# Comparing Inferential Strategies of Humans and Large Language Models in
Deductive Reasoning

Philipp Mondorf1, 2 Barbara Plank1, 2
1MaiNLP, Center for Information and Language Processing, LMU Munich, Germany
2Munich Center for Machine Learning (MCML), Munich, Germany
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Deductive reasoning plays a pivotal role in the formulation of sound and
cohesive arguments. It allows individuals to draw conclusions that logically
follow, given the truth value of the information provided. Recent progress in
the domain of large language models (LLMs) has showcased their capability in
executing deductive reasoning tasks. Nonetheless, a significant portion of
research primarily assesses the accuracy of LLMs in solving such tasks, often
overlooking a deeper analysis of their reasoning behavior. In this study, we
draw upon principles from cognitive psychology to examine inferential
strategies employed by LLMs, through a detailed evaluation of their responses
to propositional logic problems. Our findings indicate that LLMs display
reasoning patterns akin to those observed in humans, including strategies like
supposition following or chain construction. Moreover, our research
demonstrates that the architecture and scale of the model significantly affect
its preferred method of reasoning, with more advanced models tending to adopt
strategies more frequently than less sophisticated ones. Importantly, we
assert that a model’s accuracy, that is the correctness of its final
conclusion, does not necessarily reflect the validity of its reasoning
process. This distinction underscores the necessity for more nuanced
evaluation procedures in the field.

0.5pt

Comparing Inferential Strategies of Humans and Large Language Models in
Deductive Reasoning

Philipp Mondorf1, 2 and Barbara Plank1, 2 1MaiNLP, Center for Information and
Language Processing, LMU Munich, Germany 2Munich Center for Machine Learning
(MCML), Munich, Germany<EMAIL_ADDRESS><EMAIL_ADDRESS>

##  1 Introduction

Task Prompt:
Evaluate whether the given conclusion follows from the premises below: 1.
There is a pink marble if and only if there is an olive marble. 2. If there is
an olive marble then there is a brown marble. 3. Either there is a brown
marble or else there is a black marble, but not both. Conclusion: Either there
is a pink marble or else there is a black marble, but not both. Answer:
Okay, let’s say there is a pink marble. According to the first statement,
there must be an olive marble. Then, according to the second statement, there
must be a brown marble. But wait, that means there cannot be a black marble,
because of the third statement! So, the conclusion is actually true. Answer:
1. If there is a pink marble, then there must be an olive marble. 2. If there is an olive marble, then there must be a brown marble. 3. If there is a brown marble, then there cannot be a black marble. Therefore, if there is a pink marble, then there cannot be a black marble. Conclusion: True Figure 1:  Given the propositional reasoning prompt (top box), the LLM shows two different inferential strategies: supposition following (left) and chain construction (right), see Section 2 for strategy details. Note that both answers are only partially correct, as the exclusive disjunction has only been proven for one of the cases (pink and not black). Model responses are generated by LLaMA-2-Chat-70B across two random seeds.

Deductive reasoning, that is the process of drawing conclusions that logically
follow from the information at hand, is an integral aspect of human cognition
and plays a pivotal role in formulating sound and coherent arguments Leighton
(2003). Take, for example, the following statements:

If there is a blue marble in the box then there is a green marble in the box.
There is a blue marble in the box.

Even without proper training in logic, most individuals can naturally deduce
the valid conclusion:

Therefore, there is a green marble in the box.

This innate capability of drawing conclusions that invariably follow from the
truth value of available information has been a focal point of scholarly
interest for centuries Holyoak and Morrison (2005). Propositional logic, a
subfield of deductive reasoning, focuses on constructing logical arguments
based on the relationship between statements similar to those in the example
previously mentioned Hurley (2011). Extensive research has been dedicated to
examining human reasoning behavior in contexts that involve propositional
logic. For instance, Van der Henst et al. (2002) have identified _five
different strategies_ people commonly employ when navigating problems of
propositional logic (see Section 2). Such behavioral studies have been crucial
in shaping theories that shed light on the fundamental elements of cognitive
reasoning processes Rips (1994); Johnson-Laird (1986); Kahneman et al. (1982).

In parallel, recent advancements in the field of large language models have
demonstrated their potential in executing tasks involving deductive reasoning
Yang et al. (2023); Yu et al. (2023); Huang and Chang (2023). Yet, the extent
to which LLMs truly possess such abilities remains a subject of ongoing debate
Mahowald et al. (2023); Mitchell and Krakauer (2023). Unlike behavioral
studies in human reasoning that are often characterized by in-depth
examinations of the reasoners’ expressions, many studies on LLM-based
reasoning tend to focus on task performance and accuracy metrics, offering
limited insights into the underlying reasoning behavior of the models Mitra et
al. (2023); OpenAI et al. (2023); Team et al. (2023).

In this paper, we draw from the cognitive science literature Van der Henst et
al. (2002) and study inferential strategies employed by LLMs when solving
propositional logic problems (see Figure 1). We analyze the reasoning behavior
of three different language model families, varying in model size and fine-
tuning procedure, and compare them to the behavior found in humans. To the
best of our knowledge, we are the first to comprehensively compare inferential
strategies employed by large language models and humans. We analyze the
models’ output both quantitatively and qualitatively via manual inspection, to
provide insights into the soundness of their verbalized reasoning strategies.
Our findings reveal that:

  * •

All models exhibit inferential strategies akin to those observed in human
reasoning, such as supposition following and chain construction.

  * •

The inferential strategy employed is significantly influenced by the model
family, as different families favor different approaches.

  * •

Models are often right but for the wrong reasons: the accuracy of a model,
that is the number of correct final answers, does not reflect whether its
reasoning is sound, i.e. logically follows from the statements at hand.

  * •

The strategy employed by a model is closely related to the soundness of its
reasoning, where certain strategies lead to correct reasoning and others tend
to introduce errors.

  * •

In contrast to human reasoners, models occasionally adopt a symbolic strategy,
where formal logical calculus is employed to solve the problems of
propositional logic.

Through this work, we hope to advance the understanding of reasoning in LLMs.

##  2 Strategies in Propositional Reasoning

Propositional logic studies the relationships among statements (or
propositions) and the methods for constructing logical arguments based on them
Hurley (2011). At the core of propositional logic are simple statements that
can be combined through the use of logical connectives such as "not", "and",
"or", and "if… then…", thereby forming more complex compound statements.
Conclusions are logically deduced, where the truth value of the propositions
necessitates the truth of the conclusion. This form of logical reasoning
allows us to construct sound arguments that are invariably true, given the
truth value of the information provided. As such, propositional logic is
fundamental to various disciplines, including science, mathematics, and
philosophy, where it offers a structured approach to reasoning and
argumentation.

To gain insights into the inferential processes humans employ in propositional
reasoning, Van der Henst et al. (2002) conducted a series of experiments that
study the behavior of participants during propositional reasoning. They
formulated straightforward propositional logic problems with neutral content
(the presence or absence of colored marbles in a box, similar to the problem
illustrated in Figure 1) and requested participants to articulate their
thought processes while engaging with these problems. Participants were
permitted the use of paper and pencil for their workings. Both their verbal
explanations and written responses were meticulously recorded, transcribed and
analyzed thereafter. Van der Henst et al. (2002) discovered five strategies
reasoners commonly utilize to navigate the problems, offering insights into
their inferential mechanisms employed in propositional reasoning. In the
following, we give a short description of each strategy (illustrated in Figure
2). For more details and additional examples, we refer to the original study
by Van der Henst et al. (2002).

Incremental Diagram:
Blue iff white: White $\downarrow$ Blue White xor red: White Red $\downarrow$
Blue Red iff pink: White Red $\leftarrow$ Pink $\downarrow$ Blue Supposition
Following:
Blue iff white. White xor red. Red iff pink. Assuming we have no blue. So
there is no white. There is red. There is pink. Not blue and pink. Chain
Construction:
Blue iff white. White xor red. Red iff pink. If blue then white. If white then
no red. If no red then no pink. Therefore, if blue then no pink. Compound
Strategy:
Blue iff white. White xor red. Red iff pink. If red then not blue. If red or
pink, then not blue. Concatenation Strategy:
Blue iff white. White xor red. Red iff pink. Blue iff (white xor (red iff
pink)). Symbolic Strategy:
Blue iff white. White xor red. Red iff pink. First, let’s write down the
statements in a logical form: 1. $B\leftrightarrow W$ 2. $W\oplus R$ 3.
$R\leftrightarrow P$ Now, let’s derive the conclusion using these statements
[…]. Figure 2: An example for each of the five inferential strategies
identified by Van der Henst et al. (2002) (to the left of the dashed vertical
line) that human reasoners employ when solving tasks of propositional logic.
Each strategy is illustrated by a single example adopted from the transcribed
recordings published by the original study. In addition, we provide an example
of the symbolic strategy occasionally encountered in LLMs (to the right of the
dashed line). "Iff" denotes a biconditional, while "xor" indicates an
exclusive disjunction.

Incremental Diagram. This strategy involves the creation of a comprehensive
diagram that keeps track of all potential outcomes compatible with the
statements of the problem. During the reasoning process, individuals
progressively increment their diagrams to incorporate new information derived
(see left box in Figure 2). The result is a single diagram that records a
variety of possibilities compatible with the premises, often including even
those that might be irrelevant to the task.111In contrast to Van der Henst et
al. (2002), we in fact observe no single occurrence of the incremental diagram
strategy in LLMs, despite the authors finding this strategy being employed
most frequently by humans. We belief that this discrepancy stems from the use
of pen and paper in human assessments, implicitly encouraging diagrammatic
reasoning.

Supposition Following. Reasoners employing this strategy start with a
supposition, e.g. by assuming a marble of a certain color. They then trace the
consequences of that supposition following from the statements at hand, as
illustrated in the second box from the left of Figure 2. The result is a
sequence of literals (in this case, marbles of a certain color) without
logical connectives. The success of supposition following strongly depends on
the supposition made by the reasoner. Some suppositions might lead to
inferences that are relevant to the problem, while others might lead to
irrelevant conclusions.

Chain Construction. When employing this strategy, reasoners construct a chain
of conditional statements derived from the premises at hand or from
intermediate deductions. An example of chain construction is displayed in the
central box of Figure 2. Premises are converted into a chain of conditional
statements that are linked by their entities. A distinctive feature of this
chain is the interconnection between conditionals, where the consequent of one
conditional is the antecedent of the following.

Compound Strategy. Reasoners following the compound strategy combine two or
more statements to derive a new compound conclusion. This process yields a
series of novel conclusions, each building upon the preceding ones. An example
illustrating this strategy is given in the upper second box from the right of
Figure 2. Based on the first two premises, the compound conclusion: “If red
then not blue.” is inferred, and then used to draw another compound conclusion
(“If red or pink then not blue.”) together with the last premise of the
problem statement.

Concatenation Strategy. This approach entails the concatenation of two or more
statements into a single conclusion encompassing the logical implications of
each combined proposition. This strategy is subtle and has only been
infrequently observed by Van der Henst et al. (2002). An example of the
strategy is illustrated in the lower second box from the right of Figure 2.

Symbolic Strategy. We could identify an additional strategy occasionally
employed by LLMs, which has not been observed by Van der Henst et al. (2002)
in human reasoners. This strategy, which we denote as symbolic strategy, is
characterized by models employing formal logical calculus to solve the tasks
at hand. When following this strategy, models either translate logical
statements that are expressed in natural language (e.g. “If there is a white
marble then there is not a red marble.”) into formal logic ($W\rightarrow\neg
R$), and then operate on those expressions, or create a truth table from which
they aim to infer the validity of the conclusion. An illustration of this
strategy is provided in the right box of Figure 2.

##  3 Experimental Setup

Problem 1:
Statements: 1. Either there is a white marble in the box or else there is a
black marble in the box, but not both. 2. Either there is a black marble in
the box or else there is a pink marble in the box, but not both. 3. There is a
pink marble in the box if and only if there is a gray marble in the box.
Conclusion: If there is a white marble in the box then there is a gray marble
in the box. Response: Sure, I’d be happy to help! Here’s my reasoning process
for this problem: 1\. First, I’ll start by analyzing the given statements:
From Statement 1, we know that the box contains either a white marble or a
black marble, but not both. From Statement 2, we know that the box contains
either a black marble or a pink marble, but not both. From Statement 3, we
know that if there is a pink marble in the box, then there must be a gray
marble in the box. 2\. Now, let’s consider the conclusion: If there is a white
marble in the box then there is a gray marble in the box. Can we logically
deduce this conclusion from the given statements? Well, if there is a white
marble in the box, then by Statement 1, there cannot be a black marble in the
box. If there is no black marble in the box, then by Statement 2, there must
be a pink marble in the box. But if there is a pink marble in the box, then by
Statement 3, there must be a gray marble in the box. So, yes, we can logically
deduce the conclusion that if there is a white marble in the box, then there
is a gray marble in the box. Conclusion: True Annotator Comments: Placeholder
Placeholder Reformulation of statement 1. Placeholder Reformulation of
statement 2. Placeholder Reformulation of statement 3. Placeholder Repetition
of conclusion. Placeholder Chain construction (sound). W $\rightarrow\neg$ B
Placeholder $\neg$ B $\rightarrow$ P Placeholder P $\rightarrow$ G Placeholder
Conclusion: W $\rightarrow$ G Placeholder Final answer (correct). Figure 3:
The response (lower left box) of LLaMA-2-70B to problem 1 (top box) of the
problem set, demonstrating  yellow chain construction. The model correctly
constructs a chain of conditionals based on the premises (highlighted in
yellow within the model’s response), leading from the antecedent of the final
conclusion to its consequent. Comments made by the annotators are presented in
the adjacent right panel.

Task Overview. Our task setup aligns with the experiment conducted by Van der
Henst et al. (2002) to allow for a fair comparison between the inferential
strategies found in humans and those identified in LLMs.222More specifically,
experiment one of Van der Henst et al. (2002). In particular, we evaluate each
model on the 12 problems of propositional logic suggested by Van der Henst et
al. (2002) (an overview of each problem can be found in Figure 5 in the
appendix). In each problem, models are presented with a set of statements (or
premises) and must determine whether a given conclusion logically follows (for
an example, see Figure 1). Eight out of 12 problems involve three premises and
a conclusion, while the remaining four problems consist of four premises
leading to a conclusion. All premises, as well as the conclusions resemble
either biconditionals, exclusive disjunctions or conditionals. Two problems (4
and 6) include a redundant first premise. All premises are stated such that
two subsequent statements contain one proposition in common, except of two
problems (11 and 12), which are arranged in a non-sequential manner. In half
of the problems, the conclusions logically follow from the premises, whereas
in the other half, they do not. To avoid the influence of external knowledge
and ensure content neutrality, Van der Henst et al. (2002) framed the problems
around the presence of colored marbles in a box, with colors assigned randomly
to each entity within a problem.

Language Models. We aim to investigate various factors that might impact the
inferential strategies displayed by LLMs. These factors include the type of
model, its size, and the emphasis on alignment during training Tunstall et al.
(2023). Therefore, we assess a total of five models, consisting of three
prominent open-access model types: Llama 2 Touvron et al. (2023) with model
sizes of 7B, 13B, and 70B, the recently released Mistral-7B model Jiang et al.
(2023), and Zephyr-7B Tunstall et al. (2023), an extension of Mistral-7B with
a focus on intent alignment through fine-tuning with AI Feedback (AIF). For
our evaluations, we utilize the publicly accessible model weights from the
HuggingFace platform, specifically Llama-2-chat-
hf333https://huggingface.co/meta-llama(7B, 13B, and 70B),
Mistral-7B-Instruct-v0.2,444https://huggingface.co/mistralai/Mistral-7B-Instruct
and zephyr-7b-beta.555https://huggingface.co/HuggingFaceH4/zephyr-7b-beta We
consciously opt not to include proprietary models accessible via paid APIs,
despite their reported superior performance in reasoning tasks Team et al.
(2023). This methodological choice reflects our commitment to promoting
transparent and reproducible scientific research. Note that in this work, we
refer to the above models when using abbreviations such as LLaMA-2,
Mistral-7B-Instruct or Zephyr-7B-$\beta$.

Evaluation Setup. We prompt each model with a system message providing context
about the task they are about to solve and the format in which they should
answer (for the full prompt, see Figure 5 in the appendix). Analogous to Van
der Henst et al. (2002), we inform the model of its participation in an
experiment designed to explore reasoning processes, and instruct it to “think
aloud” as it tackles the problem. In addition to the system message, we
provide a user prompt that contains the problem description. In cases where
the model does not accept system messages (such as Mistral-7B-Instruct-v0.2),
we prepend the content of the system message to the user prompt. To prevent
biasing the model towards a certain strategy, we refrain from providing few-
shot examples, as done also by Leidinger et al. (2023). Instead, we elicit
reasoning through zero-shot chain-of-thought prompting (“Let’s think step by
step”) Kojima et al. (2023). Answers for each model are generated with nucleus
sampling using Llama-2-chat-hf’s default values ($\text{top-}p=0.9$,
temperature $T=0.6$), as we found this configuration to work well for all
models. To account for the statistical nature of language models, we ask each
model to solve the set of propositional problems across 5 random seeds,
resulting in a total of 60 responses per model.

We record all answers and manually evaluate them (a total of 300 responses)
for strategies employed in their reasoning (see Figure 3 for an example). For
each model response, we qualitatively evaluate for strategy and soundness.
That is, we manually label the inferential strategies identified, and the
logical validity of the model’s reasoning. In addition, we record whether the
final answer is correct. In cases of faulty reasoning, we categorize the type
of error. This comprehensive manual evaluation of model responses is
independently conducted by two hired students with expertise in manual data
annotation. To gauge the quality of the annotations, we report an overall
Cohen’s Kappa value of $\kappa=0.98$. For details on the inter-annotator
agreement of each label, we refer to Table 2 in the appendix. Further
annotated examples can be found in Appendix C. Following the recommendations
put forward by Leidinger et al. (2023), we make all input prompts, model
responses, code and manual annotations publicly available upon acceptance of
this paper.

Model |  Supposition Following |  Chain Construction |  Compound Conclusion |  Concatenation Strategy |  Symbolic Strategy |  Correct Answer |  Sound Reasoning
---|---|---|---|---|---|---|---
Zephyr-7B-$\beta$ |  $\mathbf{60.0\%}\,(\textit{55.1})$ |  $18.3\%\,(\textit{17.3})$ |  $10.0\%\,(\textit{8.9})$ |  $1.7\%\,(\textit{1.4})$ |  $20.0\%\,(\textit{17.3})$ |  $45.0\pm 15.5$ |  $25.0\pm 10.5$
Mistral-7B-Instruct |  $\mathbf{35.0}\%\,(\textit{38.4})$ |  $10.0\%\,(\textit{10.7})$ |  $\mathbf{35.0}\%\,(\textit{38.4})$ |  $3.3\%\,(\textit{3.4})$ |  $8.3\%\,(\textit{9.1})$ |  $55.0\pm 10.0$ |  $25.0\pm 7.5$
LLaMA-2-7B |  $\mathbf{20.0\%}\,(\textit{50.2})$ |  $\mathbf{20.0\%}\,(\textit{30.2})$ |  $6.7\%\,(\textit{10.9})$ |  $3.3\%\,(\textit{5.4})$ |  $1.7\%\,(\textit{3.3})$ |  $46.7\pm 6.7$ |  $0.0\pm 0.0$
LLaMA-2-13B |  $28.3\%\,(\textit{35.7})$ |  $\mathbf{36.7\%}\,(\textit{46.9})$ |  $6.7\%\,(\textit{8.7})$ |  $6.7\%\,(\textit{8.7})$ |  $0.0\%\,(\textit{0.0})$ |  $40.0\pm 8.2$ |  $15.0\pm 6.2$
LLaMA-2-70B |  $45.0\%\,(\textit{42.3})$ |  $\mathbf{50.0\%}\,(\textit{46.8})$ |  $3.3\%\,(\textit{2.9})$ |  $1.7\%\,(\textit{1.8})$ |  $6.7\%\,(\textit{6.2})$ |  $56.7\pm 6.2$ |  $31.7\pm 9.7$
Human Reasoner† |  $-\ (21.0)$ |  $-\ (25.0)$ |  $-\ (19.0)$ |  $-\ (0.0)$ |  $-\ (0.0)$ |  $100\pm 0.0$ |  $-$
Table 1: Relative occurrences of inferential strategies employed by the
different language models when solving the propositional logic problems. All
values reflect average percentages, calculated over five random seeds, with
standard deviations reported in Table 3 in the appendix. Strategies that a
model favors are highlighted in bold. Values in parentheses denote fractions
with respect to the total number of strategies employed by that model. Values
of correct answers and instances of sound reasoning are reported with their
standard deviations. †The comparison with human reasoners is based on findings
by Van der Henst et al. (2002), where dashes denote missing values.

##  4 Results and Analysis

In this section, we present the results of our evaluation. We begin with a
quantitative analysis of the inferential strategies employed by LLMs, as well
as the logical validity of their reasoning. This is followed by a qualitative
analysis providing a more in-depth examination of the models’ reasoning
behavior.

###  4.1 Quantitative Analysis

Table 1 provides an overview of the frequencies with which large language
models employ inferential strategies when navigating the problems of
propositional logic described in Section 3. Our evaluation reveals that all
models display strategies akin to those observed by Van der Henst et al.
(2002). In particular, we find that, similar to humans, models commonly employ
supposition following, chain construction and the compound strategy. In
addition, we observe that models occasionally utilize the symbolic strategy,
employing techniques from logical calculus to solve the tasks (see Section 2).
Note that, similar to humans, models might switch from one strategy to another
during a single problem, demonstrating multiple strategies within their
responses (see Figure 19 in the appendix for an example). Surprisingly, we
observe that distinct model families favor different inferential strategies.
For instance, Zephyr-7B-$\beta$ predominantly employs supposition following,
while Mistral-7B-Instruct is inclined towards drawing compound conclusions. In
comparison, models from the Llama 2 series tend to rely on supposition
following and chain construction, with negligible use of the compound
strategy. Our analysis further reveals a discrepancy between the correctness
of the models’ final answers and the logical soundness of their reasoning.
While all models achieve an answer accuracy that approximately coincides with
chance in our experimental setup, an analysis of their reasoning validity
reveals a different picture: LLaMA-2-70B outperforms the other models by
reasoning correctly in about 31.7% of cases, while Zephyr-7B-$\beta$ and
Mistral-7B-Instruct produce sound reasoning in 25% of problems. We note that
all models perform rather poorly on the propositional tasks, with LLaMA-2-7B
failing entirely to construct sound arguments.

Human Reasoning. Van der Henst et al. (2002) compute the percentages with
which human reasoners employ inferential strategies with respect to the total
number of strategies observed in their experiment, and not with respect to the
total number of problems considered. Thus, their reported values mainly
reflect which strategies are favored more or less by the reasoner, but do not
provide information about how frequently a strategy has been observed in the
overall context. To make our findings comparable to the results of Van der
Henst et al. (2002), we convert our results respectively (see values in
parentheses in Table 1). We note that almost all models seem to favor
supposition following to a higher degree than human reasoners, who employ this
strategy in only about 21% of overall use. In contrast, humans seem to draw
compound conclusions more readily, except for Mistral-7B-Instruct which shows
a tendency more than twice as high. Overall, both LLMs and humans hardly
employ the concatenation strategy. Interestingly, Van der Henst et al. (2002)
report that all reasoners successfully solve the problems of propositional
logic, though not always for the correct reasons. While the study does not
provide data on the number of problems where humans reasoned correctly, the
high success rate of human participants contrasts sharply with the performance
of the models.

Figure 4: Instances where models generate sound reasoning traces that
logically follow from the problem statement. For each inferential strategy,
the ratio of sound reasoning traces (represented by the filled portion) to the
overall application of that strategy (denoted by the unfilled bar) is
depicted. Ratios are expressed as percentages above the corresponding filled
section. Note that LLaMA-2-7B is not displayed as it does not exhibit sound
reasoning.

Effect of Model Size. Our evaluation of the Llama 2 series across three
different model sizes—7B, 13B, and 70B parameters—demonstrates that model
scale significantly influences the frequency with which strategies are
employed by the model. In particular, we observe that with increasing model
size, Llama 2 employs strategies more readily. Furthermore, larger models
within the Llama 2 framework are observed to generate a greater number of
sound reasoning traces. We interpret this trend as a result of the model’s
improving proficiency in strategic reasoning as its scale increases.

Effect of Alignment. The alignment of a model’s response with human
preferences is crucial to emulate human-like behavior Ouyang et al. (2022).
Zephyr-7B-$\beta$ is an iteration of Mistral-7B that is fine-tuned with AI
Feedback (AIF) for improved intent alignment Tunstall et al. (2023). In
comparison to the observations made by Van der Henst et al. (2002), where
besides the incremental diagram strategy (34%), chain construction was
employed most frequently by humans, Zephyr-7B-$\beta$ demonstrates a marked
preference for supposition following and significantly less engagement in
chain construction. Moreover, it is noteworthy that among the evaluated
models, Zephyr-7B-$\beta$ most frequently adopts the symbolic strategy, an
approach not reported in human reasoners.

Sound Reasoning. As previously highlighted, the accuracy of a model’s final
answer does not necessarily serve as a reliable indicator of its reasoning
capability. In particular, we observe that models often arrive at correct
answers, but through flawed reasoning processes (refer to Figure 10 in the
appendix for an illustration). Interestingly, we also find instances where
models provide incorrect final answers despite reasoning correctly (for an
example, see Figure 16 in the appendix). Our analysis reveals only a moderate
positive correlation between the accuracy of the models’ final answers and the
logical soundness of their reasoning, with a Pearson correlation coefficient
$r(298)=0.45$ and a statistically significant p-value of less than
$0.0001(p=1.6\times 10^{-16})$. This observation underscores the need for more
nuanced evaluation procedures, particularly in multiple-choice settings, where
models might select the correct answer by chance rather than through rigorous
reasoning.

In Figure 4, we explore the relationship between the inferential strategies
employed by the models and the validity of their reasoning. For each strategy,
we quantify the proportion of instances where the models’ reasoning is sound,
compared to the overall application of that strategy. Our analysis reveals
variability in the effectiveness with which different models apply various
strategies. For example, Mistral-7B-Instruct tends to reason correctly when
using approaches such as the chain, compound, or symbolic strategy, yet
frequently encounters reasoning errors with supposition following. On the
other hand, LLaMA-2-70B exhibits proficiency in supposition following, but
struggles with the symbolic strategy.

###  4.2 Qualitative Analysis

We supplement our quantitative analysis by a more detailed qualitative
analysis of the models’ reasoning behavior. Figure 3 depicts LLaMA-2-70B’s
response to problem 1 of the task set. The response illustrates a frequently
observed behavior. Initially, models tend to analyze the problem’s
propositions, often by paraphrasing each premise and the conclusion to be
evaluated. They then embark on a reasoning process, typically utilizing one of
the previously mentioned strategies. In the example, LLaMA-2-70B employs chain
construction, creating a logical chain of conditionals that leads from the
antecedent of the final conclusion to its consequent, thereby correctly
affirming the conclusion’s logical validity. A notable pitfall in such
reasoning chains is the models’ occasional misinterpretation of logical
negations, leading to erroneous chains like: A $\rightarrow\neg$ B; B
$\rightarrow$ C; therefore A $\rightarrow$ C, where the negation in the first
conditional is overlooked (for an illustrative case, refer to Figure 11 in the
appendix). This behavior can be found across all models and aligns with
previous work reporting difficulties of LLMs in understanding logical
negations Truong et al. (2023).

When employing supposition following, models often fail to consider all
implications of their assumptions. Instead, they tend to focus only on
immediate inferences, while overlooking further consequences crucial for
assessing the conclusion’s validity. This leads to models prematurely
concluding the inability to definitively determine the logical validity of the
final conclusion: “Based on our analysis, we cannot definitively say that the
conclusion logically follows from the given statements” (see Figure 7 in the
appendix for a respective example). Another source of error in supposition
following involves models making improper suppositions, such as conjecturing
about a marble not mentioned in the final conclusion, and deriving disjointed
intermediate conclusions that do not aid in solving the problem. An example of
this behavior can be found in Figure 8 in the appendix.

Finally, we identify two behaviors in models that mirror logical errors seen
in human reasoners Van der Henst et al. (2002). First, models frequently
attempt to prove an exclusive disjunction (A $\oplus$ B) by only considering a
single conditional case (A $\rightarrow\neg$ B), and second, they sometimes
engage in the logical fallacy known as denial of the antecedent: A
$\rightarrow$ B; therefore $\neg$ A $\rightarrow\neg$ B (for illustrative
examples, see Figures 12 and 13 in the appendix, respectively).

##  5 Related Work

Human Strategies in Deductive Reasoning. A considerable amount of research,
especially within psychology and cognitive science, has explored how humans
approach deductive reasoning tasks Schaeken et al. (2000). A prominent focus
of these studies is on heuristics, which are cognitive shortcuts that
individuals employ to arrive at satisfactory conclusions in deductive
reasoning despite potential flaws in the underlying logic Kahneman et al.
(1982); Evans (1989); Gigerenzer and Todd (1999); Davis (2018). For instance,
Woodworth and Sells (1935) demonstrate that individuals tend to accept
conclusions in syllogistic reasoning as valid when they share logical
quantifiers with the premises, regardless of their actual logical validity.
Nonetheless, such reliance on heuristics can result in errors and falls short
of the level of strategic reasoning necessary to develop sound and coherent
arguments Kahneman (2012). Further research has delved into more sophisticated
strategies utilized by individuals in deductive reasoning. Based on the mental
model theory Johnson-Laird (1986), Bucciarelli and Johnson-Laird (1999)
identify a variety of strategies commonly employed by individuals in
syllogistic reasoning. Byrne and Handley (1997) study strategies of
individuals in knight-and-knave puzzles, where the truthfulness of statements
made by hypothetical characters have to derived. Their experiments reveal that
humans engage in both forward and backward inferences to navigate through
potential solutions.

Human Reasoning Behavior in LLMs. Recent research has started to explore the
extent to which LLMs mirror human-like reasoning behaviors. Dasgupta et al.
(2023) demonstrate content-effects akin to those observed in human reasoning,
where the deductive process is influenced by the content of the problem
statement. Eisape et al. (2023) find that LLMs, similar to humans, exhibit
biases such as ordering effects in syllogistic reasoning tasks. Several other
studies have delved into the prevalence of biases and heuristics within LLMs
Binz and Schulz (2023); Talboy and Fuller (2023); Shaki et al. (2023); Suri et
al. (2023). However, to the best of our knowledge, we are the first who study
the presence of more sophisticated human strategies in the context of LLM-
based deductive reasoning.

##  6 Conclusion

In this paper, we examine the inferential strategies employed by LLMs in
solving problems of propositional logic. Through a comprehensive evaluation of
their reasoning behavior, we demonstrate that LLMs adopt strategies akin to
those observed in human reasoners. Our quantitative analysis reveals that the
frequency with which a model adopts a specific strategy strongly depends on
its type, size, and fine-tuning procedure. Moreover, our analysis suggests
that the accuracy of a model’s final conclusions does not adequately capture
its reasoning capabilities, underscoring the importance of a more
sophisticated evaluation framework that includes the model’s reasoning paths.
We also provide a qualitative analysis of typical reasoning behaviors among
models, pinpointing prevalent errors such as difficulties in understanding
negations or recognizing all implications of a supposition.

##  7 Limitations

While our work contributes to the understanding of reasoning processes in
large language models by demonstrating that these models, similar to humans,
employ inferential strategies in propositional logic, it encompasses several
limitations that could be addressed in future work.

Task setup. Our study is constrained by a limited set of problems, designed
within a fixed framework that revolves around hypothesis validation based on
3-4 statements of propositional logic. We employ a constant and neutral
content, disregarding potential content-effects on the models’ reasoning
behavior, as shown by Dasgupta et al. (2023). Similarly, we have not yet
examined factors such as the complexity of the problems, the differences
between hypothesis validation and generation, and the impact of logical
connectives utilized in the premises. We belief that these factors are worth
investigating and leave a detailed examination to future work.

Evaluation Framework. The extent of our manual evaluation is limited by both
the number of samples reviewed and the quantity of annotators involved.
Despite our efforts to maximize the use of available resources, these
constraints may affect the scalability and reliability of our results.
Additionally, we instruct all models through zero-shot chain-of-thought
prompting (“Let’s think step by step”) Kojima et al. (2023). Exploring
alternative reasoning frameworks, such as Tree of Thoughts Yao et al. (2023)
or Graph of Thoughts Besta et al. (2023), could provide valuable insights into
their influence on model behavior and the inferential strategies adopted.
Based on our annotated data, we endeavored to develop a classifier capable of
automatically identifying the inferential strategies employed in the models’
output, which was intended to complement our manual evaluation setup. However,
due to the complexity of the task and limited size of our annotated dataset,
our classifier struggled with generalization to new, unseen responses. In
future endeavors, we aim to allocate more resources towards expanding our
manual annotation efforts and explore this direction further. Finally, our
study predominantly offers a behavioral analysis and does not delve into the
mechanistic aspects that might explain the diversity in strategy usage by the
models. Investigating how attention patterns of models with respect to
specific logical connectives might influence their choice of reasoning
strategy presents a compelling direction for future research.

## Acknowledgements

We would like to thank the MaiNLP lab members for their insightful feedback,
specifically Diego Frassinelli, Rob van der Goot, Siyao Peng, Robert Litschko,
Jian Lan, Daniela Teodorescu, Xinpeng Wang, Verena Blaschke, Elena Senger,
Elisa Bassignana, and Max Müller-Eberstein. Furthermore, we would like to
express our gratitude to Huangyan Shan for her valuable work and support in
data annotation. Our appreciation extends to the anonymous reviewers for their
comments and suggestions. Lastly, we acknowledge the support for BP through
the ERC Consolidator Grant DIALECT 101043235.

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Rabinovitch, Piotr Stanczyk, Ye Zhang, David Steiner, Subhajit Naskar, Michael Azzam, Matthew Johnson, Adam Paszke, Chung-Cheng Chiu, Jaume Sanchez Elias, Afroz Mohiuddin, Faizan Muhammad, Jin Miao, Andrew Lee, Nino Vieillard, Sahitya Potluri, Jane Park, Elnaz Davoodi, Jiageng Zhang, Jeff Stanway, Drew Garmon, Abhijit Karmarkar, Zhe Dong, Jong Lee, Aviral Kumar, Luowei Zhou, Jonathan Evens, William Isaac, Zhe Chen, Johnson Jia, Anselm Levskaya, Zhenkai Zhu, Chris Gorgolewski, Peter Grabowski, Yu Mao, Alberto Magni, Kaisheng Yao, Javier Snaider, Norman Casagrande, Paul Suganthan, Evan Palmer, Geoffrey Irving, Edward Loper, Manaal Faruqui, Isha Arkatkar, Nanxin Chen, Izhak Shafran, Michael Fink, Alfonso Castaño, Irene Giannoumis, Wooyeol Kim, Mikołaj Rybiński, Ashwin Sreevatsa, Jennifer Prendki, David Soergel, Adrian Goedeckemeyer, Willi Gierke, Mohsen Jafari, Meenu Gaba, Jeremy Wiesner, Diana Gage Wright, Yawen Wei, Harsha Vashisht, Yana Kulizhskaya, Jay Hoover, Maigo Le, Lu Li, Chimezie Iwuanyanwu, Lu Liu, Kevin Ramirez, Andrey Khorlin, Albert Cui, Tian LIN, Marin Georgiev, Marcus Wu, Ricardo Aguilar, Keith Pallo, Abhishek Chakladar, Alena Repina, Xihui Wu, Tom van der Weide, Priya Ponnapalli, Caroline Kaplan, Jiri Simsa, Shuangfeng Li, Olivier Dousse, Fan Yang, Jeff Piper, Nathan Ie, Minnie Lui, Rama Pasumarthi, Nathan Lintz, Anitha Vijayakumar, Lam Nguyen Thiet, Daniel Andor, Pedro Valenzuela, Cosmin Paduraru, Daiyi Peng, Katherine Lee, Shuyuan Zhang, Somer Greene, Duc Dung Nguyen, Paula Kurylowicz, Sarmishta Velury, Sebastian Krause, Cassidy Hardin, Lucas Dixon, Lili Janzer, Kiam Choo, Ziqiang Feng, Biao Zhang, Achintya Singhal, Tejasi Latkar, Mingyang Zhang, Quoc Le, Elena Allica Abellan, Dayou Du, Dan McKinnon, Natasha Antropova, Tolga Bolukbasi, Orgad Keller, David Reid, Daniel Finchelstein, Maria Abi Raad, Remi Crocker, Peter Hawkins, Robert Dadashi, Colin Gaffney, Sid Lall, Ken Franko, Egor Filonov, Anna Bulanova, Rémi Leblond, Vikas Yadav, Shirley Chung, Harry Askham, Luis C. 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Mankowitz, Alex Polozov, Nate Kushman, Victoria Krakovna, Sasha Brown, MohammadHossein Bateni, Dennis Duan, Vlad Firoiu, Meghana Thotakuri, Tom Natan, Anhad Mohananey, Matthieu Geist, Sidharth Mudgal, Sertan Girgin, Hui Li, Jiayu Ye, Ofir Roval, Reiko Tojo, Michael Kwong, James Lee-Thorp, Christopher Yew, Quan Yuan, Sumit Bagri, Danila Sinopalnikov, Sabela Ramos, John Mellor, Abhishek Sharma, Aliaksei Severyn, Jonathan Lai, Kathy Wu, Heng-Tze Cheng, David Miller, Nicolas Sonnerat, Denis Vnukov, Rory Greig, Jennifer Beattie, Emily Caveness, Libin Bai, Julian Eisenschlos, Alex Korchemniy, Tomy Tsai, Mimi Jasarevic, Weize Kong, Phuong Dao, Zeyu Zheng, Frederick Liu, Fan Yang, Rui Zhu, Mark Geller, Tian Huey Teh, Jason Sanmiya, Evgeny Gladchenko, Nejc Trdin, Andrei Sozanschi, Daniel Toyama, Evan Rosen, Sasan Tavakkol, Linting Xue, Chen Elkind, Oliver Woodman, John Carpenter, George Papamakarios, Rupert Kemp, Sushant Kafle, Tanya Grunina, Rishika Sinha, Alice Talbert, Abhimanyu Goyal, Diane Wu, Denese Owusu-Afriyie, Cosmo Du, Chloe Thornton, Jordi Pont-Tuset, Pradyumna Narayana, Jing Li, Sabaer Fatehi, John Wieting, Omar Ajmeri, Benigno Uria, Tao Zhu, Yeongil Ko, Laura Knight, Amélie Héliou, Ning Niu, Shane Gu, Chenxi Pang, Dustin Tran, Yeqing Li, Nir Levine, Ariel Stolovich, Norbert Kalb, Rebeca Santamaria-Fernandez, Sonam Goenka, Wenny Yustalim, Robin Strudel, Ali Elqursh, Balaji Lakshminarayanan, Charlie Deck, Shyam Upadhyay, Hyo Lee, Mike Dusenberry, Zonglin Li, Xuezhi Wang, Kyle Levin, Raphael Hoffmann, Dan Holtmann-Rice, Olivier Bachem, Summer Yue, Sho Arora, Eric Malmi, Daniil Mirylenka, Qijun Tan, Christy Koh, Soheil Hassas Yeganeh, Siim Põder, Steven Zheng, Francesco Pongetti, Mukarram Tariq, Yanhua Sun, Lucian Ionita, Mojtaba Seyedhosseini, Pouya Tafti, Ragha Kotikalapudi, Zhiyu Liu, Anmol Gulati, Jasmine Liu, Xinyu Ye, Bart Chrzaszcz, Lily Wang, Nikhil Sethi, Tianrun Li, Ben Brown, Shreya Singh, Wei Fan, Aaron Parisi, Joe Stanton, Chenkai Kuang, Vinod Koverkathu, Christopher A. 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##  Appendix A Additional Experimental Details

In this section, we provide additional details about the experimental setup,
including supplementary information about the problem formulations and prompts
utilized.

|  Supposition Following |  Chain Construction |  Compound Conclusion |  Concatenation Strategy |  Symbolic Strategy |  Correct Answer |  Sound Reasoning
---|---|---|---|---|---|---|---
Zephyr-7B-$\beta$ |  $1.0$ |  $0.94$ |  $1.0$ |  $1.0$ |  $1.0$ |  $1.0$ |  $1.0$
Mistral-7B-Instruct |  $1.0$ |  $0.9$ |  $1.0$ |  $1.0$ |  $1.0$ |  $1.0$ |  $1.0$
LLaMA-2-7B |  $0.89$ |  $0.95$ |  $1.0$ |  $0.79$ |  $1.0$ |  $1.0$ |  $1.0$
LLaMA-2-13B |  $0.88$ |  $1.0$ |  $0.85$ |  $1.0$ |  $1.0$ |  $1.0$ |  $1.0$
LLaMA-2-70B |  $0.97$ |  $1.0$ |  $1.0$ |  $1.0$ |  $1.0$ |  $1.0$ |  $1.0$
Table 2: Cohen’s Kappa values to assess the inter-annotator agreement across
different models and label categories.

###  A.1 Task Prompts

Figure 5 displays the task prompt and problem formulations employed in
assessing the language models described in Section 3. Note that the prompt
template, i.e. special tokens and their arrangements, might vary depending on
the specific language model used. Within the task prompt (provided in the
upper box), the problem statements and conclusion for a given problem are
replaced with the corresponding problem formulations found in the lower gray
boxes. In the final version of the prompt, the phrase “colorA iff colorB” is
expanded to “There is a colorA marble in the box if and only if there is a
colorB marble in the box”. Similarly, “colorA xor colorB” is interpreted as
“There is either a colorA marble or a colorB marble in the box, but not both”,
and “If colorA then colorB” is articulated as “If there is a colorA marble in
the box, then there is a colorB marble in the box”.

###  A.2 Annotator Instructions

Our assessment of model responses involves a comprehensive independent review
by two students who are specialized in the field of natural language
processing and have expertise in manual data annotation. To ensure a high
quality of annotations, we offer comprehensive training to both annotators.
This training includes detailed explanations and extensive examples of the
strategies identified by Van der Henst et al. (2002), complemented by a
session dedicated to clarifying any questions that may emerge. Subsequently,
the annotators are tasked with independently annotating practice examples,
which serves to highlight and address any ambiguities in the annotation
process. Only when both annotators are confident in their understanding of
each strategy do we proceed. We instruct both annotators to independently go
through each model response and mark parts where they identify a certain
strategy to be employed. Each strategy is marked in a unique color code, which
is afterwards converted into labels that signify the use of a particular
strategy. In addition, we instruct both annotators to label whether the
reasoning is sound, and the final conclusion of the model is correct.
Furthermore, we ask them to classify any logical errors identified within the
reasoning process. To maintain a high standard of annotation quality,
annotators are instructed to review the model responses twice.

###  A.3 Inter-Annotator Agreement

To assess the reliability of our manual evaluation process (see Section 3), we
quantify the inter-annotator agreement by calculating Cohen’s Kappa for each
category and model, as illustrated in Table 2. Generally, the results indicate
an almost perfect level of agreement across all categories and models, with
Cohen’s Kappa values ranging from $0.81\leq\kappa\leq 1.0$. An exception is
observed in the case of the concatenation strategy applied by LLaMA-2-7B, for
which we report a substantial agreement level, with a Kappa value of
$\kappa=0.79$, slightly below the threshold for almost perfect agreement.

###  A.4 Model Details

We report further details about the models used in this study in Table 4. In
particular, we provide information about the number of parameters, context
length and fine-tuning procedure for each model.

##  Appendix B Additional Quantitative Results

In this segment, we present supplementary findings from our quantitative
evaluation. Table 3 illustrates the frequencies with which the different
language models employ inferential strategies when navigating the problems of
propositional logic, as outlined in Section 3. Values denote percentages
averaged across five distinct random seeds, accompanied by their standard
deviation. Furthermore, we detail the proportions of correct final conclusions
and sound reasoning. Note that all percentages are calculated relative to the
overall count of tasks within the experimental framework.

[INST] <<SYS>> You participate in an experiment that tries to understand how
people reason. Your task is to solve logical reasoning problems. In
particular, you are given set of statements and your task is to say whether a
conclusion logically follows from the statements. Please answer with ’True’ or
’False’ for each conclusion. In addition, it is important that you think-aloud
as you tackle the problem and report every step in your reasoning process.
<</SYS>> ### Instruction ### Explain whether the conclusion logically follows
from the set of statements below. Please report all your reasoning steps. End
your reasoning with: Conclusion: True/False. Statements: <statements and
conclusion from below> Let’s think step by step. [/INST] Problem 1:
Statements: 1. White xor black. 2. Black xor pink. 3. Pink iff gray.
Conclusion: If white then gray. Problem 2:
Statements: 1. Brown iff orange. 2. Orange xor yellow. 3. Yellow iff green.
Conclusion: If brown then green. Problem 3:
Statements: 1. Green iff purple. 2. If purple then gray. 3. Gray xor yellow.
Conclusion: Green xor yellow. Problem 4:
Statements: 1. Red xor maroon. 2. Maroon xor yellow. 3. Yellow iff orange.
Conclusion: If maroon then orange. Problem 5:
Statements: 1. Purple iff yellow. 2. Yellow iff blue. 3. Blue xor orange.
Conclusion: Purple xor orange. Problem 6:
Statements: 1. Gray iff yellow. 2. Yellow xor olive. 3. Olive iff black.
Conclusion: If yellow then black. Problem 7:
Statements: 1. Blue iff red. 2. Red xor white. 3. White iff pink. Conclusion:
If not blue then pink. Problem 8:
Statements: 1. Olive xor brown. 2. Brown iff gray. 3. Gray xor maroon.
Conclusion: If not olive then maroon. Problem 9:
Statements: 1. Purple iff blue. 2. Blue iff olive. 3. Olive xor red. 4. Red
xor green. Conclusion: If purple then green. Problem 10:
Statements: 1. Brown iff yellow. 2. Yellow xor green. 3. Green iff purple. 4.
Purple iff olive. Conclusion: If brown then olive. Problem 11:
Statements: 1. Red iff maroon. 2. Green xor olive. 3. Maroon iff green. 4.
Olive xor brown. Conclusion: If red then brown. Problem 12:
Statements: 1. Blue iff brown. 2. White iff green. 3. Brown xor white. 4.
Green iff purple. Conclusion: If blue then purple. Figure 5: The task prompt
(upper yellow box) as well as statements and conclusion for each propositional
logic problem (lower gray boxes). In the task prompt, the placeholder
“<statements and conclusion from below>” is replaced with the actual
statements and conclusion relevant to each problem. To enhance readability, we
employ abbreviations within the problem statements. In the actual prompt,
“colorA iff colorB” is replaced by “There is a colorA marble in the box if and
only if there is a colorB marble in the box”. Similarly, “colorA xor colorB”
denotes “Either there is a colorA marble in the box or else there is a colorB
marble in the box, but not both”. Lastly, “If colorA then colorB” stands for
“If there is a colorA marble in the box then there is a colorB marble in the
box”.

##  Appendix C Annotated Model Responses

Within this section, we showcase examples of model responses that exemplify
each inferential strategy identified in our study, as depicted in figures
6-19. Each figure is organized with the problem statement at the top, the
model’s response on the lower left, and the annotators’ comments to the lower
right. For an extensive array of model responses and annotations, we invite
readers to explore our data repository, which we publish upon acceptance of
this paper.

###  C.1 Supposition Following

Figures from 6 to 8 demonstrate the application of supposition following by
various models. For instance, Figure 6 presents LLaMA-2-70B’s approach to
problem 7, where the model supposes the absence of a blue marble in the box
and logically infers the implications of this assumption to reach the valid
conclusion. On the other hand, Figure 7 depicts Mistral-7B-Instruct’s response
to the same problem, where the model considers various combinations of marble
in the box, drawing immediate conclusions that follow from the premises at
hand. However, it does not explore deeper ramifications of these suppositions,
thereby failing to deduce the validity of the conclusion. This showcases a
common behavior we observe in models that employ supposition following
unsuccessfully. In Figure 8 the model approaches problem 9 by assuming the
presence of an olive marble in the box, yet inferring disjointed intermediate
conclusions that do not aid in solving the problem, thus failing to prove the
logical validity of the problem.

Model |  Supposition Following |  Chain Construction |  Compound Conclusion |  Concatenation Strategy |  Symbolic Strategy |  Correct Answer |  Sound Reasoning
---|---|---|---|---|---|---|---
Zephyr-7B-$\beta$ |  $60.0\pm 12.2$ |  $18.3\pm 6.2\phantom{0}$ |  $10.0\pm 6.2\phantom{0}$ |  $\phantom{0}1.7\pm 3.3\phantom{0}$ |  $20.0\pm 11.3$ |  $45.0\pm 15.5$ |  $25.0\pm 10.5$
Mistral-7B-Instruct |  $35.0\pm 6.2\phantom{0}$ |  $10.0\pm 3.3\phantom{0}$ |  $35.0\pm 9.7\phantom{0}$ |  $\phantom{0}3.3\pm 4.1\phantom{0}$ |  $\phantom{0}8.3\pm 7.5\phantom{0}$ |  $55.0\pm 10.0$ |  $25.0\pm 7.5\phantom{0}$
LLaMA-2-7B |  $20.0\pm 6.7\phantom{0}$ |  $20.0\pm 15.5$ |  $\phantom{0}6.7\pm 3.3\phantom{0}$ |  $\phantom{0}3.3\pm 4.1\phantom{0}$ |  $\phantom{0}1.7\pm 3.3\phantom{0}$ |  $46.7\pm 6.7\phantom{0}$ |  $\phantom{0}0.0\pm 0.0\phantom{0}$
LLaMA-2-13B |  $28.3\pm 10.0$ |  $36.7\pm 12.5$ |  $\phantom{0}6.7\pm 3.3\phantom{0}$ |  $\phantom{0}6.7\pm 6.2\phantom{0}$ |  $\phantom{0}0.0\pm 0.0\phantom{0}$ |  $40.0\pm 8.2\phantom{0}$ |  $15.0\pm 6.2\phantom{0}$
LLaMA-2-70B |  $45.0\pm 8.5\phantom{0}$ |  $50.0\pm 7.5\phantom{0}$ |  $\phantom{0}3.3\pm 4.1\phantom{0}$ |  $\phantom{0}1.7\pm 3.3\phantom{0}$ |  $\phantom{0}6.7\pm 3.3\phantom{0}$ |  $56.7\pm 6.2\phantom{0}$ |  $31.7\pm 9.7\phantom{0}$
Table 3: Relative occurrences of inferential strategies employed by the
different language models when solving the propositional problems. All values
denote percentages averaged across 5 different random seeds with standard
deviation. In addition, the percentages of correct final answers and sound
reasoning are reported.

###  C.2 Chain Construction

Figures 9 to 13 illustrate instances where models employ chain construction to
navigate the problems of propositional logic. In Figure 9, LLaMA-2-70B adeptly
forms a chain of conditional statements that bridge the antecedent of the
conclusion to its consequent, effectively validating the conclusion’s logical
soundness. Conversely, Figure 10 depicts a logical chain in which LLaMA-2-70B
erroneously concludes the nonexistence of a white marble based on the absence
of a red marble, despite an exclusive disjunction linking the two. Despite
this logical misstep, the model’s final conclusion remains accurate,
highlighting the discrepancy between the model’s final answer and the
soundness of its reasoning. In Figure 11, LLaMA-2-13B constructs a chain
correctly linking the antecedent of the final conclusion to its consequent.
Nonetheless, it overlooks the negation present in one of the conditionals,
resulting in a compromised reasoning chain. Figure 12 presents a scenario
where the model incorrectly attempts to validate an exclusive disjunction
solely through a singular conditional sequence, a reasoning error not uncommon
among human reasoners Van der Henst et al. (2002). Lastly, Figure 13
highlights LLaMA-2-70B’s engagement in the inverse fallacy, inferring $\neg$ W
$\rightarrow\neg$ G from the conditional W $\rightarrow$ G, mirroring a
logical misjudgment frequently observed in human reasoning processes.

###  C.3 Compound Strategy

The compound strategy is illustrated in Figures 14 to 16. Figure 14 presents
Mistral-7B-Instruct’s approach to problem 9, where it infers a biconditional
relationship between the purple and olive marble from the first two premises.
On the other hand, Figure 15 shows LLaMA-2-70B’s response to the same problem,
formulating a sequence of compound inferences beyond the initial biconditional
deduction, culminating in the correct final answer. Additionally, Figure 16
illustrates Mistral-7B-Instruct’s approach to problem 8, in which the model
initially generates compound conclusions derived from the problem statements,
followed by supposition following to explore the implications that the absence
of an olive marble might have. However, despite the model’s sound reasoning,
its final answer is incorrect.

###  C.4 Concatenation Strategy

Figure 17 demonstrates the concatenation strategy, where Mistral-7B-Instruct
concatenates two intermediate deductions to form a single statement. It then
uses the concatenated statement to infer the invalidity of the conclusion.

###  C.5 Symbolic Strategy

The symbolic strategy is exemplified in Figure 18, where LLaMA-2-70B employs a
truth table to assess the conclusion’s validity, albeit with errors leading to
an incorrect result. Conversely, Figure 19 shows Mistral-7B-Instruct’s
application of chain construction followed by the symbolic strategy. The model
makes false inferences while employing chain construction, and further errs in
its validation through logical calculus.

Model |  Base Model |  Parameters |  Context Length |  Tokens |  Fine-tuning
---|---|---|---|---|---
Zephyr-7B-$\beta$ |  Mistral |  7B |  8192 tokens |  - |  dSFT, AIF
Mistral-7B-Instruct |  Mistral |  7B |  8192 tokens |  - |  SFT
LLaMA-2-7B-Chat |  LLaMA-2 |  7B |  4K tokens |  2.0T |  SFT, RLHF
LLaMA-2-13B-Chat |  LLaMA-2 |  13B |  4K tokens |  2.0T |  SFT, RLHF
LLaMA-2-70B-Chat |  LLaMA-2 |  70B |  4K tokens |  2.0T |  SFT, RLHF
Table 4: Properties of the models used in this study. The context length
refers to the base model’s training. Tokens relate to the number of tokens in
the pre-training data only. We use the following abbreviations for the fine-
tuning procedure: supervised fine-tuning (SFT), reinforcement learning with
human feedback (RLHF), distilled supervised fine-tuning (dSFT), and AI
feedback through preferences (AIF). Information about the Llama 2 family is
taken from Touvron et al. (2023), specifications for Mistral-7B-Instruct are
provided by Jiang et al. (2023). For Zephyr-7B-$\beta$, we consider the work
of Tunstall et al. (2023). Dashes represent cases in which we could not find
the respective information. Problem 7:
Statements: 1. There is a blue marble in the box if and only if there is a red
marble in the box. 2. Either there is a red marble in the box or else there is
a white marble in the box, but not both. 3. There is a white marble in the box
if and only if there is a pink marble in the box. Conclusion: If there is not
a blue marble in the box then there is a pink marble in the box. Response:
Sure, I’d be happy to help! Here’s my reasoning process for this problem: 1\.
First, I’ll start by analyzing the given statements: Statement 1: "There is a
blue marble in the box if and only if there is a red marble in the box" This
statement tells me that the presence of a blue marble in the box is dependent
on the presence of a red marble in the box. If there’s no red marble, there
can’t be a blue marble. Statement 2: "Either there is a red marble in the box
or else there is a white marble in the box, but not both" This statement tells
me that the presence of a red marble and a white marble in the box are
mutually exclusive. If there’s a red marble, there can’t be a white marble,
and vice versa. Statement 3: "There is a white marble in the box if and only
if there is a pink marble in the box" This statement tells me that the
presence of a white marble in the box is dependent on the presence of a pink
marble in the box. If there’s no pink marble, there can’t be a white marble.
2\. Now, let’s consider the conclusion: "If there is not a blue marble in the
box then there is a pink marble in the box" I can start by assuming that there
is no blue marble in the box. According to statement 1, this means that there
is no red marble in the box either. Next, I can look at statement 2. Since
there’s no red marble, the alternative is that there’s a white marble in the
box. But wait, statement 3 says that if there’s a white marble, there must be
a pink marble too! So, if there’s no blue marble in the box, it means there’s
no red marble, which means there’s a white marble, which means there’s a pink
marble Therefore, the conclusion logically follows! Conclusion: True Annotator
Comments: Placeholder placeholder Reformulation of statement 1. placeholder
placeholder Reformulation of statement 2. placeholder placeholder placeholder
Reformulation of statement 3. placeholder placeholder placeholder Repetition
of conclusion. placeholder Supposition following (sound). First consequence.
Second consequence. Third consequence. placeholder Conclusion (valid).
placeholder placeholder Final answer (correct). Figure 6: The response (lower
left box) of LLaMA-2-70B to problem 7 (top box) of the problem set,
illustrating  green!20 supposition following. After reformulating the
statements of the problem at hand, the model starts to reason about the
problem by assuming the absence of a blue marble in the box. Subsequently, it
traces the consequences of that supposition, drawing intermediate conclusions
about the presence or absence of certain marbles, until it formulates a final
conclusion. In this example, the model correctly reasons about the validity of
the conclusion. Problem 7:
Statements: 1. There is a blue marble in the box if and only if there is a red
marble in the box. 2. Either there is a red marble in the box or else there is
a white marble in the box, but not both. 3. There is a white marble in the box
if and only if there is a pink marble in the box. Conclusion: If there is not
a blue marble in the box then there is a pink marble in the box. Response: To
determine if the conclusion logically follows from the given statements, let’s
first understand the given statements and the logical relationships between
them: 1\. Statement 1: "There is a blue marble in the box if and only if there
is a red marble in the box." \- This statement is equivalent to saying that
the presence of a blue marble and the presence of a red marble are mutually
exclusive and exhaustive. 2\. Statement 2: "Either there is a red marble in
the box or else there is a white marble in the box, but not both." \- This
statement tells us that there cannot be both a red and a white marble in the
box. 3\. Statement 3: "There is a white marble in the box if and only if there
is a pink marble in the box." \- This statement is equivalent to saying that
the presence of a white marble and the presence of a pink marble are mutually
exclusive and exhaustive. Now let’s reason about the conclusion: 1\.
Conclusion: "If there is not a blue marble in the box then there is a pink
marble in the box." To determine if the conclusion logically follows from the
given statements, we need to consider all possible cases: Case 1: There is no
blue marble in the box and there is a red marble. \- This case is not
possible, as statement 1 tells us that a blue marble and a red marble cannot
both be in the box. Case 2: There is no blue marble in the box and there is a
white marble. \- According to statement 3, if there is a white marble, then
there must be a pink marble as well. Case 3: There is no blue marble in the
box and there is a pink marble. \- This case is not explicitly stated in the
given statements, but it is possible based on the information provided. Based
on our analysis, we cannot definitively say that the conclusion logically
follows from the given statements because there is a possible case (Case 1)
that contradicts the conclusion. However, we have found that the conclusion is
true in at least one possible case (Case 3). Therefore, the conclusion "If
there is not a blue marble in the box then there is a pink marble in the box"
is not a logical consequence of the given statements, but it is true in at
least one possible scenario. Conclusion: False. Annotator Comments:
Placeholder Placeholder Repetition of statement 1. Placeholder Reformulation
of statement 1. Placeholder Repetition of statement 2. Placeholder
Reformulation of statement 2. Repetition of statement 3. Placeholder
Reformulation of statement 3. Placeholder Placeholder Repetition of
conclusion. Supposition following (invalid). Placeholder Supposition.
Placeholder Placeholder Supposition. Placeholder Supposition. Placeholder
Placeholder Conclusion (invalid). Placeholder Placeholder Placeholder
Placeholder Placeholder Final answer (incorrect). Figure 7: An exemplary model
response of Mistral-7B-Instruct (lower left box) to problem 7 (top box)
illustrating  green!20 supposition following. The model successively assumes
combinations of marbles in the box, and infers the immediate consequences from
the premises provided. However, it does not extend its reasoning beyond the
direct outcomes of each supposition, thereby failing to deduce the validity of
the conclusion. Problem 9:
Statements: 1. There is a purple marble in the box if and only if there is a
blue marble in the box. 2. There is a blue marble in the box if and only if
there is an olive marble in the box. 3. Either there is an olive marble in the
box or else there is a red marble in the box, but not both. 4. Either there is
a red marble in the box or else there is a green marble in the box, but not
both. Conclusion: If there is a purple marble in the box then there is a green
marble in the box. Response: Based on the given statements, here’s my
reasoning: 1\. First, let’s analyze the given statements one by one: a)
Statement 1: "There is a purple marble in the box if and only if there is a
blue marble in the box" b) Statement 2: "There is a blue marble in the box if
and only if there is an olive marble in the box" c) Statement 3: "Either there
is an olive marble in the box or else there is a red marble in the box, but
not both" d) Statement 4: "Either there is a red marble in the box or else
there is a green marble in the box, but not both" 2\. Since we are trying to
prove that "If there is a purple marble in the box then there is a green
marble in the box", let’s try to find a way to connect purple marbles with
green marbles using the given statements. 3\. Unfortunately, there doesn’t
seem to be a direct connection between purple marbles and green marbles in the
given statements. 4\. However, let’s see if we can find any indirect
connections. Based on Statement 1, we know that there is a purple marble in
the box if and only if there is a blue marble in the box. 5\. Now, let’s look
at Statement 2. We know that there is a blue marble in the box if and only if
there is an olive marble in the box. 6\. From Statement 3, we know that there
cannot be both an olive marble and a red marble in the box. However, Statement
4 tells us that there cannot be both a red marble and a green marble in the
box either. 7\. Let’s consider the possibility that there is an olive marble
in the box. If that’s the case, then, according to Statement 2, there must be
a blue marble in the box as well. 8\. Now, using Statement 1, we know that if
there is a blue marble in the box, then there must be a purple marble in the
box as well. 9\. However, Statement 3 tells us that if there is an olive
marble in the box, then there cannot be a red marble in the box. But,
Statement 4 tells us that if there is a red marble in the box, then there
cannot be a green marble in the box. 10\. Since we cannot have both an olive
marble and a red marble in the box, and we cannot have both a red marble and a
green marble in the box, it seems that the only possibility left is that there
is an olive marble in the box, and therefore, a blue marble, a purple marble,
but no red or green marbles. 11\. However, the conclusion states that "If
there is a purple marble in the box then there is a green marble in the box",
which is not logically true based on the given statements. Conclusion: False.
Annotator Comments: Placeholder Placeholder Repetition of statement 1.
Placeholder Repetition of statement 2. Placeholder Repetition of statement 3.
Placeholder Repetition of statement 4. Placeholder Repetition of conclusion.
Placeholder Placeholder Placeholder Placeholder Reformulation of statement 1.
Placeholder Reformulation of statement 2. Placeholder Reformulation of
statement 3. Reformulation of statement 4. Placeholder Supposition following
(sound). Placeholder Placeholder Placeholder Placeholder Placeholder
Placeholder Placeholder Invalid inference. Placeholder Placeholder Conclusion
(invalid). Placeholder Final answer (incorrect). Figure 8: An exemplary model
response of Mistral-7B-Instruct (lower left box) to problem 9 (top box)
illustrating  green!20 supposition following. The model supposes the presence
of an olive marble in the box and traces the consequences of that supposition.
However, it derives disjointed intermediate conclusions that do not aid in
solving the problem, failing to solve the task at hand. Problem 9:
Statements: 1. There is a purple marble in the box if and only if there is a
blue marble in the box. 2. There is a blue marble in the box if and only if
there is an olive marble in the box. 3. Either there is an olive marble in the
box or else there is a red marble in the box, but not both. 4. Either there is
a red marble in the box or else there is a green marble in the box, but not
both. Conclusion: If there is a purple marble in the box then there is a green
marble in the box. Response: Sure, I’d be happy to help! Here’s my reasoning
process for this problem: 1\. First, I’ll start by analyzing the given
statements: Statement 1: "There is a purple marble in the box if and only if
there is a blue marble in the box" This statement tells me that the presence
of a purple marble in the box is dependent on the presence of a blue marble in
the box. If there is a blue marble, then there must be a purple marble, and
vice versa. Statement 2: "There is a blue marble in the box if and only if
there is an olive marble in the box" This statement tells me that the presence
of a blue marble in the box is dependent on the presence of an olive marble in
the box. If there is an olive marble, then there must be a blue marble, and
vice versa. Statement 3: "Either there is an olive marble in the box or else
there is a red marble in the box, but not both" This statement tells me that
the presence of an olive marble and the presence of a red marble in the box
are mutually exclusive. If there is an olive marble, then there cannot be a
red marble, and vice versa. Statement 4: "Either there is a red marble in the
box or else there is a green marble in the box, but not both" This statement
tells me that the presence of a red marble and the presence of a green marble
in the box are also mutually exclusive. If there is a red marble, then there
cannot be a green marble, and vice versa. 2\. Now, let’s consider the
conclusion: "If there is a purple marble in the box then there is a green
marble in the box" My reasoning process for this conclusion is as follows: \-
If there is a purple marble in the box, then by Statement 1, there must be a
blue marble in the box. \- If there is a blue marble in the box, then by
Statement 2, there must be an olive marble in the box. \- If there is an olive
marble in the box, then by Statement 3, there cannot be a red marble in the
box. \- If there is no red marble in the box, then by Statement 4, there must
be a green marble in the box. Therefore, my conclusion is: Conclusion: True I
hope that helps! Let me know if you have any questions or if you’d like me to
explain my reasoning further. Annotator Comments: Placeholder Placeholder
Reformulation of statement 1. Placeholder Placeholder Placeholder Placeholder
Reformulation of statement 2. Placeholder Placeholder Placeholder Placeholder
Reformulation of statement 3. Placeholder Placeholder Placeholder Placeholder
Reformulation of statement 4. Placeholder Placeholder Placeholder Placeholder
Repetition of conclusion. Placeholder Chain construction (sound). P
$\rightarrow$ B Placeholder B $\rightarrow$ O Placeholder O $\rightarrow\neg$
R Placeholder $\neg$ R $\rightarrow$ G Placeholder Conclusion. Final answer
(correct). Placeholder Placeholder Figure 9: The response (lower left box) of
LLaMA-2-70B to problem 9 (top box) of the problem set, illustrating  yellow
chain construction. The model correctly constructs a chain of conditionals
leading from the antecedent of the final conclusion to its consequent. Problem
7:
Statements: 1. There is a blue marble in the box if and only if there is a red
marble in the box. 2. Either there is a red marble in the box or else there is
a white marble in the box, but not both. 3. There is a white marble in the box
if and only if there is a pink marble in the box. Conclusion: If there is not
a blue marble in the box then there is a pink marble in the box. Response:
Sure! Let’s go through the statements and see if we can determine whether the
conclusion logically follows. 1\. "There is a blue marble in the box if and
only if there is a red marble in the box" This statement tells us that the
presence of a blue marble in the box is dependent on the presence of a red
marble. If there is a red marble, then there must be a blue marble, and if
there is no red marble, then there cannot be a blue marble. 2\. "Either there
is a red marble in the box or else there is a white marble in the box, but not
both" This statement tells us that the presence of a red marble and a white
marble in the box are mutually exclusive. If there is a red marble, then there
cannot be a white marble, and if there is a white marble, then there cannot be
a red marble. 3\. "There is a white marble in the box if and only if there is
a pink marble in the box" This statement tells us that the presence of a white
marble in the box is dependent on the presence of a pink marble. If there is a
pink marble, then there must be a white marble, and if there is no pink
marble, then there cannot be a white marble. Now, let’s consider the
conclusion: "If there is not a blue marble in the box then there is a pink
marble in the box" Let’s break this down further: \- If there is not a blue
marble in the box, then we know that there is no red marble in the box (from
statement 1). \- Since there is no red marble, we know that there cannot be a
white marble in the box (from statement 2). \- If there is no white marble in
the box, then we know that there must be a pink marble in the box (from
statement 3). Therefore, we can conclude that if there is not a blue marble in
the box, then there must be a pink marble in the box. Conclusion: True.
Annotator Comments: Placeholder Placeholder Reformulation of statement 1.
Placeholder Placeholder Placeholder Reformulation of statement 2. Placeholder
Placeholder Placeholder Placeholder Reformulation of statement 3. Placeholder
Placeholder Placeholder Repetition of conclusion. Placeholder Chain
construction (invalid) $\neg$ B $\rightarrow\neg$ R Placeholder $\neg$ R
$\rightarrow\neg$ W (invalid) Placeholder $\neg$ W $\rightarrow$ P (invalid)
Placeholder Conclusion: $\neg$ B $\rightarrow$ P Placeholder Final answer
(correct). Figure 10: The response (lower left box) of LLaMA-2-70B to problem
7 (top box) of the problem set, illustrating  yellow chain construction. The
model constructs a chain of conditionals leading from the antecedent of the
final conclusion to its consequent. However, it fails to understand the
implication of the exclusive disjunction in the second statement of the
problem description, leading to a faulty reasoning trace. Despite its invalid
reasoning, the model’s final answer is correct. Problem 10:
Statements: 1. There is a brown marble in the box if and only if there is a
yellow marble in the box. 2. Either there is a yellow marble in the box or
else there is a green marble in the box, but not both. 3. There is a green
marble in the box if and only if there is a purple marble in the box. 4. There
is a purple marble in the box if and only if there is an olive marble in the
box. Conclusion: If there is a brown marble in the box then there is an olive
marble in the box. Response: Sure, I’d be happy to help with this experiment!
Here’s my reasoning process for each statement and the conclusion: 1\. There
is a brown marble in the box if and only if there is a yellow marble in the
box This statement means that if there is a brown marble in the box, then
there must also be a yellow marble in the box. Conversely, if there is a
yellow marble in the box, then there must also be a brown marble in the box.
2\. Either there is a yellow marble in the box or else there is a green marble
in the box, but not both This statement means that if there is a yellow marble
in the box, then there cannot also be a green marble in the box. Similarly, if
there is a green marble in the box, then there cannot also be a yellow marble
in the box. 3\. There is a green marble in the box if and only if there is a
purple marble in the box This statement means that if there is a green marble
in the box, then there must also be a purple marble in the box. Conversely, if
there is a purple marble in the box, then there must also be a green marble in
the box. 4\. There is a purple marble in the box if and only if there is an
olive marble in the box This statement means that if there is a purple marble
in the box, then there must also be an olive marble in the box. Conversely, if
there is an olive marble in the box, then there must also be a purple marble
in the box. Now, let’s consider the conclusion: Conclusion: If there is a
brown marble in the box then there is an olive marble in the box Based on the
statements we’ve been given, we can logically deduce the conclusion as
follows: 1\. If there is a brown marble in the box, then there must also be a
yellow marble in the box (statement 1) 2\. If there is a yellow marble in the
box, then there cannot also be a green marble in the box (statement 2) 3\. If
there is a green marble in the box, then there must also be a purple marble in
the box (statement 3) 4\. If there is a purple marble in the box, then there
must also be an olive marble in the box (statement 4) Therefore, if there is a
brown marble in the box, then there must also be an olive marble in the box,
which means the conclusion is true. Conclusion: True. Annotator Comments:
Placeholder Placeholder Reformulation of statement 1. Placeholder Placeholder
Placeholder Reformulation of statement 2. Placeholder Placeholder Placeholder
Placeholder Reformulation of statement 3. Placeholder Placeholder Placeholder
Reformulation of statement 4. Placeholder Placeholder Placeholder Repetition
of conclusion. Placeholder Chain construction (invalid). B $\rightarrow$ Y
Placeholder Y $\rightarrow\neg$ G Placeholder G $\rightarrow$ P (failure to
consider negation) Placeholder P $\rightarrow$ O Placeholder Conclusion
(invalid). Placeholder Final answer (incorrect). Figure 11: The response
(lower left box) of LLaMA-2-13B to problem 10 (top box) of the problem set,
illustrating  yellow chain construction. The model constructs a chain of
conditionals leading from the antecedent of the final conclusion to its
consequent. However, it fails to account for the negation of the second
conditional’s consequent, leading to a faulty reasoning trace. Problem 5:
Statements: 1. There is a purple marble in the box if and only if there is a
yellow marble in the box. 2. There is a yellow marble in the box if and only
if there is a blue marble in the box. 3. Either there is a blue marble in the
box or else there is an orange marble in the box, but not both. Conclusion:
Either there is a purple marble in the box or else there is an orange marble
in the box, but not both. Response: Sure! Let’s break this down step by step.
1\. "There is a purple marble in the box if and only if there is a yellow
marble in the box" This statement implies that the presence of a purple marble
in the box is dependent on the presence of a yellow marble in the box. If
there is no yellow marble, there cannot be a purple marble. 2\. "There is a
yellow marble in the box if and only if there is a blue marble in the box"
This statement implies that the presence of a yellow marble in the box is
dependent on the presence of a blue marble in the box. If there is no blue
marble, there cannot be a yellow marble. 3\. "Either there is a blue marble in
the box or else there is an orange marble in the box, but not both" This
statement implies that the presence of a blue marble and an orange marble in
the box are mutually exclusive. If there is a blue marble, there cannot be an
orange marble, and vice versa. Now, let’s consider the conclusion: "Either
there is a purple marble in the box or else there is an orange marble in the
box, but not both" Based on the first two statements, we know that if there is
a purple marble, there must be a yellow marble, and if there is a yellow
marble, there must be a blue marble. Therefore, if there is a purple marble,
there must be a blue marble. Now, let’s consider the third statement. If there
is a blue marble, there cannot be an orange marble. Therefore, if there is a
purple marble (which implies a blue marble), there cannot be an orange marble.
Conclusion: True. The conclusion logically follows from the given statements.
Annotator Comments: Placeholder Reformulation of statement 1. Placeholder
Placeholder Placeholder Reformulation of statement 2. Placeholder Placeholder
Reformulation of statement 3. Placeholder Placeholder Placeholder Repetition
of conclusion. Chain construction (sound). P $\rightarrow$ Y Y $\rightarrow$ B
P $\rightarrow$ B B $\rightarrow\neg$ O Placeholder P $\rightarrow\neg$ O
Placeholder Final answer (correct). Conclusion (invalid). Figure 12: The
response (lower left box) of LLaMA-2-70B to problem 5 (top box) of the problem
set, illustrating  yellow chain construction. The model constructs a chain of
conditionals proving one case of the exclusive disjunction. However, it fails
to account for the other conditional case, i.e. $\neg$ P $\rightarrow$ O,
therefore failing to prove the logical validity of the conclusion. Problem 12:
Statements: 1. There is a blue marble in the box if and only if there is a
brown marble in the box. 2. There is a white marble in the box if and only if
there is a green marble in the box. 3. Either there is a brown marble in the
box or else there is a white marble in the box, but not both. 4. There is a
green marble in the box if and only if there is a purple marble in the box
Conclusion: If there is a blue marble in the box then there is a purple marble
in the box. Response: Sure! Let’s break down the problem and think through
each step. 1\. First, we have the statement "There is a blue marble in the box
if and only if there is a brown marble in the box." This means that if there
is a blue marble in the box, then there must be a brown marble in the box, and
vice versa. 2\. Next, we have the statement "There is a white marble in the
box if and only if there is a green marble in the box." This means that if
there is a white marble in the box, then there must be a green marble in the
box, and vice versa. 3\. The third statement is "Either there is a brown
marble in the box or else there is a white marble in the box, but not both."
This means that if there is a brown marble in the box, then there cannot be a
white marble in the box, and vice versa. 4\. The fourth statement is "There is
a green marble in the box if and only if there is a purple marble in the box."
This means that if there is a green marble in the box, then there must be a
purple marble in the box, and vice versa. 5\. Now, let’s consider the
conclusion: "If there is a blue marble in the box then there is a purple
marble in the box." Let’s think about how we can use the given statements to
determine whether this conclusion is true or false. 1\. We know that if there
is a blue marble in the box, then there must be a brown marble in the box
(from statement 1). 2\. We also know that if there is a brown marble in the
box, then there cannot be a white marble in the box (from statement 3). 3\.
Therefore, if there is a blue marble in the box, then there cannot be a white
marble in the box. 4\. We also know that if there is a white marble in the
box, then there must be a green marble in the box (from statement 2). 5\.
Therefore, if there is a blue marble in the box, then there cannot be a green
marble in the box. 6\. However, we know that if there is a green marble in the
box, then there must be a purple marble in the box (from statement 4). 7\.
Therefore, if there is a blue marble in the box, then there must be a purple
marble in the box. Conclusion: True. In summary, we have shown that if there
is a blue marble in the box, then there must be a purple marble in the box,
using the given statements. Therefore, the conclusion logically follows from
the statements. Annotator Comments: Placeholder Reformulation of statement 1.
Placeholder Placeholder Reformulation of statement 2. Placeholder Placeholder
Reformulation of statement 3. Placeholder Placeholder Reformulation of
statement 4. Placeholder Placeholder Repetition of conclusion. Placeholder
Chain construction (invalid). Placeholder Blue $\rightarrow$ Brown Placeholder
Brown $\rightarrow\neg$ W Placeholder Blue $\rightarrow\neg$ W Placeholder W
$\rightarrow$ G Placeholder Blue $\rightarrow\neg$ G (fallacy of the inverse).
Placeholder G $\rightarrow$ P Placeholder Blue $\rightarrow$ P (invalid)
Placeholder Final answer (correct). Conclusion (invalid). Placeholder
Placeholder Figure 13: The response (lower left box) of LLaMA-2-70B to problem
12 (top box) of the problem set, illustrating  yellow chain construction. The
model constructs a chain of conditionals leading from the antecedent of the
final conclusion to its consequent. However, it makes a series of mistakes
when constructing the chain of conditionals. For instance, it infers the
absence of the green marble by denying the presence of the white marble, i.e.
Blue $\rightarrow\neg$ W; W $\rightarrow$ G; therefore Blue $\rightarrow\neg$
G by assuming that $\neg$ W $\rightarrow\neg$ G, which is a common logical
error known as the fallacy of the inverse. Problem 9:
Statements: 1. There is a purple marble in the box if and only if there is a
blue marble in the box. 2. There is a blue marble in the box if and only if
there is an olive marble in the box. 3. Either there is an olive marble in the
box or else there is a red marble in the box, but not both. 4. Either there is
a red marble in the box or else there is a green marble in the box, but not
both. Conclusion: If there is a purple marble in the box then there is a green
marble in the box. Response: First, let’s analyze the given statements: 1\.
Statement 1: There is a purple marble in the box if and only if there is a
blue marble in the box 2\. Statement 2: There is a blue marble in the box if
and only if there is an olive marble in the box 3\. Statement 3: Either there
is an olive marble in the box or else there is a red marble in the box, but
not both 4\. Statement 4: Either there is a red marble in the box or else
there is a green marble in the box, but not both Let’s examine the
relationship between the statements: From Statement 1 and Statement 2, we know
that there is a purple marble in the box if and only if there is a blue marble
in the box, and there is a blue marble in the box if and only if there is an
olive marble in the box. Therefore, we can conclude that there is a purple
marble in the box if and only if there is an olive marble in the box. However,
the conclusion states that if there is a purple marble in the box, then there
is a green marble in the box. This is not logically following from the given
statements. Conclusion: False. Annotator Comments: Placeholder Reformulation
of statement 1. Placeholder Reformulation of statement 2. Placeholder
Reformulation of statement 3. Placeholder Reformulation of statement 4.
Placeholder Compound Strategy (sound). List biconditional statements:
$P\leftrightarrow B$ $B\leftrightarrow O$ Draw compound conclusion:
$P\leftrightarrow O$ Conclusion (invalid). Placeholder Final answer
(incorrect). Figure 14: The response (lower left box) of Mistral-7B-Instruct
to problem 9 (top box) of the problem set, illustrating the compound strategy.
Based on the first two premises of the problem description, the model draws a
compound conclusion, establishing equivalence between the purple and olive
marble in the box. However, Mistral-7B-Instruct fails to draw additional
intermediate conclusions that would be required to deduce the logical validity
of the conclusion in the problem statement. Problem 9:
Statements: 1. There is a purple marble in the box if and only if there is a
blue marble in the box. 2. There is a blue marble in the box if and only if
there is an olive marble in the box. 3. Either there is an olive marble in the
box or else there is a red marble in the box, but not both. 4. Either there is
a red marble in the box or else there is a green marble in the box, but not
both. Conclusion: If there is a purple marble in the box then there is a green
marble in the box. Response: Sure! Let’s break it down step by step. 1\.
"There is a purple marble in the box if and only if there is a blue marble in
the box" This statement implies that the presence of a purple marble in the
box is dependent on the presence of a blue marble in the box. If there is no
blue marble, there cannot be a purple marble. 2\. "There is a blue marble in
the box if and only if there is an olive marble in the box" This statement
implies that the presence of a blue marble in the box is dependent on the
presence of an olive marble in the box. If there is no olive marble, there
cannot be a blue marble. 3\. "Either there is an olive marble in the box or
else there is a red marble in the box, but not both" This statement implies
that the presence of an olive marble and a red marble in the box are mutually
exclusive. If there is an olive marble, there cannot be a red marble, and vice
versa. 4\. "Either there is a red marble in the box or else there is a green
marble in the box, but not both" This statement implies that the presence of a
red marble and a green marble in the box are also mutually exclusive. If there
is a red marble, there cannot be a green marble, and vice versa. Now, let’s
evaluate the conclusion: Conclusion: If there is a purple marble in the box
then there is a green marble in the box Based on the given statements, we know
that the presence of a purple marble is dependent on the presence of a blue
marble, and the presence of a blue marble is dependent on the presence of an
olive marble. Therefore, if there is a purple marble in the box, it means that
there is also a blue marble and an olive marble in the box. Using the third
statement, we know that the presence of an olive marble means that there
cannot be a red marble in the box. Therefore, if there is a purple marble in
the box, there cannot be a red marble. Using the fourth statement, we know
that the absence of a red marble means that there must be a green marble in
the box. Therefore, if there is a purple marble in the box, there must be a
green marble in the box as well. Conclusion: True So, the conclusion logically
follows from the given statements. Annotator Comments: Placeholder
Reformulation of statement 1. Placeholder Placeholder Reformulation of
statement 2. Placeholder Placeholder Reformulation of statement 3. Placeholder
Placeholder Placeholder Reformulation of statement 4. Placeholder Placeholder
Placeholder Repetition of conclusion. Placeholder Compound Strategy (sound).
$P\leftrightarrow B$ $B\leftrightarrow O$ Draw compound conclusion:
$P\leftrightarrow O$ Derive conditional (statement 3): $O\rightarrow\neg R$
Draw compound conclusion. Derive conditional (statement 4): $\neg R\rightarrow
G$ Draw compound conclusion. Placeholder Final answer (correct). Conclusion
(valid). Figure 15: The response (lower left box) of LLaMA-2-70B to problem 9
(top box) of the problem set, illustrating the compound strategy. The model
draws a series of compound conclusions to deduce the logical validity of the
conclusion in the problem statement. Problem 8:
Statements: 1. Either there is an olive marble in the box or else there is a
brown marble in the box, but not both. 2. There is a brown marble in the box
if and only if there is a gray marble in the box. 3. Either there is a gray
marble in the box or else there is a maroon marble in the box, but not both.
Conclusion: If there is not an olive marble in the box then there is a maroon
marble in the box. Response: To determine if the conclusion logically follows
from the given statements, we need to use the principles of logical reasoning.
Let’s analyze the statements one by one and then consider the conclusion. 1\.
Statement 1: Either there is an olive marble in the box or else there is a
brown marble in the box, but not both 2\. Statement 2: There is a brown marble
in the box if and only if there is a gray marble in the box 3\. Statement 3:
Either there is a gray marble in the box or else there is a maroon marble in
the box, but not both First, let’s understand the implications of Statement 2.
It states that if there is a brown marble in the box, then there is a gray
marble in the box, and conversely, if there is a gray marble in the box, then
there is a brown marble in the box. This is an if and only if statement,
meaning both conditions are equivalent. Now, let’s consider Statement 1. It
states that there is either an olive marble or a brown marble in the box, but
not both. Since Statement 2 tells us that a brown marble implies a gray
marble, it follows that if there is no olive marble (i.e., there is a brown
marble), then there must be a gray marble. However, Statement 3 tells us that
there is either a gray marble or a maroon marble in the box, but not both.
Therefore, if there is a gray marble, then there cannot be a maroon marble.
So, based on the given statements, if there is no olive marble in the box,
then there is a brown marble (from Statement 1 and Statement 2), which implies
a gray marble (from Statement 2), and since there is no olive marble and there
is a gray marble, there cannot be a maroon marble (from Statement 3).
Therefore, the conclusion follows logically from the given statements.
Conclusion: True. Annotator Comments: Placeholder Placeholder Placeholder
Repetition of statement 1. Placeholder Repetition of statement 2. Placeholder
Repetition of statement 3. Placeholder Reformulation of statement 2.
Placeholder Placeholder Placeholder Compound strategy (valid). O $\oplus$ B B
$\leftrightarrow$ G Draw compound conclusion: $\neg$ O $\rightarrow$ G G
$\oplus$ M Draw compound conclusion: $\neg$ O $\rightarrow\neg$ M Supposition
Following (valid). Tracing consequences. Placeholder Placeholder Conclusion
(invalid). Final answer (incorrect). Figure 16: The response (lower left box)
of Mistral-7B-Instruct to problem 8 (top box) of the problem set, illustrating
the compound strategy and  green!20 supposition following. Based on the first
two premises of the problem description, the model first draws a compound
conclusion, establishing that a gray marble follows from the absence of an
olive marble. Subsequently, it uses this intermediate conclusion, together
with the third premise, to draw another compound conclusion about the absence
of the maroon marble. The model then switches to supposition following,
tracing the consequences of the absence of the olive marble, inferring the
final conclusion that there cannot be a maroon marble. However, despite the
model’s correct reasoning, it deduces the wrong answer: "True". Problem 6:
Statements: 1. There is a gray marble in the box if and only if there is a
yellow marble in the box. 2. Either there is a yellow marble in the box or
else there is an olive marble in the box, but not both. 3. There is an olive
marble in the box if and only if there is a black marble in the box.
Conclusion: If there is a yellow marble in the box then there is a black
marble in the box. Response: Based on the given statements, here is my
reasoning: 1\. The first statement tells us that there is a gray marble in the
box if and only if there is a yellow marble in the box. 2\. The second
statement tells us that either there is a yellow marble in the box or there is
an olive marble in the box, but not both. 3\. The third statement tells us
that there is an olive marble in the box if and only if there is a black
marble in the box. 4\. From the second statement, we know that if there is a
yellow marble in the box, then there cannot be an olive marble in the box. 5\.
From the third statement, we know that if there is an olive marble in the box,
then there is a black marble in the box. 6\. Since the second statement tells
us that there cannot be both a yellow marble and an olive marble in the box,
it follows that if there is a yellow marble in the box, then there cannot be
an olive marble or a black marble in the box (because of the if and only if
relationship in the third statement). 7\. Therefore, the conclusion does not
logically follow from the given statements. Conclusion: False. Annotator
Comments: Placeholder Reformulation of statement 1. Placeholder Reformulation
of statement 2. Placeholder Reformulation of statement 3. Placeholder
Conditional from statement 2. Y $\rightarrow\neg$ O Conditional from statement
3. O $\rightarrow$ B Concatenation of statements (sound). Y $\rightarrow\neg$
(O $\leftrightarrow$ B) Placeholder Placeholder Conclusion (valid). Final
answer (correct). Figure 17: The response (lower left box) of
Mistral-7B-Instruct to problem 6 (top box) of the problem set, illustrating
the concatenation strategy. Mistral-7B-Instruct concatenates the intermediate
conditional conclusion (Y $\rightarrow\neg$ O) and the third premise of the
problem statement (O $\leftrightarrow$ B) to form the concatenated conclusion
Y $\rightarrow\neg$ (O $\leftrightarrow$ B). Based on that conclusion, the
model infers that the conclusion in the problem statement does not logically
follow from the premises at hand. Problem 3:
Statements: 1. There is a green marble in the box if and only if there is a
purple marble in the box 2. If there is a purple marble in the box then there
is a gray marble in the box. 3. Either there is a gray marble in the box or
else there is a yellow marble in the box, but not both. Conclusion: Either
there is a green marble in the box or else there is a yellow marble in the
box, but not both. Response: Sure, I’d be happy to help! Here’s my reasoning
process for this problem: 1\. First, I’ll start by analyzing the given
statements: \- Statement 1: "There is a green marble in the box if and only if
there is a purple marble in the box" \- This statement establishes a direct
relationship between the presence of a green marble and a purple marble in the
box. \- Statement 2: "If there is a purple marble in the box then there is a
gray marble in the box" \- This statement establishes a conditional
relationship between the presence of a purple marble and a gray marble in the
box. \- Statement 3: "Either there is a gray marble in the box or else there
is a yellow marble in the box, but not both" \- This statement establishes an
exclusive OR relationship between the presence of a gray marble and a yellow
marble in the box. 2\. Now, let’s consider the conclusion: "Either there is a
green marble in the box or else there is a yellow marble in the box, but not
both" 3\. To determine whether the conclusion logically follows from the
statements, I’ll use a truth table analysis: Statement 1 Statement 2 Statement
3 Conclusion 1 T T T T 2 T T F F 3 T F T F 4 F T T F 5 F T F F 6 F F T T 7 F F
F F As we can see from the truth table, the conclusion "Either there is a
green marble in the box or else there is a yellow marble in the box, but not
both" is true in all cases where Statement 1 and Statement 2 are true, and
false in all cases where Statement 3 is false. Since Statement 1 and Statement
2 are always true, and Statement 3 is true in some cases, the conclusion
logically follows from the statements. Conclusion: True Annotator Comments:
Placeholder Reformulation of statement 1. Placeholder Placeholder Placeholder
Placeholder Reformulation of statement 2. Placeholder Placeholder
Reformulation of statement 3. Placeholder Placeholder Placeholder Repetition
of conclusion. Placeholder Symbolic Strategy (invalid). Placeholder Construct
truth table. Placeholder Placeholder Placeholder Placeholder Placeholder
Placeholder Placeholder Summarize truth table. Placeholder Placeholder
Conclusion (invalid). Placeholder Final answer (correct). Figure 18: The
response (lower left box) of LLaMA-2-70B to problem 3 (top) of the problem
set, illustrating the symbolic strategy. The model constructs a truth table to
infer the validity of the conclusion given in the problem statement. However,
the model produces errors in the truth table, resulting in flawed reasoning.
Problem 6:
Statements: 1. There is a gray marble in the box if and only if there is a
yellow marble in the box. 2. Either there is a yellow marble in the box or
else there is an olive marble in the box, but not both. 3. There is an olive
marble in the box if and only if there is a black marble in the box.
Conclusion: If there is a yellow marble in the box then there is a black
marble in the box. Response: First, we need to understand the relationships
between the marbles and the statements. Here’s a summary: \- There is a gray
marble in the box if and only if there is a yellow marble in the box \- This
statement creates a connection between the gray marble and the yellow marble.
If one is present, the other is as well. \- Either there is a yellow marble in
the box or else there is an olive marble in the box, but not both \- This
statement introduces two possible scenarios. Either the yellow marble is
present, or the olive marble is present, but not both. \- There is an olive
marble in the box if and only if there is a black marble in the box \- This
statement creates another connection, but this time between the olive marble
and the black marble. If one is present, the other is as well. Now, let’s see
if the conclusion follows logically from these statements. \- If there is a
yellow marble in the box then there is a black marble in the box \- To prove
this conclusion, we’ll use the second statement to eliminate the possibility
of the olive marble being present. Then, we’ll use the third statement to
connect the presence of the yellow marble with the presence of the black
marble. \- If there is a yellow marble in the box, then by the second
statement, the olive marble cannot also be present. \- If the olive marble is
not present, then by the third statement, the black marble must be present. \-
Therefore, if there is a yellow marble in the box, then there is a black
marble in the box. Conclusion: True. Explanation: \- Statement 1: Gray marble
$\leftrightarrow$ Yellow marble \- Statement 2: Yellow marble $\lor$ !Olive
marble \- Statement 3: Olive marble $\leftrightarrow$ Black marble \-
Conclusion: Yellow marble $\rightarrow$ Black marble Reasoning steps: 1\.
Given: Yellow marble $\rightarrow$ Black marble (conclusion) 2\. Given: Olive
marble $\leftrightarrow$ Black marble (statement 3) 3\. Given: Yellow marble