Conditionals and Material Implication

Classical logic treats 'if A then B' as the material conditional, true whenever A is false or B is true — but this generates paradoxes that have driven a large literature on what 'if' really means.

Definition

The material conditional 'A → B' is the truth-function that is false only when A is true and B false; a central question is whether ordinary indicative conditionals have these truth conditions or instead express a relation of conditional probability or possible-worlds dependence.

Scope

This topic covers the analysis of conditional sentences and the adequacy of the truth-functional material conditional as their logical form. It treats the paradoxes of material implication, Gricean pragmatic defences of the truth-functional account, possible-worlds (Stalnaker) semantics, the probabilistic (Adams) thesis that the assertability of a conditional goes by conditional probability, and the contrast between indicative and counterfactual conditionals.

Core questions

• Do ordinary 'if-then' sentences have the truth conditions of the material conditional?
• How should we explain the paradoxes of material implication, such as that a false antecedent makes any conditional true?
• Does the assertability of a conditional track the conditional probability of consequent given antecedent?
• How do indicative conditionals differ from counterfactual (subjunctive) conditionals?

Key concepts

• material conditional
• paradoxes of material implication
• indicative vs. counterfactual conditionals
• conditional probability
• conversational implicature
• possible-worlds selection function

Key theories

Possible-worlds semantics for conditionals

Stalnaker analyzes 'if A then B' as true iff B holds in the most similar possible world(s) in which A holds, replacing the truth-functional account with a selection function over worlds and thereby validating different inferences.

The probabilistic (Adams) thesis

Adams holds that indicative conditionals lack ordinary truth conditions and that their acceptability equals the conditional probability of the consequent given the antecedent, which explains the data better than the material conditional.

History

The truth-functional reading of 'if' goes back to the Stoics and was embedded in modern logic by Frege and Russell. Twentieth-century dissatisfaction with the resulting paradoxes prompted Grice's pragmatic defence (implicature), Stalnaker's 1968 possible-worlds semantics, and Adams's 1975 probabilistic account, with Edgington's 1995 survey crystallizing the contemporary debate.

Debates

Truth-functional vs. non-truth-functional conditionals

Whether the oddities of the material conditional can be explained away pragmatically as Gricean implicatures, preserving truth-functionality, or instead show that indicative conditionals have probabilistic or possible-worlds content rather than material truth conditions.

Key figures

• Robert Stalnaker
• Ernest Adams
• Dorothy Edgington
• H. Paul Grice
• Frank Ramsey

Seminal works

• stalnaker1968
• adams1975
• edgington1995

Frequently asked questions

What are the paradoxes of material implication?

On the truth-functional reading, a conditional is automatically true whenever its antecedent is false or its consequent is true. This makes sentences like 'If the moon is made of cheese, then 2+2=4' come out true, which clashes with ordinary judgment that the antecedent and consequent should be relevant to each other.