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Wikipedia / List of rules of inference

This is a list of rules of inference, logical laws that relate to mathematical formulae.

Introduction

Rules of inference are syntactical transform rules which one can use to infer a conclusion from a premise to create an argument. A set of rules can be used to infer any valid conclusion if it is complete, while never inferring an invalid conclusion, if it is sound. A sound and complete set of rules need not include every rule in the following list, as many of the rules are redundant, and can be proven with the other rules.

Discharge rules permit inference from a subderivation based on a temporary assumption. Below, the notation

.

Rules for propositional calculuspropositional calculus

Rules for negations

  1. Reductio ad absurdum (or Negation Introduction):

    Reductio ad absurdum (related to the law of excluded middle):

    Ex contradictione quodlibet:

Rules for conditionals

  1. Deduction theorem (or Conditional Introduction):

    Modus ponens (a type of Conditional Elimination):

    Modus tollens (a type of Conditional Elimination):

Rules for conjunctions

  1. Adjunction (or Conjunction Introduction):

    Simplification (or Conjunction Elimination):

Rules for disjunctions

  1. Addition (or Disjunction Introduction):

    Case analysis (or Proof by Cases or Argument by Cases or Disjunction elimination)

    Disjunctive syllogism:

    Constructive dilemma

Rules for biconditionals

  1. Biconditional introduction:

    Biconditional elimination:

Rules of classical First-order logicpredicate calculus

.

  1. Universal Generalization (or Universal Introduction):

. is not mentioned in any hypothesis or undischarged assumptions.

  1. Universal Instantiation (or Universal Elimination):

.

  1. Existential Generalization (or Existential Introduction):

.

  1. Existential Instantiation (or Existential Elimination):

. . is not mentioned in any hypothesis or undischarged assumptions.

Rules of substructural logicsubstructural logic

The following are special cases of universal generalization and existential elimination; these occur in substructural logics, such as linear logic.

  1. Rule of weakening (or monotonicity of entailment) (aka no-cloning theorem)

    Rule of contraction (or idempotency of entailment) (aka no-deleting theorem)

Table: Rules of Inference

The rules above can be summed up in the following table. The "Tautology" column shows how to interpret the notation of a given rule.All rules use the basic logic operators. A complete table of "logic operators" is shown by a truth table, giving definitions of all the possible (16) truth functions of 2 boolean variables (p, q):where T = true and F = false, and, the columns are the logical operators:

Each logic operator can be used in an assertion about variables and operations, showing a basic rule of inference. Examples:

  • The column-14 operator (OR), shows Addition rule: when p=T (the hypothesis selects the first two lines of the table), we see (at column-14) that p∨q=T.

    1. We can see also that, with the same premise, another conclusions are valid: columns 12, 14 and 15 are T.

    The column-8 operator (AND), shows Simplification rule: when p∧q=T (first line of the table), we see that p=T.

    1. With this premise, we also conclude that q=T, p∨q=T, etc. as shown by columns 9–15.

    The column-11 operator (IF/THEN), shows Modus ponens rule: when p→q=T and p=T only one line of the truth table (the first) satisfies these two conditions. On this line, q is also true. Therefore, whenever p → q is true and p is true, q must also be true.

Machines and well-trained people use this look at table approach to do basic inferences, and to check if other inferences (for the same premises) can be obtained.

Example 1

be "We will go on a canoe trip tomorrow". Then this argument is of the form:

Example 2

. Using the Rules of Inference table we can prove the conjecture easily:

See also

References

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