The most notorious mismatches between the meaning of a logical connective and the meaning of its natural language counterpart are found with conditional statements. The English language expression “if A then B” is used in a variety of ways, and not all of them map onto the meaning of the conditional as defined in propositional logic.

In the lecture on the logic of conditional statements I said that a conditional is any statement that is logically equivalent to a statement of the form “If A then B”, and I defined the truth conditions for such statements as follows:

“If A then B” is FALSE when the antecedent “A” is true and the consequent “B” is false; for all other combinations of truth values, the conditional is TRUE.

So, for example, given the conditional statement

“If I shoot you with this gun, you will die”

the only scenario in which this statement is FALSE is the case where I shoot you with this gun but you don’t die, i.e. the case where the antecedent is true but the consequent is false.

Now, here’s the odd bit: for all other assignments of truth values, the logical conditional is assigned the value true.

That includes cases where the antecedent is false. On this definition of the conditional, BOTH of the following statements are TRUE:

(1) “If I don’t shoot you with this gun, you will die.”

(2) “If I don’t shoot you with this gun, you won’t die.”

Most of us, looking at these two statements, will wonder how either of these follows from the original conditional claim. It’s not like we have strong intuitions the other way. It’s just … we don’t know why we should feel any particular way about them.

So, right off the bat, we can say that there is a mismatch between the semantics of the logical conditional as defined in propositional logic, and our intuitions about natural language conditionals.

To talk about this further, let’s clarify our language a bit.

The logical conditional, the conditional as defined by the truth table definition in propositional logic, is known as the MATERIAL CONDITIONAL.

The material conditional is what is represented by the various conventional symbolizations used in logic textbooks, usually one of these:

• A → B
• A ⊃ B

The material conditional, by definition, is false is when A is true and B is false, and true otherwise.

This interpretation allows us to write various logical equivalents:

(A → B) (“if A is true then B is true”) is logically equivalent to ~(A & ~B) (“it is not the case that A is true and B is false”).

Also, (A → B) (“if A is true then B is true”) is logically equivalent to (~A ∨ B) (“either A is false or B is true”).

It’s a standard logic exercise to write the truth tables for these expressions to demonstrate that they are indeed logically equivalent.

Now, the question that thousands of logic students have asked their logic instructors, and hundreds of philosophers and logicians have written about, is this:

What is the relationship between the material conditional and conditional statements expressed in natural language?

If you push this question deeply enough it opens up vast areas of research in analytic philosophy.

I’ll give you a little tour of the issues, but I can’t possibly do the topic justice. A little tour is enough to appreciate the general point, which is that there are many uses of “if-then” in natural language that are well described by the material conditional, and there are many other uses that are not.

1. Indicative vs Counterfactual Conditionals

In English there are at least two different kinds of conditional. Consider the following part of English conditionals:

(1) “If humans did not build Stonehenge, then nonhumans did.”

(2) “If humans had not built Stonehenge, then nonhumans would have.”

Conditional (1) seems clearly true.

Conditional (2) seems clearly false. There is no reason to think, for example, that if humans had not built Stonehenge, then aliens would have.

Yet it is natural to regard each of them as composed of two component propositions — “humans did not build Stonehenge” and “nonhumans built Stonehenge” — using a two-place “if-then” connective.

But if one is true and the other is false, then the conditional connective cannot be the same in both cases.

We call the conditional connective in (1) an indicative conditional. Here are some other examples of indicative conditionals:

(3) “If it rains tonight, we shall get wet.”

(4) “If the roof leaked last night, there will be water on the kitchen floor.”

We call the conditional in (2) a subjunctive or counterfactual conditional. Here are some other examples of counterfactual conditionals:

(5) “If it were to rain tonight, we would get wet.”

(6) “If the roof had leaked last night, there would be water on the kitchen floor.”

2. Counterfactual Conditionals and the Material Conditional

It is clear that the counterfactual conditional is NOT correctly translated into propositional logic as the material conditional A → B.

This is because counterfactual conditionals are not truth functional.

Consider the claim “if you had put your sandwich down, a dog would have eaten it.”

Suppose we regard this statement as a compound made up from the counterfactual conditional connective and the two propositions “you put your sandwich down” and “a dog ate your sandwich”.

In a situation in which you ate your sandwich quickly, without putting it down, while surrounded by hungry dogs, both components are false, and the counterfactual conditional is true.

But in a situation in which you ate your sandwich quickly, without putting it down, but there are no hungry dogs present, both components are again false, but the counterfactual conditional is false.

From this it follows that the counterfactual conditional is not a truth functional connective. What determines the truth of the counterfactual conditional isn’t JUST the truth of the component statements. We need to know MORE than just the truth values of the components in that situation.

To handle the semantics of counterfactual, or subjunctive, conditionals, we need more machinery than propositional logic gives us. The truth table for the material conditional won’t do the job.

3. Indicative Conditionals and the Material Conditional

The material conditional connective in propositional logic is intended to model the semantics of indicative conditionals, not counterfactual conditionals.

But even the claim that the indicative conditional is correctly translated as A → B is controversial in logic and philosophy.

On the one hand, one can argue that if the indicative conditional is truth functional at all, it has to be the material conditional, since no other truth table is a serious candidate.

And there are arguments that the semantics of the indicative conditional really does follow the truth table rules.

Consider this conditional statement: “if it rains, then the match will be cancelled”. It seems clear that this implies that “either it will not rain, or it will rain and the match will be cancelled”. But this statement is true if and only if either “it will rain” is false or “the match will be cancelled” is true. That is, if and only if the material conditional is true.

So what’s the problem with translating indicative conditionals using →?

Well, recall that the material conditional A → B is true whenever the antecedent A is false or the consequent B is true (or both). However, the following conditionals all seem quite wrong, even though the first two have false antecedents and the last has a true consequent.

(1) “If my cat has fleas, then my cat’s name is Sam.”

(false antecedent, true consequent)

(2) “If my cat has fleas, then the Earth is the largest planet in the solar system.”

(False antecedent, false consequent)

(3) “If carbon is an element, then the computer I’m typing on is an iMac.”

(true antecedent, true consequent)

If we’re following the truth table for the material conditional, all of these are TRUE statements. Yet they don’t strike us as true, at least at first glance.

The problem seems to arise from the fact that in each case the antecedent has nothing to do with the consequent: believing the antecedent is true gives us no reason to think that the consequent is true.

One might conclude that for an indicative conditional to be true, there must be some connection between the antecedent and the consequent, such that the truth of the former is relevant to the truth of the latter.

If we pursue this line of thought, we’ll be lead to the conclusion that the indicative conditional is not truth functional — whether “if A then B” is true will depend not just on whether A and B are true or false, but also on what they actually say, and in particular, on whether there is the right kind of connection between what A says and what B says.

The alternative is to defend the view that indicative conditionals have the same truth conditions as material conditionals. How? By demonstrating how the problematic examples only seem to get the semantics wrong. There are several well known defenses of this view in the literature but I won’t bother to go into them here.

The take-away point of this discussion is that conditional relationships are expressed in many different ways in natural language, and only some of those ways are well described by the material conditional of propositional logic. Counterfactual conditionals clearly do not follow the semantics of the material conditional. Whether, and what sorts of, indicative conditionals can be interpreted as material conditionals, is a subject of ongoing debate.

However, we should note that when we’re using conditional argument forms like modus ponens and modus tollens in argument analysis, we’re assuming that the conditional follows the logic of the material conditional. Otherwise the following argument forms would not be valid:

1. If A then B
2. A
Therefore, B (modus ponens)

1. If A then B
2. Not-B
Therefore, not-A (modus tollens)

This is an important point to keep in mind.