Editing Rule of inference (section)

== In various fields ==
Rules of inference are relevant to many fields, especially the [[formal science]]s, such as [[mathematics]] and [[computer science]], where they are used to prove theorems.<ref>{{multiref | {{harvnb|Fetzer|1996|pp=[https://books.google.com/books?id=JbNI3b-j0mkC&pg=PA241 241–243]}} | {{harvnb|Dent|2024|p=[https://books.google.com/books?id=NbA0EQAAQBAJ&pg=PA36 36]}} }}</ref> [[Mathematical proofs]] often start with a set of axioms to describe the logical relationships between mathematical constructs. To establish theorems, mathematicians apply rules of inference to these axioms, aiming to demonstrate that the theorems are logical consequences.<ref>{{multiref | {{harvnb|Horsten|2023|loc=Lead section, § 5.4 Mathematical Proof}} | {{harvnb|Polkinghorne|2011|p=[https://books.google.com/books?id=AJV0P1pPBNoC&pg=PA65 65]}} }}</ref> [[Mathematical logic]], a subfield of mathematics and logic, uses mathematical methods and frameworks to study rules of inference and other logical concepts.<ref>{{multiref | {{harvnb|Cook|2009|pp=174, 185}} | {{harvnb|Porta|Maillet|Mas|Martinez|2011|p=[https://books.google.com/books?id=NDMID6mWcZcC&pg=PA237 237]}} }}</ref>

Computer science also relies on deductive reasoning, employing rules of inference to establish theorems and validate [[algorithms]].<ref>{{multiref | {{harvnb|Butterfield|Ngondi|2016|loc=§ Computer Science}} | {{harvnb|Cook|2009|p=174}} | {{harvnb|Dent|2024|p=[https://books.google.com/books?id=NbA0EQAAQBAJ&pg=PA36 36]}} }}</ref> [[Logic programming]] frameworks, such as [[Prolog]], allow developers to [[Knowledge representation|represent knowledge]] and use [[computation]] to draw inferences and solve problems.<ref>{{multiref | {{harvnb|Butterfield|Ngondi|2016|loc=§ Logic Programming Languages, § Prolog}} | {{harvnb|Williamson|Russo|2010|p=[https://books.google.com/books?id=SmYu9-1HGtYC&pg=PA45 45]}} }}</ref> These frameworks often include an [[automated theorem prover]], a program that uses rules of inference to generate or verify proofs automatically.<ref>{{harvnb|Butterfield|Ngondi|2016|loc=§ Theorem proving, § Mechanical Verifier}}</ref> [[Expert system]]s utilize [[automated reasoning]] to simulate the [[decision-making]] processes of human [[experts]] in specific fields, such as [[medical diagnosis]], and assist in complex problem-solving tasks. They have a [[knowledge base]] to represent the facts and rules of the field and use an [[inference engine]] to extract relevant information and respond to user queries.<ref>{{multiref | {{harvnb|Butterfield|Ngondi|2016|loc=§ Expert System, § Knowledge Base, § Inference Engine}} | {{harvnb|Fetzer|1996|pp=[https://books.google.com/books?id=JbNI3b-j0mkC&pg=PA241 241–243]}} }}</ref>

Rules of inference are central to the [[philosophy of logic]] regarding the contrast between deductive-theoretic and [[Model theory|model-theoretic]] conceptions of [[logical consequence]]. Logical consequence, a fundamental concept in logic, is the [[Relation (philosophy)|relation]] between the premises of a deductively valid argument and its conclusion. Conceptions of logical consequence explain the nature of this relation and the conditions under which it exists. The deductive-theoretic conception relies on rules of inference, arguing that logical consequence means that the conclusion can be deduced from the premises through a series of inferential steps. The model-theoretic conception, by contrast, focuses on how the non-logical vocabulary of statements can be [[Interpretation (logic)|interpreted]]. According to this view, logical consequence means that no counterexamples are possible: under no interpretation are the premises true and the conclusion false.<ref>{{multiref | {{harvnb|McKeon|loc=Lead section, § 1. Introduction, § 2b. Logical and Non-Logical Terminology}} | {{harvnb|McKeon|2010|pp=[https://books.google.com/books?id=76SH_DI3yyoC&pg=PA24 24–25, 126–128]}} | {{harvnb|Hintikka|Sandu|2006|pp=13–14, 17–18}} | {{harvnb|Beall|Restall|Sagi|2024|loc=§ 3. Mathematical Tools: Models and Proofs}} }}</ref>

[[Cognitive psychology|Cognitive psychologists]] study mental processes, including [[logical reasoning]]. They are interested in how humans use rules of inference to draw conclusions, examining the factors that influence correctness and efficiency. They observe that humans are better at using some rules of inference than others. For example, the rate of successful inferences is higher for ''modus ponens'' than for ''modus tollens''. A related topic focuses on [[Cognitive bias|biases]] that lead individuals to mistake formal fallacies for valid arguments. For instance, fallacies of the types affirming the consequent and denying the antecedent are often mistakenly accepted as valid. The assessment of arguments also depends on the concrete meaning of the propositions: individuals are more likely to accept a fallacy if its conclusion sounds plausible.<ref>{{multiref | {{harvnb|Schechter|2013|p=227}} | {{harvnb|Evans|2005|pp=171–174}} }}</ref>