Can a conditional be both vacuously true and false?

Question

Imagine the following conditional:

If washing machines are humans, washing machines are quadrupeds.

It seems to me that the truth value of the conditional as a whole is ambiguous. Since its antecedent is false, logic tells us that the conditional is (vacuously) true. But in fact, the conditional as a whole does seem false: if we grant that washing machines are humans, then washing machines are clearly bipeds.

But how can the same conditional be both vacuously true and (at least intuitively) false?

Answer 1

But in fact, the conditional as a whole does seem false: if we grant that washing machines are humans, then washing machines are clearly bipeds.

It is indeed true that if machines are humans, then washing machines are bipeds. But from this you can not infer that the other conditional is false. They are both true at the same time, precisely because they are only vacuously true. Two vacuously true statements with the same false antecedent are not contradictory. See also Why is it that the statement "All goblins are yellow" does not contradict the statement "All goblins are pink?" Neither of the two statements is false formally logically speaking, even though this may seem unintuitive.

Answer 2

Intuition is a tricky thing and does not always play well with mathematics. Plenty of mathematical results are non-intuitive.

As lulu says in a comment: If $P$ is false then $P$ implies everything; that's just how it is in mathematics, most of the time (1).

Once I was in the office a little after the end of day and a manager said: "why is no one still here?". I replied: "they have just gone for a dinner break; any time that I have been here at midnight, it was very busy". He clearly did not want to accuse me of lying but, also, he clearly did not believe me.

Here is a simple example of day to day usage which does not match mathematical usage. If I say "do you want tea or coffee?" then I probably do not expect the answer "both"; in common usage, "or" is usually exclusive. In the world of mathematics, "both" would be acceptable; "or" is inclusive unless specified otherwise.

A less simple example: intuitively there are more positive integers than just even positive integers yet, in mathematics, we say that there are equally many.

(1) There is no standards body for mathematics so there are no absolute definitions and rules. Some are followed by most people most of time time. Others vary a lot from author to author. In this case, alternative systems of logic are studied.

Answer 3

1. This conditional—in its entirety—is not false:

   If washing machines are humans, then washing machines are quadrupeds.

2. On the other hand, given that

   • washing machines are humans,

   it is indeed false that

   • washing machines are quadrupeds.

Thus, under the assumption of the antecedent (pretending that it is actually true), we can assert that the consequent—not that the entire conditional—is false. Here, the consequent's falsity is in the context of a subroutine, so to speak.

The word ‘implication’ informally has two conflicting meanings in the English language: sometimes it means the conclusion/consequent, other times it means the entire conditional. When the antecedent and consequent are both false, these two meanings should be carefully distinguished, because in this case the consequent (the ‘implication’) and the entire conditional (also the ‘implication’) have different truth values.

In any case, these two statements don't actually contradict each other:

If washing machines are humans, then washing machines are quadrupeds.

if we grant that washing machines are humans, then washing machines are clearly bipeds.

The sentences $$H{\implies}Q$$ and $$H{\implies}\lnot Q$$ are, by definition, simultaneously (vacuously) true.

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Appending my comment from Araucaria's answer

The statements

• given event $A,$ the (conditional) probability of event $B$ is $0.6$

and

• if event $A$ happens, then the probability that event $B$ happens is $0.6$

have different meanings: the former says that $$P(B|A)=0.6$$ whereas the latter says that $$\text{A happens}\implies P(B)=0.6.$$

When $A$ denotes the empty set and $B$ denotes the sample space, the first statement is false while the second statement is vacuously true.

Answer 4

The fact that there is a mathematical theory (one amongst many) of how to represent natural language conditional statements, and it is well known by those maths students who have never studied natural language conditionals, this does not mean that that theory works as a model of the logic of natural language conditionals! It doesn't. For example, any mathematical theory of probability will disintegrate on contact with a material implication theory of natural language conditionals as proposed by Bertrand Russell.

For example, consider a situation where I throw a fair dice two times. What is the probability that, if I throw a six the first time, I will throw a six the second time?

The natural and correct answer to this question is 1/6, of course. However, iff the material implication theory of conditionals meted out to maths students is correct, then that answer is completely and utterly bananas. The answer according to the material implication theory of conditionals is 31/36.

Mathematicians will only every give you an answer to a maths question according to the material implication theory of conditionals if they have been primed to do so first. They need this priming so that they can relegate their ability to understand speech to the status of a false theory given to toddlers, and then supplant this understanding of natural language conditionals with a theory that mathematicians love to talk about but never actually use in real life or real maths.

There are other mathematical models of natural language conditionals. See for example Ernest Adams, Dorothy Eddington, Allan Gibbard. They work much better. According to theories like these conditionals have no truth values, but have assertibility conditions, where the assertibility of a conditional If P, Q is equivalent to the probability of Q given P. This kind of theory of conditionals doesn't implode on impact with either common sense or any mathematical theory of conditional probability.

On this kind of understanding of conditionals, the Original Poster's conditional might be "vacuously true" according to the material implication theory of conditionals, but in actual fact is a sentence that's incapable of having truth conditions. It would be unassertible because the conditional probability of Q being true given P in this instance is basically zero.

There are many other mathematical theories of natural language conditionals that would say that this conditional was false (see for example this famous paper by Robert Stalker: Probability and Conditionals 1970 or the paper Indicative conditionals 1975.

Answer 5

The word "if" has two different meanings. The meaning that's used in mathematical writing is different from the meaning that's used in everyday English.

In mathematical writing, the phrase "if X, then Y" is defined as meaning "either not-X, or Y." Whenever this definition conflicts with the everyday meaning of the word "if," we disregard the everyday meaning and use this definition instead.

So if we come across the sentence "If washing machines are humans, then washing machines are quadrupeds" in a mathematical text, then this sentence is defined as meaning "Either washing machines are not humans, or washing machines are quadrupeds," and this sentence is clearly true.

It's true that according to the everyday, intuitive meaning of the word "if," the sentence "If washing machines are humans, then washing machines are quadrupeds" is probably false*. But that fact is simply irrelevant, because in mathematical writing, we just don't use the everyday, intuitive meaning of the word "if."

(*Actually, I might argue that we can count the number of feet a washing machine stands on, and it's clearly four, and therefore, if washing machines are humans, then they must be humans who stand on their hands and feet, and are therefore quadrupeds. But I digress, I digress!)

Answer 6

Washing machines are not humans, so how can you know whether they would be biped or quadruped if they were? In a universe where washing machines are humans, perhaps humans are quadrupeds!

You say:

"if we grant that washing machines are humans, then washing machines are clearly bipeds."

but this is only a guess, not a certitude. It could only be a certitude if washing machines were humans, which they aren't.

This is why we say that if A then B is true if A is false: because if A then B can only be falsified if A is true.