The material conditional, also known as the material implication or truth functional conditional, expresses a property of certain conditionals in logic. Logic is the study of the principles of valid demonstration and Inference. In propositional logic, it expresses a binary truth function from truth-values to truth-values. This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic" In predicate logic, it can be viewed as a subset relation between the extension of (possibly complex) predicates. In Mathematical logic, predicate logic is the generic term for symbolic Formal systems like First-order logic, Second-order logic, Many-sorted In symbols, a material conditional is written as one of the following:

1. ,
2. , and sometimes
3. 

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. In Mathematics and in the Sciences a formula (plural formulae, formulæ or formulas) is a concise way of expressing information 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 ) 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). A consequent is the second half of a hypothetical Proposition.

The meaning of the material conditional is encapsulated in the natural language English "if condition then consequence" construction, where condition and consequence are to be filled with English sentences. In the Philosophy of language, a natural language (or ordinary language) is a Language that is spoken or written in phonemic-alphabetic or phonemically-related However, this construction also implies a "reasonable" connection between the condition (protasis) and consequence (apodosis) (see Connexive logic). In Linguistics, a protasis is the subordinate Clause (the if -clause in a Conditional sentence. Connexive logic names one class of alternative or non-classical logics designed to exclude the so-called Paradoxes of material implication. So, although a material conditional from a contradiction is always true, in natural language, "If there are three hydrogen atoms in H₂O then the government will lose the next election" is interpreted as false by most speakers, since assertions from chemistry are considered irrelevant conditions for proposing political consequences. "If P then Q", in natural language, appears to mean "P and Q are connected and P→Q". Just what kind of connection is meant by the natural language is not clearly defined.

• The statement "if (B) all bachelors are unmarried then (C) the speed of light is constant" is false, because there is no discernible connection between (B) and (C), even though (B)→(C) is true. A bachelor is a man above the Age of majority who has never been married (see single)
• The statement "if (S) Socrates was a woman then (T) 1+1=3" is false, for the same reason; even though (S)→(T) is true. SOCRATES is the European Community action programme in the field of Education.

When protasis and apodosis are connected, the truth functionality of linguistic and logical conditionals coincide; the distinction is only apparent when the material conditional is true, but its antecedent and consequent are perceived to be unconnected.

The modifier material in material conditional makes the distinction from linguistic conditionals explicit. It isolates the underlying, unambiguous truth functional relationship. Therefore, exact natural language encapsulation of the material conditional X → Y, in isolation, is seen to be "it's false that X be true while Y false" — i. e. in symbols, . Arguably this is more intuitive than its logically equivalent disjunction .

Contents 1 Definition 1.1 Truth table 1.2 Johnston diagram 2 Formal properties 3 Philosophical problems with material conditional 4 References 5 See also 5.1 Conditionals 5.2 Related topics

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 Logic and Mathematics, logical implication is a logical relation that holds between a set T of formulae and a formula B when every Table of logic symbolsIn Logic, two sentences (either in a formal language or a natural language may be joined by means of a logical connective to form a compound sentence In Logic and Mathematics, a logical value, also called a truth value, is a value indicating the extent to which a Proposition is true In Logic and Philosophy, proposition refers to either (a the content or Meaning of a meaningful Declarative sentence

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:

p	q	→
T	T	T
T	F	F
F	T	T
F	F	T

Johnston diagram

The Johnston diagram of - "If A then B" - where the white portion indicates the space in which the relation is false. Johnston diagrams, which look similar to Euler or Venn diagrams illustrate formal propositional logic in a visual manner

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. Classical logic identifies a class of Formal logics that have been most intensively studied and most widely used For example, the following principles hold:

• If then for some . (This is a particular form of the deduction theorem. In Mathematical logic, the deduction theorem states that if a formula F is deducible from E then the implication E → F is demonstrable (i )

• 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. Monotonicity of Entailment is a property of many Logical systems that states that the hypotheses of any derived fact may be freely extended with additional assumptions )

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. A non-monotonic logic is a Formal logic whose consequence relation is not monotonic. Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications be relevantly related

Other properties of implication:

• 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, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law In Mathematics, commutativity is the ability to change the order of something without changing the end result Idempotence ˌaɪdɨmˈpoʊtəns describes the property of operations in Mathematics and Computer science which means that multiple applications of the operation

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. Implication, in logic describes conditional if-then statements e

There are various kinds of conditionals in English; e. g. , there is the indicative conditional and the subjunctive or counterfactual conditional. In Natural languages an indicative conditional is the logical operation given by statements of the form "If A then B" A counterfactual conditional, Subjunctive conditional or remote conditional is a conditional (or "if-then" statement indicating what would be 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.

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. Willard Van Orman Quine (June 25 1908 Akron, Ohio &ndash December 25 2000 (known to intimates as "Van" 1950), (2nd ed. 1959), (3rd ed. 1972), 4th edition, Harvard University Press, Cambridge, MA.

• Stalnaker, Robert. Robert Culp Stalnaker is Laurance S Rockefeller Professor of Philosophy at the Massachusetts Institute of Technology. 'Indicative Conditionals'. Philosophia 5 (1975): 269–286.

See also

Conditionals

Counterfactual conditional Indicative conditional Corresponding conditional (logic) Strict conditional Logical implication

Related topics

Ampheck Boolean algebra Boolean domain Boolean function

Boolean logic Laws of Form Logic gate Logical graph

Peirce's law Propositional logic Sole sufficient operator Conditional quantifier

A counterfactual conditional, Subjunctive conditional or remote conditional is a conditional (or "if-then" statement indicating what would be In Natural languages an indicative conditional is the logical operation given by statements of the form "If A then B" In Logic, a strict conditional is a Material conditional that is acted upon by the necessity operator from Modal logic. In Logic and Mathematics, logical implication is a logical relation that holds between a set T of formulae and a formula B when every In Boolean logic, logical nor or joint denial is a truth-functional operator which produces a result that is the inverse of logical or. Boolean algebra (or Boolean logic) is a logical calculus of truth values, developed by George Boole in the late 1830s In Mathematics and Abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include false and In Mathematics, a (finitary Boolean function is a function of the form f: B k &rarr B, where B  = {0 1} Boolean logic is a complete system for Logical operations It was named after George Boole, who first defined an algebraic system of Laws of Form (hereinafter LoF) is a book by G Spencer-Brown, published in 1969 that straddles the boundary between Mathematics and of A logic gate performs a logical operation on one or more logic inputs and produces a single logic output A logical graph is a special type of diagramatic structure in any one of several systems of graphical syntax that Charles Sanders Peirce developed for Peirce's law in Logic is named after the Philosopher and Logician Charles Sanders Peirce. This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic" A sole sufficient operator or a sole sufficient connective is an operator that is sufficient by itself to generate all of the operators in a specified class of operators In Logic, a conditional quantifier is a kind of Lindström quantifier (or Generalized quantifier) Q_A that relative to a classical model A

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