Material Conditional

The material conditional is a truth-functional operator used in logic. It is generally expressed by the horseshoe symbol, ⊃.

Its truth values are given by the following table:

Material Conditional

A	C	A⊃C	¬A∨C
T	T	T	T
T	F	F	F
F	T	T	T
F	F	T	T

That is to say, its truth value is entirely determined by the truth values of A and C. As, demonstrated by the above table, A⊃C is equivalent to ¬A∨C. The material conditional is well understood. Its truth-functions determine how it works.

The Equivalence Thesis

Some philosophers have attempted to defend the Equivalence Thesis: the indicative conditional is equivalent to the material conditional, or A⊃C=A→C. This is also referred to as "the horseshoe analysis". This position has famously been held by H. P. Grice and Frank Jackson. There are strong motivations for the equivalence thesis. We know all there is to know about the material conditional. If the indicative conditional is simply the material conditional, then we have powerful insights into everyday usage.

The Equivalence Thesis also has some intuitive appeal, as pointed out by Frank Jackson. Consider the following example: I tell Claudia, "Either they served coffee, or they served tea." Claudia seems perfectly justified in reporting to Lucy, "If they didn't serve coffee, then they served tea." So, A∨C entails ¬A→C. But, Jonathan Bennett shows us that by substituting ¬A for A, I get ¬A∨C entails ¬¬A→C, which means simply A⊃C entails A→C. And we already know that A→C, if anything, is stronger than A⊃C. If A→C entails A⊃C and A⊃C entails A→C, then the two are equivalent, which is precisely what the Equivalence Thesis suggests.

Despite its apparent appeal, most philosophers are now convinced that the Equivalence Thesis is false.