Material conditional
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The material conditional expresses a property of certain conditionals in logic. In propositional logic, it expresses a binary truth function ⊃ from truth-values to truth-values. In predicate logic, it can be viewed as a subset relation between the extension of (possibly complex) predicates. In symbols, a material conditional is written as one of the following:
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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 formulas of a formal theory.) 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 distant approximation to the material conditional is the English construction 'if...then...', where the ellipses are to be filled with English sentences. However, this is the most common reading of the material conditional in English. A closer approximation to X → Y is 'it's false that X be true while Y false'—i.e., in symbols, ¬(X ∧ ¬Y). Arguably this is more intuitive than its logically equivalent disjunction ¬X ∨ Y.
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[edit] 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. For example, the following principles hold:
- If
then 
for some 
. (This is a particular form of the deduction theorem.)
- 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.)
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.
[edit] 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 "2 is odd implies 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.
[edit] 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.
