Material conditional

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

$p\rightarrow q$ ;
$p\supset q$ ;
$p\Rightarrow q$ (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.

Definitions of the material conditional

Mathematicians have many different views on the material conditional and approaches to explain its sense.^[1]

As a truth function

In classical logic, the material conditional is associated with a truth function of 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. 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, 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.

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

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 truth-functional one, this approach to logical connectives permits to examine 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, $(p \to q) \Rightarrow \neg p \lor q$ is not a propositional theorem, but the material conditional is used to define negation.

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 →.

Note also that comparison of truth table shows that $a\rightarrow b$ is equivalent to $\neg a\lor b$ , and it is sometimes convenient to replace one by the other in proofs. Such a replacement can be viewed as a 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).^[2] 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.^[2] Examples of non-truth-functional statements include: "p because q", "p before q" and "it is possible that p".^[2] “[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.”^[2]

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”^[2]

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).^{[citation needed]}

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.

References

^ Clarke, Matthew C. (March 1996). "A Comparison of Techniques for Introducing Material Implication". Cornell University. Retrieved March 04, 2012. {{cite web}}: Check date values in: |accessdate= (help)
^ ^a ^b ^c ^d ^e 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.
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.

[Clarke-1] Clarke, Matthew C. (March 1996). "A Comparison of Techniques for Introducing Material Implication". Cornell University. Retrieved March 04, 2012. {{cite web}}: Check date values in: |accessdate= (help)

[sep-conditionals-2] Edgington, Dorothy (2008). Edward N. Zalta (ed.). "Conditionals". The Stanford Encyclopedia of Philosophy (Winter 2008 ed.).

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v t e Common logical connectives
Tautology/True $\top$
Alternative denial (NAND gate) $\uparrow$ Converse implication $\leftarrow$ Implication (IMPLY gate) $\rightarrow$ Disjunction (OR gate) $\lor$
Negation (NOT gate) $\neg$ Exclusive or (XOR gate) $\not \leftrightarrow$ Biconditional (XNOR gate) $\leftrightarrow$ Statement (Digital buffer)
Joint denial (NOR gate) $\downarrow$ Nonimplication (NIMPLY gate) $\nrightarrow$ Converse nonimplication $\nleftarrow$ Conjunction (AND gate) $\land$
Contradiction/False $\bot$
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