ and each moment $w$ in its domain, we have:

$\mathfrak{M},w\models t(\Delta^{\alpha_{i}})\text{ {iff} }w\in
W_{t(\Delta^{\alpha_{i}})}$

Hence, we obtain the following semantic interpretation of our previously
defined operator $[\Delta^{\alpha_{i}}]^{would}\varphi$ (page $\mathsf{d1}$):

$\mathfrak{M},w\models[\Delta^{\alpha_{i}}]^{would}\varphi\text{ {iff} for all
}v\in W\text{ s.t. }R_{\Box}wv,\text{ if }v\in
W_{t(\Delta^{\alpha_{i}})}\text{ then }\mathfrak{M},v\models\varphi$

In other words, every immediate successor witnessing the performance of
$\Delta$ by an agent $\alpha_{i}$ guarantees the truth of $\varphi$.

######  Definition 6 ($\mathsf{TLAE}$-frame, $\mathsf{TLAE}$-model).

We define a _$\mathsf{TLAE}$ -frame_ to be a
$\mathcal{L}_{\mathsf{TLAE}}$-frame satisfying the following properties:

p(A3) |  For all $w,u,v\in W$, if $R_{\scalebox{0.8}{ \raisebox{0.90417pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip 0.6pt\raisebox{-0.54247pt}{{A}}\hskip 0.6pt}$\bm{]}$}} }}wu$ and $R_{\scalebox{0.8}{ \raisebox{0.90417pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip 0.6pt\raisebox{-0.54247pt}{{A}}\hskip 0.6pt}$\bm{]}$}} }}wv$, then $w=v$;777Item $p(A3)$ expresses the functionality of the operator  $\bm{[}$ A $\bm{]}$ , which is a widespread property among operators used in the literature on agency logic in order to make reference to the actual future. See, e.g., approaches offering a fusion of the next-time operator and STIT operators, such as [19] and [10].
---|---
p(A4) |  For all $w,u\in W$, if $R_{\scalebox{0.8}{ \raisebox{0.90417pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip 0.6pt\raisebox{-0.54247pt}{{A}}\hskip 0.6pt}$\bm{]}$}} }}wu$, then $R_{\Box}wu$;
p(A5) |  For all $w\in W$ and all distinct agents $\alpha_{1}$, $\ldots$, $\alpha_{n}$, if there are (not necessarily distinct) action types $\Delta_{1}$, $\ldots$, $\Delta_{n}$ s.t. for $1\leq i\leq n$ there is a $u_{i}\in W$ s.t. $R_{\Box}wu_{i}$ and $u_{i}\in W_{t(\Delta_{i}^{\alpha_{i}})}$, there is a $v\in W$ s.t. $R_{\Box}wv$ and $v\in W_{t(\Delta_{1}^{\alpha_{1}})}\cap\cdots\cap W_{t(\Delta_{n}^{\alpha_{n}})}$;
p(A6) |  For all $w\in W$ and $\alpha_{i}\in\mathtt{Agent}$, if there is a $v\in W$ s.t. $R_{\Box}wv$ and $v\in W_{\mathfrak{e}^{\alpha_{i}}}$, then there is also a $u\in W$ s.t. $R_{\Box}wu$ and $u\not\in W_{\mathfrak{e}^{\alpha_{i}}}$.
p(A10;A11) |  For all $w,v\in W$, $R_{\Box}wv$ iff $R_{\blacksquare}vw$;
p(A12) |  For all $w,u,v\in W$, if $R_{\blacksquare}wu$ and $R_{\blacksquare}wv$, then $u=v$;
p(A9;A14) |  $R_{\mathsf{H}}$ is the transitive closure of $R_{\blacksquare}$ (that we will occasionally represent as $R_{\blacksquare}^{+}$);
p(A13) |  For all $w\in W$ either (i) there is no $v$ s.t. $R_{\mathsf{H}}wv$ or (ii) there is some $u$ s.t. $R_{\mathsf{H}}wu$ and there is no $z$ s.t. $R_{\mathsf{H}}uz$.

A $\mathsf{TLAE}$-model $\mathfrak{M}=\langle\mathfrak{F},V\rangle$ is a
$\mathcal{L}_{\mathsf{TLAE}}$-model such that $\mathfrak{F}$ is a
$\mathsf{TLAE}$-frame.

For the sake of comparability, we have named the frame properties in reference
to the axioms of $\mathsf{TLAE}$ introduced below (see Definition 7). Some of
the properties presented above correspond to a set of axioms as opposed to a
single axiom; e.g., $p(A9;A14)$ is a property characterised through a
combination of the axioms $A9$ and $A14$.

Let us briefly explain the intuitive meaning of each property: $p(A3)$ ensures
that every moment has at most one immediate actual successor. This conveys a
simplification of the framework that we exploit to discuss various examples in
the rest of the article and that allows us to remain more closely related to
the original framework formulated in [6]. However, reference to $p(A3)$, and
the corresponding axiom discussed below, $A3$, can be safely omitted from the
technical part of the article (basically, one only has to remove reference to
it in the proof of Lemma 4). $p(A4)$ states that this successor must be a
possible successor. $p(A5)$ expresses the agency property known as
independence of agents.888Independence of agents is a fundamental property in
the setting of STIT logic, where it expresses that any choice available to an
agent at a certain moment is compatible with any combination of choices
available to the other agents at that moment. In STIT, choices are primitive
objects and no object-language reference is made to actions. Here, following
[6], we adopt independence of agents in a setting in which we have explicit
actions available to agents. Such a property ensures that, if an action
$\Delta$ is a $\varphi$-instrument for $\alpha$, then failing to produce
$\varphi$ by performing $\Delta$ cannot be caused by the interference of other
agents. In other words, $\Delta$ is a proper $\varphi$ instrument for
$\alpha$. Nevertheless, we will discuss possible failures due to the agent’s
(false) expectations about the action in question. Future work may be directed
to investigating instrumentality in the light of interfering agents, i.e., by
dropping independence of agents. See [5] for a discussion. This property
ensures that if an agent can perform a certain action at a certain moment that
agent can perform the action at issue irrespective of the actions performed by
the other agents. $p(A6)$ ensures that if an agent expects a certain next
moment to arise, then there will be another possible next moment that the
agent does not expect to arise. $p(A10;A11)$ defines immediate future moments
as the inverse of immediate past moments. $p(A12)$ states that the past is
linear. $p(A9;A14)$ defines $R_{\mathsf{H}}$ as the transitive closure of
$R_{\blacksquare}$. Last, $p(A13)$ ensures that the past is finite, that is,
time has a beginning. This last property will prove useful to comparing
candidate instruments serving the same purpose.

We point out the following fact:

######  Theorem 1.

$\mathsf{TLAE}$-frames are irreflexive, that is, for all $w\in W$ of a
$\mathsf{TLAE}$-frame $\mathfrak{F}$, we have $(w,w)\not\in R_{\blacksquare}$,
$(w,w)\not\in R_{\Box}$, $(w,w)\not\in R_{\mathsf{H}}$, and $(w,w)\not\in
R_{\scalebox{0.8}{
\raisebox{0.90417pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip
0.6pt\raisebox{-0.54247pt}{{A}}\hskip 0.6pt}$\bm{]}$}} }}$.

###### Proof.

Suppose that $R_{\mathsf{H}}ww$. By $p(A13)$, there is some $u\neq w$ s.t.
$R_{\mathsf{H}}wu$ and there does not exist a $z$ such that
$R_{\mathsf{H}}uz$. By $p(A9;A14)$ there is a sequence of moments
$\sigma=v_{1},\dots,v_{n}$ s.t. $v_{1}=w$, $v_{n}=u$ and, for $1\leq i\leq
n-1$, $R_{\blacksquare}v_{i}v_{i+1}$. Furthermore, there is a sequence of
moments $\sigma^{\prime}=v_{1}^{\prime},\ldots,v_{m}^{\prime}$ s.t.
$v_{1}^{\prime}=v_{m}^{\prime}=w$ and for $1\leq i\leq m-1$,
$R_{\blacksquare}v_{i}^{\prime}v_{i+1}^{\prime}$. We now have three cases to
consider due to $p(A12)$ and the fact that $v_{1}=v_{1}^{\prime}$: either (i)
$\sigma$ is a proper sub-sequence of $\sigma^{\prime}$, (ii) $\sigma^{\prime}$
is a proper sub-sequence of $\sigma$, or (iii) $\sigma=\sigma^{\prime}$. We
show (ii) as (i) and (iii) are simple. Since $\sigma^{\prime}$ is a sub-
sequence of $\sigma$, we know that for some $1\leq i\leq n-1$,
$v_{m}^{\prime}=w=v_{i}$. It follows that $R_{\blacksquare}wv_{i+1}$.
Moreover, we have that $R_{\blacksquare}wv_{2}^{\prime}$ by the definition of
$\sigma^{\prime}$, which implies that $v_{i+1}=v_{2}^{\prime}$ by $p(A12)$.
Continuing in this way, one can show that $v_{i+2}=v_{3}^{\prime}$,
$v_{i+3}=v_{4}^{\prime}$, etc. It follows that for some $1\leq j\leq m$,
$v_{j}^{\prime}=v_{n}=u$. However, since $\sigma^{\prime}$ forms a cycle, this
implies that there is some $z$ (namely, $v_{j+1}^{\prime}$) such that
$R_{\blacksquare}uv_{j+1}$, which further entails that
$R_{\mathsf{H}}uv_{j+1}$ by $p(A9;A14)$. This gives a contradiction.

By $p(A9;A14)$ we can infer that for each $w$, $(w,w)\not\in
R_{\blacksquare}$, whence, by $p(A10;A11)$, that $(w,w)\not\in R_{\Box}$ and
finally, by $p(A4)$, that $(w,w)\not\in R_{\scalebox{0.8}{
\raisebox{0.90417pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip
0.6pt\raisebox{-0.54247pt}{{A}}\hskip 0.6pt}$\bm{]}$}} }}$. ∎

######  Corollary 1.

The relations $R_{\Box},R_{\blacksquare}$, $R_{\mathsf{H}}$, and
$R_{\scalebox{0.8}{
\raisebox{0.90417pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip
0.6pt\raisebox{-0.54247pt}{{A}}\hskip 0.6pt}$\bm{]}$}} }}$ of
$\mathsf{TLAE}$-frames are acyclic.

Our axiomatisation for the logic $\mathsf{TLAE}$ is given below.

######  Definition 7 ($\mathsf{TLAE}$ axiomatisation).

The axiomatisation of $\mathsf{TLAE}$ consists of the following axiom schemes
and rules:

A0 |  Any propositional tautology
---|---
R0 |  $\varphi,\varphi\rightarrow\psi/\psi$
A1 |  $\Box(\varphi\rightarrow\psi)\rightarrow(\Box\varphi\rightarrow\Box\psi)$
R1 |  $\varphi/\Box\varphi$
A2 |  $\raisebox{1.29167pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip 0.6pt\raisebox{-0.77496pt}{{A}}\hskip 0.6pt}$\bm{]}$}}(\varphi\rightarrow\psi)\rightarrow(\raisebox{1.29167pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip 0.6pt\raisebox{-0.77496pt}{{A}}\hskip 0.6pt}$\bm{]}$}}\varphi\rightarrow\raisebox{1.29167pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip 0.6pt\raisebox{-0.77496pt}{{A}}\hskip 0.6pt}$\bm{]}$}}\psi)$
A3 |  $\raisebox{0.55974pt}{\scalebox{0.83}{$\langle$\scalebox{0.9}{{A}}$\rangle$}}\varphi\rightarrow\raisebox{1.29167pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip 0.6pt\raisebox{-0.77496pt}{{A}}\hskip 0.6pt}$\bm{]}$}}\varphi$
A4 |  $\Box\varphi\rightarrow\raisebox{1.29167pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip 0.6pt\raisebox{-0.77496pt}{{A}}\hskip 0.6pt}$\bm{]}$}}\varphi$
A5 |  For any distinct $\alpha_{1},\dots,\alpha_{n}\in\mathtt{Agent}$ and non-necessarily distinct
|  $\Delta_{1},\dots,\Delta_{n}\in\mathcal{L}_{\mathsf{Act}}$, $(\Diamond
t(\Delta_{1}^{\alpha_{1}})\wedge\dots\wedge\Diamond
t(\Delta_{n}^{\alpha_{n}}))\rightarrow\Diamond(t(\Delta_{1}^{\alpha_{1}})\wedge\dots\wedge
t(\Delta_{n}^{\alpha_{n}}))$
A6 |  For any $\alpha_{j}\in\mathtt{Agent}$, $\Diamond\mathfrak{e}^{\alpha_{j}}\rightarrow\Diamond\neg\mathfrak{e}^{\alpha_{j}}$

A7 |  $\mathsf{H}(\varphi\rightarrow\psi)\rightarrow(\mathsf{H}\varphi\rightarrow\mathsf{H}\psi)$
---|---
A8 |  $\blacksquare(\varphi\rightarrow\psi)\rightarrow(\blacksquare\varphi\rightarrow\blacksquare\psi)$
A9 |  $\mathsf{H}\varphi\leftrightarrow(\blacksquare\varphi\land\blacksquare\mathsf{H}\varphi)$
A10 |  $\varphi\rightarrow\Box\blacklozenge\varphi$
A11 |  $\varphi\rightarrow\blacksquare\Diamond\varphi$
A12 |  $\blacklozenge\varphi\rightarrow\blacksquare\varphi$
A13 |  $\mathsf{H}\bot\vee\mathsf{P}\mathsf{H}\bot$
A14 |  $\mathsf{H}(\varphi\rightarrow\blacksquare\varphi)\rightarrow(\blacksquare\varphi\rightarrow\mathsf{H}\varphi)$
R2 |  $\varphi/\mathsf{H}\varphi$

For any formula $\varphi\in\mathcal{L}_{\mathsf{TLAE}}$, we define $\varphi$
to be a _theorem_ , and write $\vdash\varphi$, iff (i) $\varphi$ is an axiom
instance, or (ii) $\varphi$ is derivable from $\psi$ (and $\chi$) via one of
the inference rules where $\vdash\psi$ (and $\vdash\chi$, resp.). We say that
$\psi$ is _derivable_ from $\Gamma$ in $\mathsf{TLAE}$, written
$\Gamma\vdash\psi$, iff there exist $\varphi_{1},\cdots,\varphi_{n}\in\Gamma$
s.t. $\vdash\varphi_{1}\land\cdots\land\varphi_{n}\rightarrow\psi$.

In Definition 7 above, we define the notion of a theorem recursively as in
Blackburn et al. [8, Section 4.8], that is, a theorem is a formula which can
be derived via a sequence of axiom instances and rule applications to
previoulsy derived theorems. We remark that axioms A1, A2, A7 and A8, together
with rules R1, R2, and the derivable rules
$A/\raisebox{1.29167pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip
0.6pt\raisebox{-0.77496pt}{{A}}\hskip 0.6pt}$\bm{]}$}}A$ and $A/\blacksquare
A$, qualify the modal operators $\Box$,  $\bm{[}$ A $\bm{]}$ , $\blacksquare$,
and $\mathsf{H}$ as normal. Axioms A3 and A12 qualify the accessibility
relations associated with  $\bm{[}$ A $\bm{]}$ and $\blacksquare$ as
functional. Axiom A4 makes the accessibility relation associated with
$\square$ a superset of the accessibility relation associated with  $\bm{[}$ A
$\bm{]}$ . Axiom A5 corresponds to the ‘independence of agents’ principle in
the STIT-literature [5], and states that if each agent can perform a
particular action, then all agents can jointly perform such actions. Axioms A9
and A14 are used to express the fact that the accessibility relation
associated with $\mathsf{H}$ is the transitive closure of the accessibility
relation associated with $\blacksquare$. Axioms A10 and A11 are used to
express the fact that the accessibility relations associated with $\Box$ and
$\blacksquare$ are reciprocally converse. Finally, axiom A13 says that for any
state there is a finite sequence of states related to it via the accessibility
relation for $\mathsf{H}$. Taken together, these axioms ensure that the
accessibility relations for $\mathsf{H}$, $\Box$, $\blacksquare$ and  $\bm{[}$
A $\bm{]}$ are irreflexive. However, none of the axioms can ensure this
property if taken alone—this follows from general results in correspondence
theory for multi-modal, normal logics (cf. [8]). We note that the logic
$\mathsf{LAE}$ originally presented in [6] is the fragment of $\mathsf{TLAE}$
without operators $\mathsf{H}$ and $\blacksquare$ and axiomatized with the
deductive principles A0-A6 and R0-R1.

###  3.4 Agentive Modals in $\mathsf{TLAE}$

In order to get an impression of the expressiveness of $\mathsf{TLAE}$, we
provide some formal definitions of agentive modals in the spirit of von
Wright’s analysis. Our basic building blocks will be the agentive modals
‘would’ ($\mathsf{d1}$), ‘could’ ($\mathsf{d2}$), and ‘will’ ($\mathsf{d3}$)
defined in Section 3.1. First, consider the four elementary action types as
presented in Figure 1:

  1. Produce
$[\Delta^{\alpha_{i}}]^{prod}\varphi:=\lnot\varphi\land[\Delta^{\alpha_{i}}]^{will}\varphi\land\lozenge\lnot\varphi$
($\mathsf{d4}$)

  2. Destroy
$[\Delta^{\alpha_{i}}]^{destr}\varphi:=\varphi\land[\Delta^{\alpha_{i}}]^{will}\lnot\varphi\land\lozenge\varphi$
($\mathsf{d5}$)

  3. Suppress
$[\Delta^{\alpha_{i}}]^{supp}\varphi:=\lnot\varphi\land[\Delta^{\alpha_{i}}]^{will}\lnot\varphi\land\lozenge\varphi$
($\mathsf{d6}$)

  4. Preserve
$[\Delta^{\alpha_{i}}]^{pres}\varphi:=\varphi\land[\Delta^{\alpha_{i}}]^{will}\varphi\land\lozenge\lnot\varphi$
($\mathsf{d7}$)

Von Wright’s action types are often referred to as deliberative in nature;
that is, they exclude outcomes which are trivial (e.g., $\top$) and ensure
that outcomes are about contingent states of affairs $\varphi$, namely, for
which $\lozenge\varphi$ and $\lozenge\lnot\varphi$ hold. As discussed in
Section 2, von Wright’s reading of these actions may be too strong, namely,
the agent’s action decides the faith of $\varphi$ completely. For instance, in
the case of ‘producing’, through acting the agent ensures $\varphi$ whereas
through not-acting the agent is able to ensure $\lnot\varphi$. In other words,
von Wright’s account takes the agent’s agency as causally sufficient in both
directions. In line with our discussion, taking a slightly weaker standpoint
(cf. causal contribution in Figure 2), we alternatively formalise that the
agent has the ability to bring about $\varphi$ through performing $\Delta$,
but does not bring about $\varphi$ by refraining from performing $\Delta$.
This is reflected in definitions ($\mathsf{d4}$), ($\mathsf{d5}$),
($\mathsf{d6}$), and ($\mathsf{d7}$). (NB. Observe that this position has
already been adopted in [3].)

By making use of the notions of ‘would’ ($\mathsf{d1}$) and ‘could’
($\mathsf{d2}$), one can provide new versions of the four action types
presented above. We call such variations volitional concepts. For instance,
($\mathsf{d8}$) expresses the idea that an agent $\alpha_{i}$ could destroy
$\varphi$ by performing the action $\Delta$. In particular, the first conjunct
of ($\mathsf{d8}$) states that $\varphi$ is presently the case, the second
ensures that by performing $\Delta$ the agent $\alpha_{i}$ would bring about
$\lnot\varphi$ and $\Delta$ can be performed by $\alpha_{i}$, and last it is
possible that $\varphi$ will not be destroyed.

  1. Could Destroy
$[\Delta^{\alpha_{i}}]^{could}_{destr}\varphi:=\varphi\land[\Delta^{\alpha_{i}}]^{could}\lnot\varphi\land\lozenge\varphi$
($\mathsf{d8}$)

Last, consider the notion of forbearance, which is a stronger agentive notion
than merely not acting according to von Wright. To be more precise, forbearing
assumes the agent’s ability to perform the action that is forborne. The formal
definition of forbearance (irrespective of its outcome, denoted by ‘$\top$’)
is presented in ($\mathsf{d9}$).

  1. Forbear
$[\Delta^{\alpha_{i}}]^{forb}\top:=[\Delta^{\alpha_{i}}]^{could}\top\land[\overline{\Delta}^{\alpha_{i}}]^{will}\top$
($\mathsf{d9}$)

In words, ($\mathsf{d9}$) reads ‘the agent $\alpha$ forbears performing action
$\Delta$ whenever $\alpha$ could perform action $\Delta$, but will instead
perform the action’s complement $\overline{\Delta}$’.999We note that von
Wright’s concept of forbearing is different from Belnap et al’s [5] notion of
refraining. Briefly, von Wright considers ‘zero action’ or ‘passivity’ as a
meaningful notion, occurring when an agent does not act and lets the course of
nature take over (see page 1). This is captured in the definition of
forbearing ($\mathsf{d9}$), by taking an action corresponding to the negation
of all primitive actions. This contrasts with Belnap et al’s account where
refraining from acting necessarily corresponds to the agent actively
performing some other action.  One can see how the notion of forbearance can
be extended to incorporate the four elementary action types. The definition
provided in ($\mathsf{d10}$) gives an example.

  1. Forbear to Produce
$[\Delta^{\alpha_{i}}]^{forb}_{prod}\varphi:=\varphi\land[\Delta^{\alpha_{i}}]^{could}\varphi\land[\overline{\Delta}^{\alpha_{i}}]^{will}\top\land\lozenge\lnot\varphi$
($\mathsf{d10}$)

So far, we only considered those temporal operators referring to the future.
We briefly point out that the modularity of combining complex modals of agency
extends to reasoning about the past. For instance, one can combine the four
elementary action types and the notion of forbearance, with the three notions
of ‘would’, ‘could’, and ‘will’, while referring to the agent’s past. We
investigate reasoning about the past when we formalise instrumentality in
Section 5.

The aim of this section was to demonstrate the high versatility in defining
formal notions of agency in the language of $\mathsf{TLAE}$. One of the
reasons for this expressiveness relates to the use of action constants.
Namely, the use of constants referring to actions allows us to distinctively
reason about actions and states of affairs in a highly modular way, combining
them freely with the available temporal operators: future, past, and actual
future. In Section 5, we demonstrate how this language can be employed to
express various instrumentality notions. Furthermore, we will consider
agentive modals that arise by involving the notion of expectations. Before
moving to our analysis of instrumentality, we demonstrate that $\mathsf{TLAE}$
is consistent, sound, and weakly complete.

##  4 Soundness and Weak Completeness of $\mathsf{TLAE}$

Due to the interaction between the two new operators for past reference,
$\blacksquare$ and $\mathsf{H}$ (in particular, the fact that we want one to
be the transitive closure of the other), proving the completeness of
$\mathsf{TLAE}$ requires a much more complex construction than the one for
$\mathsf{LAE}$ provided in [6]. As a matter of fact, the usual canonical model
construction cannot be used since the logic $\mathsf{TLAE}$ is not compact
(and hence, not _strongly_ complete, cf. [8]): one can prove that the infinite
set $\Sigma=\\{\blacksquare^{n}p:n\in\mathbb{N}\\}\cup\\{\neg\mathsf{H}p\\}$
has no $\mathsf{TLAE}$-model, whereas each of its finite subsets has some
$\mathsf{TLAE}$-model. The strategy followed in this section consists of
adapting the Fischer-Ladner construction for the completeness of propositional
dynamic logic (illustrated in [8, Section 4.8]) in order to obtain a weak
completeness result for our logic $\mathsf{TLAE}$.

First, $\mathsf{TLAE}$ is sound with respect to the class of
$\mathsf{TLAE}$-frames:

######  Theorem 2 (Soundness).

For any formula $\varphi\in\mathcal{L}_{\mathsf{TLAE}}$, if $\vdash\varphi$,
then $\models\varphi$.

###### Proof.

Straightforward by demonstrating that all axioms of $\mathsf{TLAE}$ are valid
for the class of $\mathsf{TLAE}$-frames and all rules of $\mathsf{TLAE}$
preserve validity (see [8]). ∎

Furthermore, we observe that the logic $\mathsf{TLAE}$ is consistent.

######  Theorem 3 (Consistency).

The logic $\mathsf{TLAE}$ is consistent.

###### Proof.

To show $\mathsf{TLAE}$ consistent, we show that the class of models for
$\mathsf{TLAE}$ is non-empty. We define a $\mathsf{TLAE}$-model $\mathfrak{M}$
as follows: the set of moments $W:=\\{w_{i}\ |\ i\in\mathbb{N}\\}$, for each
$\mathfrak{d}^{\alpha_{i}}_{j}\in\mathtt{Wit}$,
$W_{\mathfrak{d}^{\alpha_{i}}_{j}}:=\emptyset$, for each
$\alpha_{i}\in\mathtt{Agent}$, $W_{\mathfrak{e}^{\alpha_{i}}}:=\emptyset$,
$R_{\scalebox{0.8}{
\raisebox{0.90417pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip
0.6pt\raisebox{-0.54247pt}{{A}}\hskip 0.6pt}$\bm{]}$}}
}}:=R_{\Box}:=\\{(w_{i},w_{i+1})\ |\ i\in\mathbb{N}\\}$, $R_{\blacksquare}$ is
taken to be the converse of $R_{\Box}$, $R_{\mathsf{H}}$ is the transitive
closure of $R_{\blacksquare}$, and $V$ is taken to be an arbitrary valuation.
It is straightforward to verify that $\mathfrak{M}$ is a
$\mathsf{TLAE}$-model. ∎

We now define a sequence of concepts that will assist us in establishing our
weak completeness result.

######  Definition 8 ($\mathsf{TLAE}$-Closure).

Let $\Sigma$ be a finite set of formulae. The _$\mathsf{TLAE}$ -closure_ of
$\Sigma$ is the smallest set $Cl({\Sigma})$ satisfying the conditions below:

  * –

if $\varphi\in\Sigma$ or $\varphi$ is a subformula of some $\psi\in\Sigma$,
then $\varphi\in Cl({\Sigma})$

  * –

if $\mathsf{P}\varphi\in\Sigma$, then
$\blacklozenge\mathsf{P}\varphi,\blacklozenge\varphi\in Cl({\Sigma})$

  * –

each constant $\mathfrak{d}^{\alpha_{j}}_{i}$ and $\mathfrak{e}^{\alpha_{j}}$
is in $Cl({\Sigma})$

  * –

if $\Diamond t(\Delta_{1}^{\alpha_{1}}),\ldots,\Diamond
t(\Delta_{n}^{\alpha_{n}})\in\Sigma$, then
$\Diamond(t(\Delta_{1}^{\alpha_{1}})\wedge\cdots\wedge
t(\Delta_{n}^{\alpha_{n}}))\in Cl({\Sigma})$

  * –

if $\Diamond\mathfrak{e}^{\alpha_{j}}\in\Sigma$, then
$\Diamond\neg\mathfrak{e}^{\alpha_{j}}\in Cl({\Sigma})$

  * –

$\mathsf{H}\bot,\mathsf{P}\mathsf{H}\bot\in Cl({\Sigma})$

######  Definition 9 (Negation $\sim$).

${\sim}\varphi:=\begin{cases}\psi&\text{if }\varphi\text{ is of the form
}\neg\psi,\\\ \neg\varphi&\text{otherwise.}\end{cases}$

For a given finite set of formulae $\Sigma$, we define $\neg Cl({\Sigma})$ to
be the smallest extension of $Cl({\Sigma})$ closed under $\sim$ (i.e., under
single negations).

######  Definition 10 (Atomic Set).

Let $\Sigma$ be a finite set of formulae. We say that a set $X$ of formulae is
_$\mathsf{TLAE}$ -consistent_ iff $X\not\vdash\bot$, and say that a set $X$ of
formulae is _maximally $\mathsf{TLAE}$-consistent_ iff $X$ is consistent and
for any formula $\varphi\not\in X$, $X\cup\\{\varphi\\}\vdash\bot$. A set of
formulae $X$ is an _atomic set_ over $\Sigma$ iff it is a maximal
$\mathsf{TLAE}$-consistent subset of $\neg Cl({\Sigma})$. We use
$At({\Sigma})$ to denote the set of all atomic sets over $\Sigma$.

######  Lemma 1.

Let $\Sigma$ be a finite set of formulae and $X\in At({\Sigma})$. Then,

  * (i)

For all $\varphi\in\neg Cl({\Sigma})$, either $\varphi\in X$ or
${\sim}\varphi\in X$, but not both.

  * (ii)

For all $\varphi\in\neg Cl({\Sigma})$, if $X\vdash\varphi$, then $\varphi\in
X$.

  * (iii)

For all $\varphi\lor\psi\in\neg Cl({\Sigma})$, $\varphi\lor\psi\in X$ iff
either $\varphi\in X$ or $\psi\in X$.

  * (iv)

For all $\mathsf{P}\varphi\in\neg Cl({\Sigma})$, $\mathsf{P}\varphi\in X$ iff
either $\blacklozenge\varphi\in X$ or $\blacklozenge\mathsf{P}\varphi\in X$.

###### Proof.

Claims (i)–(iii) are relatively straightforward, so we present the proof of
claim (iv) and assume that $\mathsf{P}\varphi\in\neg Cl({\Sigma})$.

For the forwards direction, assume that $\mathsf{P}\varphi\in X$. By the
condition on the $\mathsf{TLAE}$ closure of a set,
$\blacklozenge\mathsf{P}\varphi,\blacklozenge\varphi\in\neg Cl({\Sigma})$.
Since
$\mathsf{H}\varphi\leftrightarrow(\blacksquare\varphi\wedge\blacksquare\mathsf{H}\varphi)$
is an instance of axiom A9, it follows that
$\vdash\mathsf{P}\varphi\leftrightarrow(\blacklozenge\mathsf{P}\varphi\vee\blacklozenge\varphi)$.
Then, since $\mathsf{P}\varphi\in X$, one can infer that
$X\vdash\blacklozenge\varphi\vee\blacklozenge\mathsf{P}\varphi$. Suppose that
neither $\blacklozenge\varphi$ nor $\blacklozenge\mathsf{P}\varphi$ are in
$X$, then, due to the definition of $\neg Cl(\Sigma)$ and claim (i),
$\neg\blacklozenge\varphi,\neg\blacklozenge\mathsf{P}\varphi\in X$ and one can
infer $X\vdash\neg(\blacklozenge\varphi\vee\blacklozenge\mathsf{P}\varphi)$,
whence $X\vdash\bot$, which contradicts the fact that $X$ is an atomic set
over $\Sigma$.

For the other direction, assume that either $\blacklozenge\mathsf{P}\varphi\in
X$ or $\blacklozenge\varphi\in X$. It follows from axiom A9 that
$\vdash(\blacklozenge\mathsf{P}\varphi\vee\blacklozenge\varphi)\rightarrow\mathsf{P}\varphi$,
which further implies that $X\vdash\mathsf{P}\varphi$, regardless of which
case holds. By claim (ii), the assumption that $\mathsf{P}\varphi\in\neg
Cl({\Sigma})$, and the assumption that $X\in At({\Sigma})$, we have that
$\mathsf{P}\varphi\in X$. ∎

######  Lemma 2.

If $\varphi\in\neg Cl({\Sigma})$ and $\varphi$ is consistent, then there is an
$X\in At(\Sigma)$ such that $\varphi\in X$.

###### Proof.

Similar to [8, Lemma 4.83]. ∎

We note that given a finite set $X$ of formulae, we define $\widehat{X}$ to be
a conjunction of all of its elements. Since all such conjunctions are
equivalent according to our axiomatisation and semantics, we are free to use
any conjunction of the elements of $X$ for $\widehat{X}$.

######  Definition 11 (Canonical Model over $\Sigma$).

Let $\Sigma$ be a finite set of formulae. The _canonical model over $\Sigma$_
is define to be the tuple

$\mathfrak{M}(\Sigma):=\langle W,\\{W_{\mathfrak{d}^{\alpha_{i}}_{j}}\ |\
\mathfrak{d}^{\alpha_{i}}_{j}\in\mathtt{Wit}\\},\\{W_{\mathfrak{e}^{\alpha_{i}}}\
|\ \alpha_{i}\in\mathtt{Agent}\\},R_{\Box},R_{\scalebox{0.8}{
\raisebox{0.90417pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip
0.6pt\raisebox{-0.54247pt}{{A}}\hskip 0.6pt}$\bm{]}$}}
}},R_{\blacksquare},R_{\mathsf{H}}\rangle$

such that:

  * –

$W:=At(\Sigma)$

  * –

$Y\in W_{\mathfrak{d}^{\alpha_{i}}_{j}}$ iff $\mathfrak{d}^{\alpha_{i}}_{j}\in
Y$

  * –

$Y\in W_{\mathfrak{e}^{\alpha_{i}}}$ iff $\mathfrak{e}^{\alpha_{i}}\in Y$

  * –

$YR_{\Box}Z$ iff $\widehat{Y}\land\Diamond\widehat{Z}$ is consistent

  * –

$YR_{\scalebox{0.8}{
\raisebox{0.90417pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip
0.6pt\raisebox{-0.54247pt}{{A}}\hskip 0.6pt}$\bm{]}$}} }}Z$ iff
$\widehat{Y}\land\raisebox{0.55974pt}{\scalebox{0.83}{$\langle$\scalebox{0.9}{{A}}$\rangle$}}\widehat{Z}$
is consistent

  * –

$YR_{\blacksquare}Z$ iff $\widehat{Y}\land\blacklozenge\widehat{Z}$ is
consistent

  * –

$YR_{\mathsf{H}}Z$ iff $YR_{\blacksquare}^{+}Z$

  * –

$V(p):=\\{Y\in At(\Sigma)\ |\ p\in Y\\}$

We take $R^{+}_{\blacksquare}$ to be the transitive closure of
$R_{\blacksquare}$, and the usual definition of sets of moments associated
with complex action types is defined as usual:

  * –

$Y\in W_{t(\overline{\Delta}^{\alpha_{i}})}$ iff $Y\notin
W_{t(\Delta^{\alpha_{i}})}$

  * –

$Y\in W_{t(\Delta^{\alpha_{i}}\cup\Gamma^{\alpha_{j}})}$ iff $Y\in
W_{t(\Delta^{\alpha_{i}})}\cup W_{t(\Gamma^{\alpha_{j}})}$.

######  Lemma 3.

Let $\Sigma$ be a finite set of formulae and $X\in At({\Sigma})$.

  * (i)

If
$[?]\in\\{\Box,\blacksquare,\raisebox{1.29167pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip
0.6pt\raisebox{-0.77496pt}{{A}}\hskip 0.6pt}$\bm{]}$}}\\}$ and
$\langle?\rangle\in\\{\Diamond,\blacklozenge,\raisebox{0.55974pt}{\scalebox{0.83}{$\langle$\scalebox{0.9}{{A}}$\rangle$}}\\}$,
then for all $\langle?\rangle\varphi\in\neg Cl({\Sigma})$,
$\langle?\rangle\varphi\in X$ iff there exists a $Y\in At(\Sigma)$ such that
$XR_{[?]}Y$ and $\varphi\in Y$.

  * (ii)

If $\mathsf{P}\varphi\in\neg Cl({\Sigma})$, then $\mathsf{P}\varphi\in X$ iff
there exists a $Y$ such that $XR_{\mathsf{H}}Y$ and $\varphi\in Y$.

###### Proof.

The two claims are proven similar to Lemma 4.86 and Lemma 4.89 in Blackburn et
al. [8], respectively. ∎

######  Lemma 4.

Let $\Sigma$ be a finite set of formulae. Then, the canonical model
$\mathfrak{M}$ over $\Sigma$ is a $\mathsf{TLAE}$-model.

###### Proof.

We know that $\mathfrak{M}(\Sigma)$ is an $\mathcal{L}_{\mathsf{TLAE}}$-model
by definition. To prove the claim, it suffices to argue that
$\mathfrak{M}(\Sigma)$ satisfies the properties of a $\mathsf{TLAE}$-model. We
only show the p(A3) and p(A6) cases as the remaining cases are similar or
routine.

  * p(A3)

Assume that $XR_{\scalebox{0.8}{
\raisebox{0.90417pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip
0.6pt\raisebox{-0.54247pt}{{A}}\hskip 0.6pt}$\bm{]}$}} }}Y$ and
$XR_{\scalebox{0.8}{
\raisebox{0.90417pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip
0.6pt\raisebox{-0.54247pt}{{A}}\hskip 0.6pt}$\bm{]}$}} }}Z$ hold. We want to
show that $Y=Z$, that is, we want to show that $\widehat{Y}\land\widehat{Z}$
is consistent (note that since $Y$ and $Z$ are atoms, $\widehat{Y}$ and
$\widehat{Z}$ are consistent iff $Y=Z$). We therefore assume that
$\widehat{Y}\land\widehat{Z}$ is inconsistent and derive a contradiction. If
$\widehat{Y}\land\widehat{Z}$ is inconsistent, then it follows that
$\vdash\widehat{Y}\land\widehat{Z}\rightarrow\bot$. By modal reasoning, this
implies that
$\vdash\raisebox{0.55974pt}{\scalebox{0.83}{$\langle$\scalebox{0.9}{{A}}$\rangle$}}(\widehat{Y}\land\widehat{Z})\rightarrow\raisebox{0.55974pt}{\scalebox{0.83}{$\langle$\scalebox{0.9}{{A}}$\rangle$}}\bot$.
Observe that
$\vdash\raisebox{0.55974pt}{\scalebox{0.83}{$\langle$\scalebox{0.9}{{A}}$\rangle$}}\widehat{Y}\land\raisebox{0.55974pt}{\scalebox{0.83}{$\langle$\scalebox{0.9}{{A}}$\rangle$}}\widehat{Z}\rightarrow\raisebox{0.55974pt}{\scalebox{0.83}{$\langle$\scalebox{0.9}{{A}}$\rangle$}}(\widehat{Y}\land\widehat{Z})$
holds, as it is a consequence of the axiom
$\raisebox{0.55974pt}{\scalebox{0.83}{$\langle$\scalebox{0.9}{{A}}$\rangle$}}\varphi\rightarrow\raisebox{1.29167pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip
0.6pt\raisebox{-0.77496pt}{{A}}\hskip 0.6pt}$\bm{]}$}}\varphi$; hence,
$\vdash\raisebox{0.55974pt}{\scalebox{0.83}{$\langle$\scalebox{0.9}{{A}}$\rangle$}}\widehat{Y}\land\raisebox{0.55974pt}{\scalebox{0.83}{$\langle$\scalebox{0.9}{{A}}$\rangle$}}\widehat{Z}\rightarrow\raisebox{0.55974pt}{\scalebox{0.83}{$\langle$\scalebox{0.9}{{A}}$\rangle$}}\bot$.
By modal and propositional reasoning, we have that
$\vdash(\widehat{X}\land\raisebox{0.55974pt}{\scalebox{0.83}{$\langle$\scalebox{0.9}{{A}}$\rangle$}}\widehat{Y})\land(\widehat{X}\land\raisebox{0.55974pt}{\scalebox{0.83}{$\langle$\scalebox{0.9}{{A}}$\rangle$}}\widehat{Z})\rightarrow\bot$,
meaning that
$\vdash(\widehat{X}\land\raisebox{0.55974pt}{\scalebox{0.83}{$\langle$\scalebox{0.9}{{A}}$\rangle$}}\widehat{Y})\rightarrow\bot\lor(\widehat{X}\land\raisebox{0.55974pt}{\scalebox{0.83}{$\langle$\scalebox{0.9}{{A}}$\rangle$}}\widehat{Z})\rightarrow\bot$.
The last theorem implies that either $XR_{\scalebox{0.8}{
\raisebox{0.90417pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip
0.6pt\raisebox{-0.54247pt}{{A}}\hskip 0.6pt}$\bm{]}$}} }}Y$ or
$XR_{\scalebox{0.8}{
\raisebox{0.90417pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip
0.6pt\raisebox{-0.54247pt}{{A}}\hskip 0.6pt}$\bm{]}$}} }}Z$ does not hold (by
the definition or $R_{\scalebox{0.8}{
\raisebox{0.90417pt}{\scalebox{0.74}{$\bm{[}$\scalebox{1.0}{\hskip
0.6pt\raisebox{-0.54247pt}{{A}}\hskip 0.6pt}$\bm{]}$}} }}$), thus
contradicting our assumption.

  * p(A6)

Let $X\in W$, $\alpha_{i}$ an agent, and suppose that there exists a $Y$ such
that $XR_{\Box}Y$ and $Y\in W_{\mathfrak{e}^{\alpha_{i}}}$. We want to show
that there exists a $Z\in W$ such that $XR_{\Box}Z$ and $Z\not\in
W_{\mathfrak{e}^{\alpha_{i}}}$. By Definition 8, we know that
$\Diamond\neg\mathfrak{e}^{\alpha_{i}}\in\neg Cl({\Sigma})$. We aim to show
that $\widehat{X}\land\Diamond\neg\mathfrak{e}^{\alpha_{i}}$ is consistent,
since this will imply that $\Diamond\neg\mathfrak{e}^{\alpha_{i}}\in X$, due
to the fact that $X$ is an atomic set. Thus, we assume that
$\vdash\widehat{X}\land\Diamond\neg\mathfrak{e}^{\alpha_{i}}\rightarrow\bot$
to derive a contradiction. By axiom A6, we have
$\vdash\widehat{X}\land\Diamond\mathfrak{e}^{\alpha_{i}}\rightarrow\bot$ as a
consequence, which implies
$\vdash\widehat{X}\land\Diamond\mathfrak{e}^{\alpha_{i}}\land\Diamond\widehat{Y}\rightarrow\bot$
by propositional reasoning. Modal and propositional reasoning may then be
applied to derive the following

$\vdash\widehat{X}\land\Diamond(\mathfrak{e}^{\alpha_{i}}\land\widehat{Y})\rightarrow\bot$

We know that $\widehat{Y}\land\mathfrak{e}^{\alpha_{i}}$ is consistent because
$Y\in W_{\mathfrak{e}^{\alpha_{i}}}$. However, since $Y$ is an atomic set and
$\mathfrak{e}^{\alpha_{i}}\in\neg Cl({\Sigma})$, it follows that
$\widehat{Y}\land\mathfrak{e}^{\alpha_{i}}$ is equivalent to $\widehat{Y}$.
Hence,

$\vdash\widehat{X}\land\Diamond\widehat{Y}\rightarrow\bot$

contradicting our assumption that $XR_{\Box}Y$. Consequently,
$\Diamond\neg\mathfrak{e}^{\alpha_{i}}\in X$, so by Lemma 3, there exists a
$Z\in At(\Sigma)$ such that $XR_{\Box}Z$ and $\neg\mathfrak{e}^{\alpha_{i}}\in
Z$. The latter fact implies that $Z\not\in W_{\mathfrak{e}^{\alpha_{i}}}$ by
Definition 11. ∎

######  Lemma 5 (Truth Lemma).

Let $\mathfrak{M}(\Sigma)$ be the canonical model over $\Sigma$. For all
atomic sets $Y$ and all $\varphi\in\neg Cl({\Sigma})$,
$\mathfrak{M}(\Sigma),Y\models\varphi$ iff $\varphi\in Y$.

###### Proof.

By induction on the complexity of $\varphi$. ∎

######  Theorem 4 (Weak Completeness).

For any formula $\varphi$, if $\models\varphi$, then $\vdash\varphi$.

###### Proof.

Follows from Lemma 4 and Lemma 5. ∎

##  5 Formal Notions of Instrumentality

In this section, we formalize a variety of instrumentality notions
corresponding to the philosophical analysis of Section 2. As will be
demonstrated, the logic $\mathsf{TLAE}$ suffices to capture many of the
desired nuances present in judgments of instrumentality. In Section 5.1, we
show how time intervals can be utilised to evaluate an action’s suitability
for serving a particular purpose (criterion I). In Section 5.2, using the
various definitions of instrumentality, we semantically define a collection of
value judgments qualifying instruments as ‘better’, ‘best’, ‘worst’, ‘good’,
and ‘poor’ (criterion II). In Section 5.3, we show that, as a result of using
past and future references, the obtained formal definitions capture the
inherent defeasible nature of judgments of instrumentality (criterion III).

###  5.1 Elementary Instrumentality Notions

In [6], four formal definitions of instrumentality are provided: a ‘basic’ and
a ‘proper’ notion of instrumentality, which are either agent-dependent or
agent-independent. (NB. The terms ‘basic’ and ‘proper’ refer to our account of
instruments in parts (1) and (2) of Definition 3, respectively, i.e., ‘plain’
and ‘excellent’ instruments.) As remarked above, the logical system
$\mathsf{LAE}$, introduced in [6], is closely related to the logic
$\mathsf{TLAE}$ presented in this paper. The principal difference between
$\mathsf{LAE}$ and $\mathsf{TLAE}$ is that the former’s language does not
include modalities that refer to the past, whereas the latter does. To be
precise, the language of $\mathsf{TLAE}$ includes the additional
$\blacksquare$ and $\mathsf{H}$ operators. These operators allow us to
syntactically define the notion of ‘candidate instrument’ and ‘excellent
candidate instrument’ (see Definition 1), referring either to a finite
interval of time preceding the moment of evaluation or to the entire past of
the moment of evaluation. These notions were merely semantically defined in
[6]. The logical characterisation of the $\blacksquare$ operator enables us to
count successful applications of candidate instruments and syntactically
capture intervals of time.

In ($\mathsf{d11}$) below, we formalise the agent-dependent notion of
candidate $\varphi$-instrument as presented in item (1) of Definition 1.
Observe that the definition is relativised to a past time interval with length
$n$.

  1. Candidate Instrument for $\alpha$ (with a past interval of length $n$)
$[\Delta^{\alpha}]_{n}^{c-instr}\varphi:=\bigvee\limits_{0\leq i\leq
n}\blacklozenge^{i}(t(\Delta^{\alpha})\land\blacklozenge[\Delta^{\alpha}]^{would}\varphi)$
($\mathsf{d11}$)

The formula $[\Delta^{\alpha}]_{n}^{c-instr}\varphi$ ($\mathsf{d13}$) reads as
‘somewhere within the past interval of length $n$ there is a moment $v$ at a
distance of $i$ units of time that witnessed the successful performance of
$\Delta$ by agent $\alpha$ such that at $v$’s immediate predecessor, the
performance of $\Delta$ by that agent would have guaranteed $\varphi$’.

Agent-dependent excellent candidate $\varphi$-instruments are, then,
formalized by combining the above definition with the idea that every past
performance of the relevant action type has led to the intended outcome (item
(2) of Definition 1).

  1. Excellent Candidate Instrument for $\alpha$ (with a past interval of length $n$)
$[\Delta^{\alpha}]_{n}^{ex{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}.}c-instr}\varphi:=[\Delta^{\alpha}]_{n}^{c-instr}\varphi\wedge\bigwedge\limits_{1\leq
k\leq n}\blacksquare^{k}[\Delta^{\alpha}]^{would}\varphi$ ($\mathsf{d12}$)

We read ($\mathsf{d12}$) as ‘action type $\Delta$ has proven to be a candidate
instrument for $\varphi$ for $\alpha$ at least once in the interval, and every
other performance of $\Delta$ by $\alpha$ within the interval would have also
guaranteed $\varphi$’. One can say that, within the past interval $n$, the
action type $\Delta$ has a one hundred percent success rate for $\alpha$ in
obtaining $\varphi$. In the sequel, we introduce ways of refining these
definitions.101010Observe that in ($\mathsf{d11}$) and ($\mathsf{d12}$)
reference to past experience—i.e., $\blacklozenge^{i}$ and
$\blacksquare^{i}$—also includes reference to outcomes obtained at the moment
of evaluation $w$. Otherwise, a performance of $\Delta$ immediately before $w$
might fail to deliver $\varphi$ at $w$.

Agent-independent generalizations of ($\mathsf{d11}$) and ($\mathsf{d12}$) are
captured by ($\mathsf{d13}$), respectively ($\mathsf{d14}$) (cf. the agent-
dependent definitions of [6]).

  1. $[\Delta]_{n}^{c-instr}\varphi:=\bigvee\limits_{\alpha\in\mathtt{Agent}}\bigvee\limits_{0\leq i\leq n}\blacklozenge^{i}(t(\Delta^{\alpha})\land\blacklozenge[\Delta^{\alpha}]^{would}\varphi)$ ($\mathsf{d13}$)

  1. $[\Delta]_{n}^{ex{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}.}c-instr}\varphi:=[\Delta]_{n}^{c-instr}\varphi\wedge\bigwedge\limits_{1\leq k\leq n}\blacksquare^{k}\bigwedge\limits_{\alpha\in\mathtt{Agent}}[\Delta^{\alpha}]^{would}\varphi$ ($\mathsf{d14}$)

Henceforth, we focus on agent-dependent notions. The agent-independent
versions of each of the definitions below can be straightforwardly obtained.

The more refined definitions of instruments and excellent
instruments—corresponding to items (1) and (2) of Definition 3—are obtained by
adding the pivotal conjectural element, reflecting the agent’s expectations
about the instrument’s suitability in the immediate future. To capture these
notions, we first need to alter the agentive operator would ($\mathsf{d1}$)
(Section 3.2).

  1. Expected Would
$[\Delta^{\alpha}]^{would}_{ex}\varphi:=\square((t(\Delta^{\alpha})\land\mathfrak{e}^{\alpha})\rightarrow\varphi)$.
($\mathsf{d15}$)

The ‘expected would’ operator ($\mathsf{d15}$) restricts the formula’s
evaluation to immediate future moments that the agent expects as continuations
of the present. Using ($\mathsf{d15}$), we formalize items (1) and (2) of
Definition 3 as follows:

  1. Instrument for $\alpha$ (with a past interval of length $n$)
$[\Delta^{\alpha}]_{n}^{instr}\varphi:=\bigvee\limits_{0\leq i\leq
n}\blacklozenge^{i}(t(\Delta^{\alpha})\land\blacklozenge[\Delta^{\alpha}]^{would}\varphi)\land[\Delta^{\alpha}]^{would}_{ex}\varphi$
($\mathsf{d16}$)

  2. Excellent Instrument for $\alpha$ (with a past interval of length $n$)
$[\Delta^{\alpha}]_{n}^{ex-
instr}\varphi:=[\Delta^{\alpha}]_{n}^{instr}\varphi\wedge\bigwedge\limits_{1\leq
k\leq n}\blacksquare^{k}[\Delta^{\alpha}]^{would}\varphi$ ($\mathsf{d17}$)

The final formalisations ($\mathsf{d16}$) and ($\mathsf{d17}$)—to which we
sometimes refer as ‘proper instruments’—differ from their candidate
counterparts ($\mathsf{d11}$) and ($\mathsf{d12}$) through the additional
conjunct expressing that the agent expects that, at the moment of evaluation,
she would guarantee $\varphi$ by performing $\Delta$. Stronger notions of
(excellent) instruments are straightforwardly obtained by using a definition
of ‘expected could’ instead of ‘expected would’ as the last conjunct in
($\mathsf{d16}$).111111Such stronger notions are used in [6]. As pointed out
by a reviewer, the use of expected could can be problematic. For instance,
suppose that at $w$ agent $\alpha$ acknowledges a relation between past
performances of action $\Delta$ (e.g., turning a car’s ignition key) and an
outcome $\varphi$ (the car’s motor is running). Now, suppose that at $w$,
$\alpha$ expects that $\Delta$ will serve purpose $\varphi$ in the immediate
future. This may suffice for $\alpha$ to consider $\Delta$ an instrument for
$\varphi$ at $w$, even if $\Delta$ cannot be performed at $w$ for reasons
unknown to $\alpha$ (e.g., the car’s battery is down). The following
implications are theorems of $\mathsf{TLAE}$ and show the various relations
between the four notions defined so far.

  1. $[\Delta^{\alpha}]_{n}^{ex-instr}\varphi\rightarrow[\Delta^{\alpha}]_{n}^{ex{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>$ and $\ [\Delta^{\alpha}]_{n}^{instr}\varphi\rightarrow[\Delta^{\alpha}]_{n}^{c-instr}\varphi$

  2. $[\Delta^{\alpha}]_{n}^{ex{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>$ and $\ [\Delta^{\alpha}]_{n}^{ex-instr}\varphi\rightarrow[\Delta^{\alpha}]_{n}^{instr}\varphi$

Before moving to comparative notions of instrumentality, we make three
remarks. First, purposes may include action formulae or action witnesses. For
instance, agent $\alpha$’s purpose may be to ensure that agent $\beta$ could
bring about $\varphi$ by performing $\Delta$. In that case, $\alpha$ is
looking for the action that will ensure that, at the next moment,
$[\Delta^{\beta}]\varphi$ holds. We will not further pursue this here.

Second, we did not consider deliberative versions of instrumentality.
Definitions ($\mathsf{d11}$), ($\mathsf{d12}$), ($\mathsf{d16}$), and
($\mathsf{d17}$) will qualify each action type as an instrument for bringing
about tautologous propositions. Deliberative variants can be straightforwardly
obtained in the spirit of [5]’s deliberative STIT operator.

Third, the volitional concepts of Section 3.4 (e.g., ‘producing’ and
‘destroying’) can also be employed in the context of instrumentality. Such
definitions can be straightforwardly given in the framework of
$\mathsf{TLAE}$. For instance, ($\mathsf{d18}$) (below) expresses that, for
agent $\alpha$, $\Delta$ is an instrument for producing $\varphi$ because (i)
$\alpha$ produced $\varphi$ through performing $\Delta$ at least once in the
past (with an interval of length $n$) and (ii) $\alpha$ expects to produce
$\varphi$ through $\Delta$ at present.

  1. $[\Delta^{\alpha}]_{prod-n}^{instr}\varphi$ | $:=$ | $(i)\ \bigvee\limits_{0\leq i\leq n}\blacklozenge^{i}(t(\Delta^{\alpha})\land\blacklozenge[\Delta^{\alpha}]^{would}_{prod}\varphi)\ \land$
---|---|---
|  | $(ii)\quad[\Delta^{\alpha}]^{would}_{ex}\varphi\land\lnot\varphi\land\lozenge(\lnot\varphi\land\mathfrak{e}^{\alpha})$

($\mathsf{d18}$)

That ($\mathsf{d18}$) is a stronger notion than ($\mathsf{d16}$) follows from
the following theorem:

  1. $[\Delta^{\alpha}]_{prod-n}^{instr}\varphi\rightarrow[\Delta^{\alpha}]_{n}^{instr}\varphi\ $

###  5.2 Good Instruments and Comparative Instrumentality

As shown in the previous section, our language $\mathcal{L}_{\mathsf{TLAE}}$
is suitable for the definition of candidate instruments, as per
($\mathsf{d11}$) and ($\mathsf{d12}$), as well as proper instruments, as per
($\mathsf{d16}$) and ($\mathsf{d17}$). We now discuss how to evaluate
different instruments serving the same purpose, giving rise to notions such as
‘better’ and ‘good’ instruments (see Definition 2 and items (3) and (4) of
Definition 3). In our framework, we can use $\mathsf{TLAE}$-models to
determine whether an available (candidate) instrument is regarded as better
than another. In this section, we proceed with a semantic analysis of
comparative instrumentality, providing various notions of better (worse), best
(worst), and good (poor) instruments.

We adopt the following methodology:

  1. 1.

Collect all the instruments that serve $\varphi$;

  2. 2.

Determine the success ratio of the obtained instruments w.r.t. $\varphi$.

For comparative judgments of betterness, we proceed accordingly:

  1. 3.

Order the available instruments on the basis of their success ratio;

  2. 4.

Identify the best and the worst;

  3. 5.

Identify better instruments by comparing instruments within their order.

For comparative judgments of instrumental goodness, we proceed as follows:

  1. 6.

Identify good instruments by checking whether their success ratio satisfies a
certain threshold ratio;

  2. 7.

Identify good instruments by checking whether they satisfy certain additional
thresholds, such as a minimum amount of past witnesses.

We addressed step (1) in the previous section via ($\mathsf{d16}$). In Section
5.2.1, we deal with (3)-(5), and in Section 5.2.2, we deal with (6) and (7).
First, let us address the notion of success ratio from step (2). For
readability, in what follows, we omit explicit reference to a past interval of
length $n$ without a loss of generality. Recall that $\mathsf{TLAE}$-frames
are rooted tree-like structures, meaning each moment $w$ has a finite past.
Thus, there are only finitely many witnesses of an instrument’s application.
We use ‘$\ast$’ to denote consideration of the entire past. That is, for a
moment $w\in W$, $w\models[\Delta^{\alpha}]^{instr}_{\ast}\varphi$ signifies
that we evaluate the past of $w$ up to the root of the model.

In the sequel, we provide our definitions for proper instruments only
($\mathsf{d16}$) (variations can be straightforwardly obtained). Let $\varphi$
be the purpose endorsed by agent $\alpha$ at moment $w$. Then, we let
$\textit{Instr}^{w}_{\alpha}({\varphi})$ denote the set containing all
$\varphi$-instruments available to $\alpha$ at $w$ with respect to the entire
past of $w$, i.e., all actions that served $\varphi$ at least once, and of
which $\alpha$ expects that it will serve $\varphi$ again.121212Note that even
if $\textit{Instr}^{w}_{\alpha}({\varphi})$ contains infinitely many action
types, up to logical equivalence, the set will only contain finitely many. We
come back to this in Section 5.2.1. Definition ($\mathsf{d19}$) below makes
this formally precise.

  1. Available $\varphi$-instruments for $\alpha\in\mathtt{Agent}$ at $w$
$\textit{Instr}^{w}_{\alpha}({\varphi}):=\\{\Delta\in\mathcal{L}_{\mathsf{Act}}\
|\ w\models[\Delta^{\alpha}]^{instr}_{\ast}\varphi\\}$ ($\mathsf{d19}$)

For each collected available instrument
$\Delta\in\textit{Instr}^{w}_{\alpha}({\varphi})$, we must determine how
‘successful’ it is at securing $\varphi$. Here, we opt for defining success in
terms of an achievement/failure ratio. Below, achievement ($\mathsf{d20}$)
refers to the fact that by performing $\Delta$ agent $\alpha$ would secure
$\varphi$ and, in fact, did secure $\varphi$.

  1. Achievement of $\Delta\in\textit{Instr}^{w}_{\alpha}({\varphi})$
$\textit{Achiev}^{w}_{\alpha}({\varphi},{\Delta}):=\\{\theta\ |\
\theta=\blacklozenge^{i}(t(\Delta^{\alpha})\land\blacklozenge[\Delta^{\alpha}]^{would}\varphi),0{\leq}i$,
and $w\models\theta\\}$ ($\mathsf{d20}$)

In ($\mathsf{d20}$), we use $0\leq i$ to indicate that the moment of
evaluation may serve as a witness of a successful application of $\Delta$
initiated in its immediate preceding moment. Furthermore, the first conjunct
in $\theta$ witnesses the actual performance of $\Delta$ and attainment of
$\varphi$, whereas the second conjunct ensures that the simultaneous
occurrence of $\Delta$ and $\varphi$ is not merely coincidental.

We define failure ($\mathsf{d21}$) in terms of the expectations of the agent
involved. A failure to secure $\varphi$ by $\alpha$’s performance of $\Delta$
at $w$ means that (i) at $w$, $\Delta$ has just been performed by $\alpha$,
(ii) $\varphi$ does not hold at $w$, and (iii) at the immediately preceding
moment $\alpha$ expected that by performing $\Delta$, $\varphi$ would be
attained.

  1. Failure of $\Delta\in\textit{Instr}^{w}_{\alpha}({\varphi})$
$\textit{Fail}^{w}_{\alpha}({\varphi},{\Delta}){:=}\\{\theta\ |\
\\!\theta=\blacklozenge^{i}(t(\Delta^{\alpha})\land\\!\lnot\varphi\land\blacklozenge[\Delta^{\alpha}]^{would}_{ex}\varphi)$,
$0{\leq}i$, and $w\models\theta\\}$ ($\mathsf{d21}$)

Through the use of $\blacklozenge^{i}$, all members of
$\textit{Achiev}^{w}_{\alpha}({\varphi},{\Delta})$ and
$\textit{Fail}^{w}_{\alpha}({\varphi},{\Delta})$ represent unique past
moments. In what follows, we sometimes abuse notation and let these sets stand
for the sets of moments witnessing the respective achievements and failures.
Furthermore, the two sets are finite since the past is finite, and so, the
cardinality of the two sets in ($\mathsf{d20}$) and ($\mathsf{d21}$) can be
utilised to count the amount of achieved or failed applications of the
instrument under consideration. Thus, we have the means to calculate a success
ratio for the collected instruments, i.e., ($\mathsf{d22}$). We denote the
cardinality of a set $S$ by $|S|$.

  1. Success ratio of $\Delta\in\textit{Instr}^{w}_{\alpha}({\varphi})$

  2. $\textit{Succ}^{w}_{\alpha}({\varphi},{\Delta}):=\displaystyle\frac{|\textit{Achiev}^{w}_{\alpha}({\varphi},{\Delta})|}{|\textit{Achiev}^{w}_{\alpha}({\varphi},{\Delta})\cup\textit{Fail}^{w}_{\alpha}({\varphi},{\Delta})|}$ ($\mathsf{d22}$)

To exemplify the above machinery, consider Figure 4 and suppose there is an
agent $\alpha$ at moment $w_{6}$ who aims to bring about $\varphi$. There are
two available instruments at $w_{6}$, namely, $\Delta$ and $\Gamma$, whose
potential was witnessed at $w_{2}$. We find that $\alpha$ expects both
instruments to serve $\varphi$ at $w_{6}$ (see $w_{8}$), although $\alpha$’s
current expectations about $\Gamma$ are incorrect (see $w_{9}$). Furthermore,
notice that, from the vantage point of $w_{6}$, $\Gamma$ and $\Delta$ both
have two achievement witnesses, namely,
$\textit{Achiev}^{w_{6}}_{\alpha}({\varphi},{\Gamma})=\\{w_{2},w_{5}\\}$, and
$\textit{Achiev}^{w_{6}}_{\alpha}({\varphi},{\Delta})=\\{w_{1},w_{2}\\}$.
Furthermore, $\Delta$ also has a witness of a failed application at $w_{6}$,
that is, $\textit{Fail}^{w_{6}}_{\alpha}({\varphi},{\Gamma})=\\{w_{6}\\}$. At
$w_{5}$, $\alpha$ had expectations that turned out to be wrong at $w_{6}$.
Hence, the success ratios of $\Gamma$ and $\Delta$ at $w_{6}$ are
$\textit{Succ}^{w_{6}}_{\alpha}({\varphi},{\Gamma})=1$ and
$\textit{Succ}^{w_{6}}_{\alpha}({\varphi},{\Delta})=\frac{2}{3}$,
respectively.

$w_{1}$ $[\Gamma^{\alpha}]^{would}\varphi$
---
$[\Delta^{\alpha}]^{would}\varphi$
$t(\Delta^{\alpha})\land\varphi$
$w_{0}$ $[\Delta^{\alpha}]^{would}\varphi$
---
$w_{2}$ $t(\Delta^{\alpha})\land\varphi$
---
$t(\Gamma^{\alpha})$
$w_{3}$ $[\Gamma^{\alpha}]_{ex}^{would}\varphi$
---
$w_{4}$ $t(\Gamma^{\alpha})\land\varphi\land\mathfrak{e}^{\alpha}$
---
$w_{5}$ $t(\Gamma^{\alpha})\land\varphi$
---
$[\Delta^{\alpha}]_{ex}^{would}\varphi$
$w_{6}$ $t(\Delta^{\alpha})\land\lnot\varphi$
---
$w_{7}$ $t(\Delta^{\alpha})\land\varphi\land\mathfrak{e}^{\alpha}$
---
$w_{8}$ $t(\Delta^{\alpha})\land\varphi$
---
$t(\Gamma^{\alpha})\land\mathfrak{e}^{\alpha}$
$w_{9}$ $\lnot t(\Delta^{\alpha})\land\lnot\varphi$
---
$t(\Gamma^{\alpha})$

Figure 4: An example of evaluating $\varphi$-instruments $\Gamma$ and $\Delta$
for $\alpha$ at $w_{6}$.

####  5.2.1 Comparative Judgments of Betterness

The notion of success ratio is pivotal for defining various types of
axiological—i.e., value—judgment concerning instrumentality. In particular, we
are interested in the following axiological concepts: best, worst, better,
good, and poor. The latter two are formally addressed in the next section.

Recall that the language $\mathcal{L}_{\mathsf{TLAE}}$ allows only for
finitely many action types, up to provable equivalence of the formulas that
witness their performance by an agent. That is, since
$\mathcal{L}_{\mathsf{Act}}$ is constructed over a finite number of atomic
action types in $\mathtt{Action}$, for each agent in $\mathtt{Agent}$, there
will be finitely many equivalence classes
$[\\![\Delta^{\alpha}]\\!]~{}:=~{}\\{\Gamma^{\alpha}\ |\
\vdash_{\mathsf{TLAE}}t(\Delta^{\alpha})\leftrightarrow t(\Gamma^{\alpha})\\}$
of equivalent actions. We let $\mathsf{EqAct}=\\{[\\![\Delta^{\alpha}]\\!]\ |\
$ for $\Delta\in\mathcal{L}_{\mathsf{Act}}$ and $\alpha\in\mathtt{Agent}\\}$
be the set of all such equivalence classes. Consequently, we obtain a finite
ordering of actions (up to equivalence) when ordering candidate instruments.
This observation enables us to identify those instruments at the ordering’s
upper- and lower-bound.

Naive best, worst, and better. We define $\succeq^{w}_{\varphi}$ to be a
success ratio ordering over actions such that, for each
$\Delta,\Gamma\in\textit{Instr}^{w}_{\alpha}({\varphi})$, we have

  1. $\Delta^{\alpha}\succeq^{w}_{\varphi}\Gamma^{\alpha}\text{ {iff} }\textit{Succ}^{w}_{\alpha}({\varphi},{\Delta})\geq\textit{Succ}^{w}_{\alpha}({\varphi},{\Gamma}).$ ($\mathsf{d23}$)

We define $\Delta^{\alpha}\succ^{w}_{\varphi}\Gamma^{\alpha}$ as the
conjunction $\Delta^{\alpha}\succeq^{w}_{\varphi}\Gamma^{\alpha}$ and
$\Gamma^{\alpha}\not\succeq^{w}_{\varphi}\Delta^{\alpha}$. We interpret
$\succeq^{w}_{\varphi}$ as a “betterness” relation: i.e., we read
$\Delta^{\alpha}\succeq^{w}_{\varphi}\Gamma^{\alpha}$ as ‘at $w$, $\Delta$ is
a weakly better instrument for agent $\alpha$ to secure $\varphi$ than the
instrument $\Gamma$’ (i.e., $\Delta$ is at least as good as $\Gamma$ for
$\alpha$ in order to get $\varphi$). Then,
$\Delta^{\alpha}\succ^{w}_{\varphi}\Gamma^{\alpha}$ expresses that $\Delta$ is
a (strictly) better $\varphi$-instrument than $\Gamma$, for $\alpha$ at $w$.

Likewise, we can define notions of best and worst because, in the ordering, we
find an upper and lower bound. Since we have a finite ordering of classes of
equivalent actions, we know that if
$\textit{Instr}^{w}_{\alpha}({\varphi})\neq\emptyset$, there are
$\Delta,\Gamma\in\textit{Instr}^{w}_{\alpha}({\varphi})$ such that
$\Delta^{\alpha}$ is an upper bound, and $\Gamma^{\alpha}$ is a lower bound of
$\succeq^{w}_{\varphi}$. In other words, there are no
$\Theta,\Sigma\in\textit{Instr}^{w}_{\alpha}({\varphi})$ such that
$\Theta^{\alpha}\succ^{w}_{\alpha}\Delta^{\alpha}$, respectively
$\Gamma^{\alpha}\succ^{w}_{\alpha}\Sigma^{\alpha}$. Note that it can be that
$\textit{Succ}^{w}_{\alpha}({\varphi},{\Delta})=\textit{Succ}^{w}_{\alpha}({\varphi},{\Gamma})$
or that $\Delta=\Gamma$. Naively, we may say that an instrument
$\Delta\in\textit{Instr}^{w}_{\alpha}({\varphi})$ in an upper bound of
$\succeq^{w}_{\varphi}$ is among the best instruments for $\alpha$ at $w$,
whereas an instrument $\Gamma\in\textit{Instr}^{w}_{\alpha}({\varphi})$ in a
lower bound of $\succeq^{w}_{\varphi}$ is among the worst instruments for
$\alpha$ at $w$ for securing $\varphi$.

There is an obvious objection to this naive approach: Suppose there are only
two available $\varphi$-instruments
$\Delta,\Gamma\in\textit{Instr}^{w}_{\alpha}({\varphi})$ for $\alpha$ at $w$
such that $\textit{Succ}^{w}_{\alpha}({\varphi},{\Delta})=1$ and
$\textit{Succ}^{w}_{\alpha}({\varphi},{\Gamma})=0.999$. Our naive definition
tells us: (i) $\Delta$ is better than $\Gamma$ (for $\alpha$, in securing
$\varphi$ at $w$), (ii) $\Delta$ is among the best available
$\varphi$-instruments, and (iii) $\Gamma$ is among the worst available
$\varphi$-instruments. That $\Gamma$ is the ‘worst’ instrument is arguably an
overstatement. In fact, the argument can even be made that $\Gamma$ is an
exceptionally better $\varphi$-instrument than $\Delta$. For example, suppose
that $\Delta$ was performed only once in the past by $\alpha$, securing
$\varphi$ and that $\Gamma$ was performed 1000 times, securing $\varphi$ 999
times. Then, one should be able to conclude that $\Gamma$ is (by far) the best