Material conditional

"Logical conditional" redirects here. For other related meanings, see Conditional statement (disambiguation).

For the rule of replacement of the same name, see Material implication.

The material conditional, (also known as material implication) is a logical binary compound using a logical connective often symbolized by a forward arrow "→", 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 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, the compound p→q 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.

A material conditional is often simply called a conditional. For convenience, p→q is typically read "If p, then q" or "q if p". In everyday English, saying "It is false that if p then q" is not often taken as flatly equivalent to saying "p is true and q is false" but, when used within logic, those phrasings are 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
T	T	T
T	F	F
F	T	T
F	F	T

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 (following expressions are allways true, for any logical values of variables):

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

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

• reflexivity: $a\rightarrow a$

• 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 $(a\land b)\rightarrow c$; 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

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: "p because q", "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.

— Dorothy Edgington, The Stanford Encyclopedia of Philosophy, “Conditionals”

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, $\neg (X\land \neg Y)$.

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.