||This article may be confusing or unclear to readers. (May 2013)|
The material conditional (also known as material implication, material consequence, or simply implication, implies, or conditional) is a logical connective (or a binary operator) that is often symbolized by a forward arrow "→". The material conditional is used to form statements of the form 𝑝 → 𝑞 (termed a conditional statement) which is read as "if 𝑝 then 𝑞" or "𝑝 only if 𝑞". It is conventionally compared to the English construction "If...then...". However, unlike the English construction, the material conditional statement 𝑝 → 𝑞 does not specify a causal relationship between 𝑝 and 𝑞. It is merely to be understood to mean "if 𝑝 is true, then 𝑞 is also true" such that the statement 𝑝 → 𝑞 is false only when 𝑝 is true and 𝑞 is false. Intuitively, consider that the statement "if 𝑝 is true, 𝑞 is always also true" is false when 𝑝 is true and 𝑞 is false — even when "if 𝑝 then 𝑞" does not represent a causal relationship between 𝑝 and 𝑞. Instead, the statement describes 𝑝 and 𝑞 as each only being true when the other is true and makes no claim that 𝑝 causes 𝑞. However, note that such a general and informal way of thinking about the material conditional is not always acceptable, as will be discussed. 
The material conditional is also symbolized using:
- 𝑝 ⊃ 𝑞 (Although this symbol may be used for the superset symbol in set theory.);
- 𝑝 ⇒ 𝑞 (Although this symbol is often used for logical consequence (i.e., logical implication) rather than for material conditional.)
- C𝑝𝑞 (using Łukasiewicz notation)
With respect to the material conditionals above:
Conditional statements may be nested such that either or both of the antecedent or the consequent may themselves be conditional statements. In the example "(p→q) → (r→s)" both the antecedent and the consequent are conditional statements.
In classical logic is logically equivalent to and by De Morgan's Law logically equivalent to . Whereas, in minimal logic (and therefore also intuitionistic logic) only logically entails ; and in intuitionistic logic (but not minimal logic) entails .
Definitions of the material conditional
Logicians have many different views on the nature of material implication and approaches to explain its sense.
As a truth function
In classical logic, the compound p→q is logically equivalent to the negative compound: not both p and not q. Thus the compound p→q is false if and only if both p is true and q is false. By the same stroke, p→q is true if and only if either p is false or q is true (or both). Thus → is a function from pairs of truth values of the components p, q to truth values of the compound p→q, whose truth value is entirely a function of the truth values of the components. Hence, this interpretation is called truth-functional. The compound p→q is logically equivalent also to ¬p∨q (either not p, or q (or both)), and to ¬q→¬p (if not q then not p). But it is not equivalent to ¬p→¬q, which is equivalent to q→p.
The truth table associated with the material conditional p→q is identical to that of ¬p∨q. It is as follows:
It may also be useful to note that in Boolean algebra, true and false can be denoted as 1 and 0 respectively with an equivalent table.
As a formal connective
The material conditional can be considered as a symbol of a formal theory, taken as a set of sentences, satisfying all the classical inferences involving →, in particular the following characteristic rules:
Unlike the truth-functional one, this approach to logical connectives permits the examination of structurally identical propositional forms in various logical systems, where somewhat different properties may be demonstrated. For example, in intuitionistic logic which rejects proofs by contraposition as valid rules of inference, (p → q) ⇒ ¬p ∨ q is not a propositional theorem, but the material conditional is used to define negation.
When studying logic formally, the material conditional is distinguished from the semantic consequence relation . We say if every interpretation that makes A true also makes B true. However, there is a close relationship between the two in most logics, including classical logic. For example, the following principles hold:
- If then for some . (This is a particular form of the deduction theorem. In words, it says that if Γ models ψ this means that ψ can be deduced just from some subset of the theorems in Γ.)
- 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.)
Other properties of implication (the following expressions are always true, for any logical values of variables):
- 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.
- commutativity of antecedents:
Comparison of Boolean truth tables shows that is equivalent to , and one is an equivalent replacement for the other in classical logic. See material implication (rule of inference).
Philosophical problems with material conditional
Outside of mathematics, it is a matter of some controversy as to whether the truth function for material implication provides an adequate treatment of conditional statements in English (a sentence in the indicative mood with a conditional clause attached, i.e., an indicative conditional, or false-to-fact sentences in the subjunctive mood, i.e., a counterfactual conditional). That is to say, critics argue that in some non-mathematical cases, the truth value of a compound statement, "if p then q", is not adequately determined by the truth values of p and q. Examples of non-truth-functional statements include: "q because p", "p before q" and "it is possible that p". "[Of] the sixteen possible truth-functions of A and B, material implication is the only serious candidate. First, it is uncontroversial that when A is true and B is false, "If A, B" is false. A basic rule of inference is modus ponens: from "If A, B" and A, we can infer B. If it were possible to have A true, B false and "If A, B" true, this inference would be invalid. Second, it is uncontroversial that "If A, B" is sometimes true when A and B are respectively (true, true), or (false, true), or (false, false)… Non-truth-functional accounts agree that "If A, B" is false when A is true and B is false; and they agree that the conditional is sometimes true for the other three combinations of truth-values for the components; but they deny that the conditional is always true in each of these three cases. Some agree with the truth-functionalist that when A and B are both true, "If A, B" must be true. Some do not, demanding a further relation between the facts that A and that B."
The truth-functional theory of the conditional was integral to Frege's new logic (1879). It was taken up enthusiastically by Russell (who called it "material implication"), Wittgenstein in the Tractatus, and the logical positivists, and it is now found in every logic text. It is the first theory of conditionals which students encounter. Typically, it does not strike students as obviously correct. It is logic's first surprise. Yet, as the textbooks testify, it does a creditable job in many circumstances. And it has many defenders. It is a strikingly simple theory: "If A, B" is false when A is true and B is false. In all other cases, "If A, B" is true. It is thus equivalent to "~(A&~B)" and to "~A or B". "A ⊃ B" has, by stipulation, these truth conditions.
The meaning of the material conditional can sometimes be used in the natural language English "if condition then consequence" construction (a kind of conditional sentence), where condition and consequence are to be filled with English sentences. However, this construction also implies a "reasonable" connection between the condition (protasis) and consequence (apodosis) (see Connexive logic).
The material conditional can yield some unexpected truths when expressed in natural language. For example, any material conditional statement with a false antecedent is true (see vacuous truth). 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 I have a penny in my pocket then Paris is in France" is always true, regardless of whether or not there is a penny in my pocket. 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. These unexpected truths arise because speakers of English (and other natural languages) are tempted to equivocate between the material conditional and the indicative conditional, or other conditional statements, like the counterfactual conditional and the material biconditional. It is not surprising that a rigorously defined truth-functional operator does not correspond exactly to all notions of implication or otherwise expressed by 'if...then...' sentences in English (or their equivalents in other natural languages). For an overview of some the various analyses, formal and informal, of conditionals, see the "References" section below.
- Magnus, P.D (January 6, 2012). "forallx: An Introduction to Formal Logic" (PDF). Creative Commons. p. 25. Retrieved 28 May 2013.
- Teller, Paul (January 10, 1989). "A Modern Formal Logic Primer: Sentence Logic Volume 1" (PDF). Prentice Hall. p. 54. Retrieved 28 May 2013.
- Clarke, Matthew C. (March 1996). "A Comparison of Techniques for Introducing Material Implication". Cornell University. Retrieved March 4, 2012.
- Edgington, Dorothy (2008). Edward N. Zalta, ed. "Conditionals". The Stanford Encyclopedia of Philosophy (Winter 2008 ed.).
- 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.
- Quine, W.V. (1982), Methods of Logic, (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), 4th edition, Harvard University Press, Cambridge, MA.
- Stalnaker, Robert, "Indicative Conditionals", Philosophia, 5 (1975): 269–286.