Material conditional

The material conditional, also known as material implication, is a binary truth function, such that the compound sentence p→q (typically read "if p then q" or "p implies q") is logically equivalent to the negative compound: not (p and not q). A material conditional compound itself is often simply called a conditional. By definition of "→", the compound p→q is false if and only if both p is true and q is false. That is to say that 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. Thus p→q is said to be truth-functional. 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 not to ¬p → ¬q. For convenience, p→q is typically read "If p, then q" or "q if p". Saying "It is false that if p then q" does not always sound logically equivalent in everyday English to saying "both p and not q" but, when used in logic, it is taken as logically equivalent. (Other senses of English "if...then..." require other logical forms.) The material implication between two sentences p, q is typically symbolized as

1. $p \rightarrow q$;
2. $p \supset q$;
3. $p \Rightarrow q$ (Typically used for logical implication rather than for material implication.)

As placed within the material conditionals above, p is known as the antecedent, and q as the consequent, of the conditional. One can also use compounds as components, for example pq → (r→s). There, the compound pq (short for "p and q") is the antecedent, and the compound r→s is the consequent, of the larger conditional of which those compounds are components.

The material conditional may also be viewed, not as a truth function, but 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:

1. Modus ponens;
2. Conditional proof;
3. Classical contraposition;
4. Classical reductio.

Definition

The material conditional is associated with an operation on two logical values, typically the values of two propositions, that produces a value of false just in the case when the first operand is true and the second operand is false.

Truth table

The truth table associated with the material conditional not p or q (symbolized as p → q) and the logical implication p implies q (symbolized as p → q, or Cpq) is as follows:

p	q	p → q
1	1	1
1	0	0
0	1	1
0	0	1

Formal properties

When studying logic formally, the material conditional is distinguished from the entailment relation $\models$ which is typically defined semantically: $A \models B$ 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 $\Gamma\models\psi$ then $\emptyset\models\phi_1\land\dots\land\phi_n\rightarrow\psi$ for some $\phi_1,\dots,\phi_n\in\Gamma$. (This is a particular form of the deduction theorem.)

• The converse of the above

• Both $\rightarrow$ and ⊨ are monotonic; i.e., if $\Gamma\models\psi$ then $\Delta\cup\Gamma\models\psi$, and if $\phi\rightarrow\psi$ then $(\phi\land\alpha)\rightarrow\psi$ 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.

Other properties of implication:

• distributivity: $(s \rightarrow (p \rightarrow q)) \rightarrow ((s \rightarrow p) \rightarrow (s \rightarrow q))$

• transitivity: $(a \rightarrow b) \rightarrow ((b \rightarrow c) \rightarrow (a \rightarrow c))$

• idempotency: $a \rightarrow a$

• 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: $(a \rightarrow (b \rightarrow c)) \equiv (b \rightarrow (a \rightarrow c))$

Note that $a \rightarrow (b \rightarrow c)$ is logically equivalent to $(a \and b) \rightarrow c$; this property is sometimes called currying. Because of these properties, it is convenient to adopt a right-associative notation for →.

Philosophical problems with material conditional

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).

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 in a vacuum is constant" may be considered false, because there is no discernible connection between (B) and (C), even though (B)→(C) is true.
• The statement "if (S) Socrates was a woman then (T) 1+1=3" may be considered false, for the same reason; even though (S)→(T) is true.

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" or "it cannot be that X AND not-Y" — i.e. in symbols, $\neg(X \and \neg Y)$.

The truth function $\rightarrow$ corresponds to 'not ... or ...' and does not correspond 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.

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.

See also

Ampheck Boolean algebra Boolean domain Boolean function Boolean logic

Implicational propositional calculus Laws of Form Logic gate Logical graph Paradoxes of material implication

Peirce's law Propositional logic Sole sufficient operator Conditional quantifier

Conditionals

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

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.
• Stalnaker, Robert, "Indicative Conditionals", Philosophia, 5 (1975): 269–286.