The material conditional, also known as the material implication or truth functional conditional, expresses a property of certain conditionals in logic. In propositional logic, it expresses a binary truth function ⊃ from truth-values to truth-values. In predicate logic, it can be viewed as a subset relation between the extension of (possibly complex) predicates. In symbols, a material conditional is written as one of the following: Look up conditional in Wiktionary, the free dictionary. ... Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. ... Propositional logic or sentential logic is the logic of propositions, sentences, or clauses. ... ...

1.
2.

The material conditional is false when X is true and Y is false - otherwise, it is true. (Here, X and Y are variables ranging over formulæ of a formal theory.) We call X the antecedent, and Y the consequent. The material conditional is also commonly referred to as material implication with the understanding that the antecedent (X) materially implies the consequent (Y). In mathematics and in the sciences, a formula (plural: formulae, formulæ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ... In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. ...

A distant approximation to the material conditional is the English construction 'if...then...', where the ellipses are to be filled with English sentences. However, this is the most common reading of the material conditional in English. A closer approximation to X → Y is 'it's false that X be true while Y false'—i.e., in symbols, . Arguably this is more intuitive than its logically equivalent disjunction ¬X ∨ Y.

Definition

Logical implication and the material conditional are both associated with an operation on two logical values, typically the values of two propositions, that produces a value of false just in case the first operand is true and the second operand is false. In logical calculus of mathematics, the logical conditional (also known as the material implication, sometimes material conditional) is a binary logical operator connecting two statements, if p then q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ... In mathematics, a finitary boolean function is a function of the form f : Bk → B, where B = {0, 1} is a boolean domain and where k is a nonnegative integer. ... In logic and mathematics, a logical value, also called a truth value, is a value indicating to what extent a proposition is true. ... This article is about the word proposition as it is used in logic, philosophy, and linguistics. ...

Truth table

The truth table associated with the material conditional if p then q (symbolized as p → q) and the logical implication p implies q (symbolized as p ⇒ q) is as follows:

Johnston diagram

The Johnston diagram of "If A then B" Johnston diagrams, which look similar to Euler or Venn diagrams, illustrate formal propositional logic in a visual manner. ...

Image File history File links Johnston_Diagram-_A_implies_B.svg‎ I, the copyright holder of this work, hereby release it into the public domain. ...

Formal properties

The material conditional is not to be confused with the entailment relation ⊨ (which is used here as a name for itself). But there is a close relationship between the two in most logics, including classical logic which we only consider here. For example, the following principles hold: Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. ...

• If then for some . (This is a particular form of the deduction theorem.)

• The converse of the above

• Both ⊃ and ⊨ are monotonic; i.e., if then , and if then for any α, Δ. (In terms of structural rules, this is often referred to as weakening or thinning.)

These principles do not hold in all logics, however. Obviously they do not hold in non-monotonic logics, nor do they hold in relevance logics. In mathematical logic, the deduction theorem states that if a formula F is deducible from E then the implication E → F is demonstrable (i. ... In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ... A structural rule or structural principle of mathematical logic that states that the hypotheses of any derived fact may be freely extended with additional assumptions. ... A non-monotonic logic is a formal logic whose consequence relation is not monotonic. ... Relevance logic, also called relevant logic, is any of a family of non-classical substructural logics that impose certain restrictions on implication. ...

Other properties of implication:

• associativity:

• distributivity:

• transitivity: (

• commutativity: (

• idempotency:

• truth preserving : The interpretation under which all variables are assigned a truth value of 'true' produces a truth value of 'true' as a result of material implication.

In mathematics, associativity is a property that a binary operation can have. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ... In grammar, a verb is transitive if it takes an object. ... Example showing the commutativity of addition (3 + 2 = 2 + 3) For other uses, see Commute (disambiguation). ... In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ...

Philosophical problems with material conditional

The truth function ⊃ does not correspond exactly to the English 'if...then...' construction. For example, any material conditional statement with a false antecedent is true. So the statement "if 2 is odd then 2 is even" is true. Similarly, any material conditional with a true consequent is true. So the statement, "if Pigs fly then Paris is in France" is true. These problems are known as the paradoxes of material implication, though they are not really paradoxes in the strict sense; that is, they do not elicit logical contradictions. The paradoxes of material implication are a group of formulas recognized as logical truths in classical logical theory, but which strike common intuition as somewhat questionable or even downright wrong. ...

There are various kinds of conditionals in English; e.g., there is the indicative conditional and the subjunctive or counterfactual conditional. The latter do not have the same truth conditions as the material conditional. For an overview of some the various analyses, formal and informal, of conditionals, see the "References" section below. The indicative conditional is the logical operation given by statements of the form If A then B in ordinary English (or similar natural languages). ... A counterfactual conditional (sometimes called a subjunctive conditional) is a logical conditional statement whose antecedent is (ordinarily) taken to be contrary to fact by those who utter it. ...

References

• Brown, Frank Markham (2003), Boolean Reasoning: The Logic of Boolean Equations, 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003.

• Edgington, Dorothy (2001), "Conditionals", in Lou Goble (ed.), The Blackwell Guide to Philosophical Logic, Blackwell.

• Edgington, Dorothy (2006), "Conditionals", in Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy, Eprint.

• Quine, W.V. (1982), Methods of Logic, (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), 4th edition, Harvard University Press, Cambridge, MA.

W. V. Quine Willard Van Orman Quine (June 25, 1908 - December 25, 2000) was one of the most influential American philosophers and logicians of the 20th century. ...

See also

Conditionals

A counterfactual conditional (sometimes called a subjunctive conditional) is a logical conditional statement whose antecedent is (ordinarily) taken to be contrary to fact by those who utter it. ... The indicative conditional is the logical operation given by statements of the form If A then B in ordinary English (or similar natural languages). ... In logic a corresponding conditional is a statement whose principal connective is the material implication symbol, and whose antecedent is the conjunction of the premises or an argument and whose consequent is the conclusion of that argument. ... In logic, a strict conditional is a material conditional that is acted upon by the necessity operator from modal logic. ... In logical calculus of mathematics, the logical conditional (also known as the material implication, sometimes material conditional) is a binary logical operator connecting two statements, if p then q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ...

Related topics