Rule of Inference

Monday, 27 August 2012

The Rules of Inference, part 1: Conjunctions & Disjunctions

The Rules of Inference, part 2: Conditionals

The Rules of Inference, part 3: Dilemmas

EECS 203 Project 2 (Rules of Inference).m4v

Logic: Argument Forms / Rules of Inference

4. Logic Lecture: Predicate Logic: Formal Proofs of Validity: Universal Instantiation

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Alan Ross Anderson
Alan Ross Anderson (1925 – 1973) was an American logician and professor of philosophy at Yale University and the University of Pittsburgh.
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David Hilbert
David Hilbert (; January 23, 1862 –
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Duns Scotus
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Giuseppe Peano
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Peter Geach
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Saul Kripke
Saul Aaron Kripke (born November 13, 1940) is an American philosopher and logician. He is a professor emeritus at Princeton and teaches as a Distinguished Professor of Philosophy at CUNY Graduate Center. Since the 1960s Kripke has been a central figure in a number of fields related to mathematical logic, philosophy of language, philosophy of mathematics, metaphysics, epistemology, and set theory. Much of his work remains unpublished or exists only as tape-recordings and privately circulated manuscripts (see "Unpublished Manuscripts and Online Lectures" below). Kripke was the recipient of the 2001 Schock Prize in Logic and Philosophy. A recent poll conducted among philosophers ranked Kripke among the top ten most important philosophers of the past 200 years.
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Stephen Cole Kleene
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Thoralf Skolem
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William of Ockham
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The Rules of Inference, part 1: Conjunctions & Disjunctions

Part of a series on logic, this video provides further details on validity, discusses the ...

published: 16 May 2011

Author: fortheloveofwisdom

Rules of Inference I

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published: 24 Feb 2012

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This section may be confusing or unclear to readers. Please help clarify the section; suggestions may be found on the talk page. (October 2010)

Transformation rules
Propositional calculus
Rules of inference Modus ponens Modus tollens Biconditional introduction Biconditional elimination Conjunction introduction Simplification Disjunction introduction Disjunction elimination Disjunctive syllogism Hypothetical syllogism Constructive dilemma Destructive dilemma Absorption Rules of replacement Associativity Commutativity Distributivity Double negation De Morgan's laws Transposition Material implication Exportation Tautology
Predicate logic
Universal generalization Universal instantiation Existential generalization Existential instantiation
v t e

In logic, a rule of inference, inference rule, or transformation rule is the act of drawing a conclusion based on the form of premises interpreted as a function which takes premises, analyses their syntax, and returns a conclusion (or conclusions). For example, the rule of inference modus ponens takes two premises, one in the form of "If p then q" and another in the form of "p" and returns the conclusion "q". The rule is valid with respect to the semantics of classical logic (as well as the semantics of many other non-classical logics), in the sense that if the premises are true (under an interpretation) then so is the conclusion.

Typically, a rule of inference preserves truth, a semantic property. In many-valued logic, it preserves a general designation. But a rule of inference's action is purely syntactic, and does not need to preserve any semantic property: any function from sets of formulae to formulae counts as a rule of inference. Usually only rules that are recursive are important; i.e. rules such that there is an effective procedure for determining whether any given formula is the conclusion of a given set of formulae according to the rule. An example of a rule that is not effective in this sense is the infinitary ω-rule.^[1]

Popular rules of inference include modus ponens, modus tollens from propositional logic and contraposition. First-order predicate logic uses rules of inference to deal with logical quantifiers. See List of rules of inference for examples.

1 Overview
2 Admissibility and derivability
3 Other considerations
4 See also
5 References

Overview [link]

In formal logic (and many related areas), rules of inference are usually given in the following standard form:

  Premise#1
  Premise#2
        ...
  Premise#n
  Conclusion

This expression states, that whenever in the course of some logical derivation the given premises have been obtained, the specified conclusion can be taken for granted as well. The exact formal language that is used to describe both premises and conclusions depends on the actual context of the derivations. In a simple case, one may use logical formulae, such as in:

  A→B
  A
  ∴B

This is just the modus ponens rule of propositional logic. Rules of inference are usually formulated as rule schemata by the use of universal variables. In the rule (schema) above, A and B can be instantiated to any element of the universe (or sometimes, by convention, some restricted subset such as propositions) to form an infinite set of inference rules.

A proof system is formed from a set of rules chained together to form proofs, or derivations. Any derivation has only one final conclusion, which is the statement proved or derived. If premises are left unsatisfied in the derivation, then the derivation is a proof of a hypothetical statement: "if the premises hold, then the conclusion holds."

Admissibility and derivability [link]

Main article: Admissible rule

In a set of rules, an inference rule could be redundant in the sense that it is admissible or derivable. A derivable rule is one whose conclusion can be derived from its premises using the other rules. An admissible rule is one whose conclusion holds whenever the premises hold. All derivable rules are admissible. To appreciate the difference, consider the following set of rules for defining the natural numbers (the judgment Failed to parse (Missing texvc executable; please see math/README to configure.): n\,\,\mathsf{nat}

asserts the fact that Failed to parse (Missing texvc executable; please see math/README to configure.): n
is a natural number):

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{matrix} \frac{}{\mathbf{0} \,\,\mathsf{nat}} & \frac{n \,\,\mathsf{nat}}{\mathbf{s(}n\mathbf{)} \,\,\mathsf{nat}} \\ \end{matrix}

The first rule states that 0 is a natural number, and the second states that s(n) is a natural number if n is. In this proof system, the following rule demonstrating that the second successor of a natural number is also a natural number, is derivable:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{n \,\,\mathsf{nat}}{\mathbf{s(s(}n\mathbf{))} \,\,\mathsf{nat}}

Its derivation is just the composition of two uses of the successor rule above. The following rule for asserting the existence of a predecessor for any nonzero number is merely admissible:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mathbf{s(}n\mathbf{)} \,\,\mathsf{nat}}{n \,\,\mathsf{nat}}

This is a true fact of natural numbers, as can be proven by induction. (To prove that this rule is admissible, assume a derivation of the premise and induct on it to produce a derivation of Failed to parse (Missing texvc executable; please see math/README to configure.): n \,\,\mathsf{nat} .) However, it is not derivable, because it depends on the structure of the derivation of the premise. Because of this, derivability is stable under additions to the proof system, whereas admissibility is not. To see the difference, suppose the following nonsense rule were added to the proof system:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{}{\mathbf{s(-3)} \,\,\mathsf{nat}}

In this new system, the double-successor rule is still derivable. However, the rule for finding the predecessor is no longer admissible, because there is no way to derive Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{-3} \,\,\mathsf{nat} . The brittleness of admissibility comes from the way it is proved: since the proof can induct on the structure of the derivations of the premises, extensions to the system add new cases to this proof, which may no longer hold.

Admissible rules can be thought of as theorems of a proof system. For instance, in a sequent calculus where cut elimination holds, the cut rule is admissible.

Other considerations [link]

Inference rules may also be stated in this form: (1) some (perhaps zero) premises, (2) a turnstile symbol Failed to parse (Missing texvc executable; please see math/README to configure.): \vdash , which means "infers", "proves" or "concludes", (3) a conclusion. This usually embodies the relational (as opposed to functional) view of a rule of inference, where the turnstile stands for a deducibility relation holding between premises and conclusion.

Rules of inference must be distinguished from axioms of a theory. In terms of semantics, axioms are valid assertions. Axioms are usually regarded as starting points for applying rules of inference and generating a set of conclusions. Or, in less technical terms:

Rules are statements ABOUT the system, axioms are statements IN the system. For example:

The RULE that from Failed to parse (Missing texvc executable; please see math/README to configure.): \vdash p

you can infer Failed to parse (Missing texvc executable; please see math/README to configure.): \vdash\text{Provable}(p)
is a statement that says if you've proven p, then it is provable that p is provable. This holds in Peano arithmetic, for example.

The Axiom Failed to parse (Missing texvc executable; please see math/README to configure.): p \to \text{Provable}(p)

would mean that every true statement is provable. This does not hold in Peano arithmetic.

Rules of inference play a vital role in the specification of logical calculi as they are considered in proof theory, such as the sequent calculus and natural deduction.

References [link]

^ Boolos, George; Burgess, John; Jeffrey, Richard C. (2007). Computability and logic. Cambridge: Cambridge University Press. p. 364. ISBN 0-521-87752-0.

Logic

Overview

Academic areas	Argumentation theory Axiology Critical thinking Computability theory Formal semantics History of logic Informal logic Logic in computer science Mathematical logic Mathematics Metalogic Metamathematics Model theory Philosophical logic Philosophy Philosophy of logic Philosophy of mathematics Proof theory Set theory

Foundational concepts	Abduction Analytic truth Antinomy A priori Deduction Definition Description Entailment Induction Inference Logical consequence Logical form Logical implication Logical truth Name Necessity Meaning Paradox Possible world Presupposition Probability Reason Reasoning Reference Semantics Statement Strict implication Substitution Syntax Truth Truth value Validity

Philosophical logic

Critical thinking and Informal logic	Analysis Ambiguity Argument Belief Bias Credibility Evidence Explanation Explanatory power Fact Fallacy Inquiry Opinion Parsimony Premise Propaganda Prudence Reasoning Relevance Rhetoric Rigor Vagueness

Theories of deduction	Constructivism Dialetheism Fictionalism Finitism Formalism Intuitionism Logical atomism Logicism Nominalism Platonic realism Pragmatism Realism

Metalogic and metamathematics

Cantor's theorem Church's theorem Church's thesis Consistency Effective method Foundations of mathematics Gödel's completeness theorem Gödel's incompleteness theorems Soundness Completeness Decidability Interpretation Löwenheim–Skolem theorem Metatheorem Satisfiability Independence Type–token distinction Use–mention distinction

Mathematical logic

General	Formal language Formation rule Formal system Deductive system Formal proof Formal semantics Well-formed formula Set Element Class Classical logic Axiom Natural deduction Rule of inference Relation Theorem Logical consequence Axiomatic system Type theory Symbol Syntax Theory

Traditional logic	Proposition Inference Argument Validity Cogency Syllogism Square of opposition Venn diagram

Propositional calculus and Boolean logic	Boolean functions Propositional calculus Propositional formula Logical connectives Truth tables

Predicate	First-order Quantifiers Predicate Second-order Monadic predicate calculus

Set theory	Set Empty set Enumeration Extensionality Finite set Function Subset Power set Countable set Recursive set Domain Range Ordered pair Uncountable set

Model theory	Model Interpretation Non-standard model Finite model theory Truth value Validity

Proof theory	Formal proof Deductive system Formal system Theorem Logical consequence Rule of inference Syntax

Computability theory	Recursion Recursive set Recursively enumerable set Decision problem Church–Turing thesis Computable function Primitive recursive function

Non-classical logic

Modal logic	Alethic Axiologic Deontic Doxastic Epistemic Temporal

Intuitionism	Intuitionistic logic Constructive analysis Heyting arithmetic Intuitionistic type theory Constructive set theory

Fuzzy logic	Degree of truth Fuzzy rule Fuzzy set Fuzzy finite element Fuzzy set operations

Substructural logic	Structural rule Relevance logic Linear logic

Paraconsistent logic	Dialetheism

Description logic	Ontology Ontology language

Logicians

Anderson Aristotle Averroes Avicenna Bain Barwise Bernays Boole Boolos Cantor Carnap Church Chrysippus Curry De Morgan Frege Geach Gentzen Gödel Hilbert Kleene Kripke Leibniz Löwenheim Peano Peirce Putnam Quine Russell Schröder Scotus Skolem Smullyan Tarski Turing Whitehead William of Ockham Wittgenstein Zermelo

Lists

Topics	Outline of logic Index of logic articles Mathematical logic Boolean algebra Set theory

Other	Logicians Rules of inference Paradoxes Fallacies Logic symbols

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Branches of Mathematics

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Applied mathematics

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Contents

Overview [link]

Admissibility and derivability [link]

Other considerations [link]

See also [link]

References [link]

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