Page 275 - Encyclopedia of Philosophy of Language

dition the main objective is to gain insight into the processes by which conditionals come to be expressed in new ways and how they come to express new func- tions. Traugott (1985) suggests the following set of nonconditional sources of conditional markers: (a) modals of possibility, doubt, and wish; (b) inter- rogatives; (c) copulas, typically of the existential kind; (d) topic markers and demonstratives; and (e) temporals. How could the change from any of these sources to conditionals take place? As already indi- cated above, the antecedent of a conditional raises the possibility of some alternative situation, which is subsequently treated as a conditional constraint on the consequent. It seems plausible that such alternative situations were originally indicated by the diacritics listed above, which then came to be the conventional means for expressing the fact that a conditional con- straint was posited.
2. Truth Conditional Semantics
In the logical-philosophical tradition the problem of conditionals is addressed by abstracting away from the way conditionals are expressed in everyday language. Here the aim is to give a systematic account of their logical properties.
2.1 Material Implication
The oldest theory of conditionals states that / / . . . then is just the so-called material implication. According to this theory, first proposed by the Megarian Philo (fourth century BC),a conditional sentence is true if and only if its antecedent is false or its consequent is true:
Ifp thenqistrueifandonlyifpisfalseorqistrue. (3)
In introductory logic courses this truth condition is usually motivated by pointing out that classical logic leaves one with no alternative. Given that the language of classical logic is truth-functional, and that within this framework every sentence has exactly one of the truth values 'true' and 'false,' there isjust no room for if... then to mean anything else than (3) says.
This motivation will only appeal to those who, for some reason or other, believe that classical logic is the only correct logic. Others might prefer the conclusion that natural language conditionals cannot properly be analyzed within the framework of classical logic. There are strong arguments in favor of this position.
quent false (as (5) wants to have it), but at best that this is possibly so.
2.2 Strict Implication
Examples like the above naturally lead to the idea that theif... thenofnaturallanguageisastrictimplication rather than the material one. The example suggests that a conditional sentence is false if it is possible for its antecedent to be true while its consequent is false. By adding to this that it is true otherwise one gets:
Ifp then q is true iff it is necessarily so that p is (6) false or q is true.
Restated in the language of possible worlds semantics (see Intension), this becomes (7):
Ifp then q is true if q is true in every possible (7) world in which p is true.
There are various ways to make this truth definition more precise, depending on how one interprets 'poss- ible.' But whatever interpretation one prefers—logi- cally possible, or physically possible, or whatever else—difficulties arise. The theory of strict implication runs into similar problems as the theory of material implication by validating patterns of inference of which it is not immediately evident that they accord with the actual use of conditionals. One such pattern is the principle of 'strengthening the antecedent.' As a counterexample against the latter, one could suggest that from (8):
If I put sugar in my coffee, it will taste better. (8) it does not follow that (9):
If I put sugar and diesel oil in my coffee, it will (9) taste better.
An analysis of this example along the lines suggested by (7) yields, however, that this argument is valid. Hence (7) calls for refinement.
2.3 Variable Strict Implication
The next truth condition was first proposed in the late 1960s by Robert Stalnaker (10):
Ifp then q is true if q is true in every possible (10) world in which (i) p is true, and which (ii) otherwise differs minimallyfrom the actual world.
It is easy to see how this amendment to (7) blocks the inference from Ifp, then r to Ifp andq, then r. Consider the set S of worlds in which (i) p is true and which (ii) in other respects differ minimally from the actual world. It could very well be that q is false in all these worlds. If so, the set T of worlds in which (i) both p and q are true, but which (ii) in other respects differ minimally from the actual world will not be a subset of 5. So, r could be true in every world in S, but false in some of the worlds in T.
Definition (10) is the heart of what is the most 253
For example, within classical logic the propositions a
—\(.P~*<J) »d (/JA—#) are equivalent. But the fol- lowing two sentences clearly are not:
It is not true that if you study hard, you will pass (4) your exam.
You will study hard, and you will not pass (5) your exam.
Sentence (4) does not say that it is in fact the case that the antecedent of the conditional is true and its conse-
Conditionals