Introduction

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The Material Conditional

In its simplest form in English, a conditional is a sentence of the form “If dots then dots,” where the dots are themselves sentences, such as “If the butler did it, then the gardener is innocent.” In introductory logic courses, we earn to symbolize conditionals using the $\lif$source connective: symbolize the parts indicated by dots, e.g., by formulas $!A$source and $!B$source, and the entire conditional is symbolized by $!A \lif !B$source.

The connective $\lif$source is truth-functional, i.e., the truth value---$\True$source or $\False$source---of $!A \lif !B$source is determined by the truth values of $!A$source and $!B$source: $!A \lif !B$source is true iff $!A$source is false or $!B$source is true, and false otherwise. Relative to a truth value assignment $\pAssign v$source, we define $\pSat{v}{!A \lif !B}$source iff $\pSat/{v}{!A}$source or $\pSat{v}{!B}$source. The connective $\lif$source with this semantics is called the material conditional.

This definition results in a number of elementary logical facts. First of all, the deduction theorem holds for the material conditional:

It is truth-functional: $!A \lif !B$source and $\lnot !A \lor !B$source are equivalent:

These are all useful and unproblematic inferences in mathematical reasoning. However, the philosophical and linguistic literature is replete with purported counterexamples to the equivalent inferences in non-mathematical contexts. These suggest that the material conditional $\lif$source is not---or at least not always---the appropriate connective to use when symbolizing English “if dots then dots” statements.

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Paradoxes of the Material Conditional

One of the first to criticize the use of $!A \lif !B$source as a way to symbolize “if dots then dots” statements of English was C. I. Lewis. Lewis was criticizing the use of the material condition in Whitehead and Russell's Principia Mathematica, who pronounced $\lif$source as “implies.” Lewis rightly complained that if $\lif$source meant “implies,” then any false proposition $p$source implies that $p$source implies $q$source, since $p \lif (p \lif q)$source is true if $p$source is false, and that any true proposition $q$source implies that $p$source implies $q$source, since $q \lif (p \lif q)$source is true if $q$source is true.

Logicians of course know that implication, i.e., logical entailment, is not a connective but a relation between formulas or statements. So we should just not read $\lif$source as “implies” to avoid confusion.Footnote: Reading “$\lif$source” as “implies” is still widely practised by mathematicians and computer scientists, although philosophers try to avoid the confusions Lewis highlighted by pronouncing it as “only if.” As long as we don't, the particular worry that Lewis had simply does not arise: $p$source does not “imply” $q$source even if we think of $p$source as standing for a false English sentence. To determine if $p \Entails q$source we must consider all valuations, and $p \Entails/ q$source even when we use $p$source to symbolize a sentence which happens to be false.

But there is still something odd about “if dots thendots” statements such as Lewis's

and about the inferences

Yet, if “if dots then dots” were just $\lif$source, the sentence would be unproblematically true, and the inferences unproblematically valid.

Another example of concerns the tautology $(!A \lif !B) \lor (!B \lif !A)$source. This would suggest that if you take two indicative sentences $S$source and $T$source from the newspaper at random, the sentence “If $S$source then $T$source, or if $T$source then $S$source” should be true.

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The Strict Conditional

Lewis introduced the strict conditional $\strictif$source and argued that it, not the material conditional, corresponds to implication. In alethic modal logic, $!A \strictif !B$source can be defined as $\Box(!A \lif !B)$source. A strict conditional is thus true (at a world) iff the corresponding material conditional is necessary.

How does the strict conditional fare vis-a-vis the paradoxes of the material conditional? A strict conditional with a false antecedent and one with a true consequent, may be true, or it may be false. Moreover, $(!A \strictif !B) \lor (!B \strictif !A)$source is not valid. The strict conditional $!A \strictif !B$source is also not equivalent to $\lnot !A \lor !B$source, so it is not truth functional.

We have:

However, the strict conditional still has its own “paradoxes.” Just as a material conditional with a false antecedent or a true consequent is true, a strict conditional with a necessarily false antecedent or a necessarily true consequent is true. Moreover, any true strict conditional is necessarily true, and any false strict conditional is necessarily false. In other words, we have

These are not problems if you think of $\strictif$source as “implies.” Logical entailment relationships are, after all, mathematical facts and so can't be contingent. But they do raise issues if you want to use $\strictif$source as a logical connective that is supposed to capture “if dots then dots,” especially the last two. For surely there are “if dots then dots” statements that are contingently true or contingently false---in fact, they generally are neither necessary nor impossible.

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Counterfactuals

A very common and important form of “if dots then dots” constructions in English are built using the past subjunctive form of to be: “if it were the case that dots then it would be the case that dots” Because usually the antecedent of such a conditional is false, i.e., counter to fact, they are called counterfactual conditionals (and because they use the subjunctive form of to be, also subjunctive conditionals. They are distinguished from indicative conditionals which take the form of “if it is the case that dots then it is the case that dots” Counterfactual and indicative conditionals differ in truth conditions. Consider Adams's famous example:

The first is indicative, the second counterfactual. The first is clearly true: we know President John F. Kennedy was killed by someone, and if that someone wasn't (contrary to the Warren Report) Lee Harvey Oswald, then someone else killed Kennedy. The second one says something different. It claims that if Oswald hadn't killed Kennedy, i.e., if the Dallas shooting had been avoided or had been unsuccessful, history would have subsequently unfolded in such a way that another assassination would have been successful. In order for it to be true, it would have to be the case that powerful forces had conspired to ensure JFK's death (as many JFK conspiracy theorists believe).

It is a live debate whether the indicative conditional is correctly captured by the material conditional, in particular, whether the paradoxes of the material conditional can be “explained” in a way that is compatible with it giving the truth conditions for English indicative conditionals. By contrast, it is uncontroversial that counterfactual conditionals cannot be symbolized correctly by the material conditionals. That is clear because, even though generally the antecedents of counterfactuals are false, not all counterfactuals with false antecedents are true---for instance, if you believe the Warren Report, and there was no conspiracy to assassinate JFK, then Adams's counterfactual conditional is an example.

Counterfactual conditionals play an important role in causal reasoning: a prime example of the use of counterfactuals is to express causal relationships. E.g., striking a match causes it to light, and you can express this by saying “if this match were struck, it would light.” Material, and generally indicative conditionals, cannot be used to express this: “the match is struck $\lif$source the match lights” is true if the match is never struck, regardless of what would happen if it were. Even worse, “the match is struck $\lif$source the match turns into a bouquet of flowers” is also true if it is never struck, but the match would certainly not turn into a bouquet of flowers if it were struck.

It is still debated What exactly the correct logic of counterfactuals is. An influential analysis of counterfactuals was given by Stalnaker and Lewis. According to them, a counterfactual “if it were the case that $S$source then it would be the case that $T$source” is true iff $T$source is true in the counterfactual situation (“possible world”) that is closest to the way the actual world is and where $S$source is true. This is called an “ontic” analysis, since it makes reference to an ontology of possible worlds. Other analyses make use of conditional probabilities or theories of belief revision. There is a proliferation of different proposed logics of counterfactuals. There isn't even a single Lewis--Stalnaker logic of counterfactuals: even though Stalnaker and Lewis proposed accounts along similar lines with reference to closest possible worlds, the assumptions they made result in different valid inferences.