What is the reasoning behind vacuous truths? [duplicate]

Question

Vacuous truths assert that an entire proposition is true even if the antecedent is false.

Take the following proposition.. "If I am a toilet paper roll then I am a flying guitar"

The antecedent is false, I am not a toilet paper roll. Therefore, how could the entire proposition be true? I asserted that if the antecedent is true, then the consequent is true (that is what makes the proposition true, both the antecedent and consequent must be true). Yet it is said that the entire proposition is (vacuously) true even if the antecedent is false. What is true about the proposition? It certainly is not true in the sense that I understand truth.

I'd greatly appreciate it if someone is willing to explain it without technical mathematical notation (which I am unfamiliar with).

Answer 1

The concept of vacuous truth arises mainly with two kinds of statement. One is with conditionals, e.g. "if A then B", and the other with universal statements, e.g. "all F's are G's". The two are actually related, because on a popular account of understanding universal statements, they can be construed as a conditional, e.g. as something like, "for any thing, if it is F then it is G". Vacuous truth is concerned with the issue of what are the truth conditions of "if A then B" when A is false, and of "all F's are G's" when there are no F's. We could possibly take the view that under those circumstances such statements don't have a truth value. But if we want to use a logic that is bivalent, i.e. its statements all come out true or false, then we don't want to leave gaps unless there is a really compelling reason. The example you give is a conditional.

The short answer is that the conditional you are using is a special kind of conditional called a material conditional, or material implication. It just expresses the statement that there is no counterexample, i.e. that it is not the case that the antecedent is true and the consequent false. "Not (A and not B)" comes out true whenever A is false, so this kind of conditional is always true under those circumstances. Such conditionals turn out to be very useful, so they are commonly used in logic and mathematics. It does not mean that all, or even many, ordinary uses of 'if' in English are conditionals of this kind. It is mainly just a specialised kind of conditional used in logic, and usually the first conditional you are introduced to when studying logic.

The name 'material' is perhaps not the most apt, but it has been in use for over 100 years now, so it's here to stay. It does not mean 'material' in the way a lawyer would use the term, but relates instead to the term 'material validity' as used by the medieval logician Abelard. The material conditional is not a modern invention: it was first described by the stoic philosopher Philo.

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For a longer answer, we need to dig a little deeper into what we use conditionals for. Conditionals are used in a wide variety of ways in ordinary language, e.g. to express a rule, a hypothesis, a causal claim, an evidential claim, a proviso, an assumption, a disposition, etc. In many cases, such conditionals can be true or false quite independently of whether their antecedents are true or false. "If I let go of this pen it would fall to the ground" is not true just because I never let go of the pen. If it were, "if I let go of this pen it would fly off to the moon" would also be true, which it isn't. Likewise for counterfactual conditionals, such as "if I hadn't stopped for coffee I wouldn't have missed my train". Ditto for conditionals that express dispositional properties, such as "if you put common salt in water, it dissolves". Such things are not vacuously true when their antecedents are false.

Natural language conditionals like this are often doing something fairly complex. They are expressing a relationship between the antecedent and consequent, or they are implicitly quantified in some way, or they express a modality, or they express a hypothetical possibility, or they might involve uncertainty, or they might tacitly make all kinds of default assumptions that are relevant to the circumstances of their use. Making sense of the logic of conditionals in general is difficult and there is a huge literature on it.

Material implication is a simple kind of conditional that bypasses all these difficulties and attempts to do nothing more than express the fact that the truth of A is a sufficient condition for the truth of B. It is a truth function, which means that its truth value depends only on the truth values of A and B, and nothing more. Truth functional logic is pretty useful, and if you do lots of exercises in expressing simple natural language sentences using 'and', 'or' and 'not' you will find that this truth functional material conditional is also just as useful.

Material implication can be formally defined in any of several different ways, which are provably equivalent to each other. It can be defined using replacement rules, being equivalent to "¬(A & ¬B)" and also "¬A v B". It can be defined using rules of logical implication: it is the connective whose introduction rule is the rule of conditional proof, and whose elimination rule is the rule of modus ponens in classical logic. It can be defined using a truth table, in which case its truth values are T/F/T/T, i.e. it is true except when A is true and B is false.

If you really want to get into the weeds with conditionals, there is a lot of published material. You can start with the SEP article on indicative conditionals, and then try some books, such as Jonathan Bennett "A Philosophical Guide to Conditionals", David Sanford "If P then Q", and Ernest Adams "The Logic of Conditionals".

Answer 2

The notion of vacuous truth has been entirely manufactured by mathematicians.

Mathematical logic eventually adopted the material implication as the best mathematical model of the logic of (natural language) conditionals. Despite its name, the material implication is not an implication. It is the logical operation ¬ϕ ∨ ψ. Mathematicians adopted it because they could not find anything better than the material implication as mathematical model of the ordinary conditional, which is the usual syntactic form used in logical reasoning, including by mathematicians themselves.

However, a material implication ¬ϕ ∨ ψ is made true if the first term ϕ is false, and this simply because if ϕ is false, then ¬ϕ is true, and then the disjunction ¬ϕ ∨ ψ is true, whatever the truth value of ψ.

Initially, the people who conceived of this solution were very much aware of its limitations, which is why they called it "material" in the first place. They understood that the material implication was not the logical implication. They also discovered what they themselves called "paradoxes of the material implication", which is evidence that they were very much aware of the fact that the material implication seems wrong in some cases.

Perhaps for this reason, they adopted initially a special symbol, '⊃', called the "horseshoe". Unfortunately, subsequent generations of mathematicians at some point dropped the horseshoe, probably because it was an unpleasant reminder that the material implication was not the logical implication. They started to use instead the left-to-right arrow, '→', a symbol which everybody understands as the symbol of the logical implication.

As most mathematicians have a superficial training in mathematical logic, many of them have come to believe that the material implication is the logical implication. Thus, typically, mathematicians will use the expression "ϕ → ψ" to denote the material implication ϕ ⊃ ψ, material implication which is in reality nothing but ¬ϕ ∨ ψ. And they will just call it, "an implication".

The consequence of this is that mathematicians will routinely assert that an implication with a false antecedent is ipso facto true, which is plain nonsense, since while the material implication ¬ϕ ∨ ψ is ipso facto true if ϕ is false, the corresponding conditional, viz., "If ϕ, then ψ", may well be false, depending on the relation between ϕ and ψ.

The problem is compounded by the fact that mathematicians generally tend to use the same vocabulary as we all use but often with a different meaning. In particular, they use the words "implication", "proof", "logical validity", and even the word "logic" itself, to mean something very different from what most people mean, and yet without ever acknowledging that this is what they are doing.

So much so in fact, that there is today a widespread confusion directly caused by this. One interesting aspect of the situation is that many academics who are not mathematicians themselves are nonetheless interested in logic, and many of them still use the logical terminology with the same semantics as the Aristotelian tradition. The result is that there is often a profound misunderstanding as to what exactly people are saying when they talk about "logic", if this is really what they are doing to begin with.

Your question is one which is recurrent on all discussion forums where logic is sometimes discussed. Many people are just puzzled by the mathematical logic tenet that an implication with a false antecedent is ipso facto true.

Many of them are in fact students training for mathematical logic, and some of them voice their puzzlement to their teachers. The problem is so recurrent that logic textbooks now often include arguments to justify the idea that it is as it should be that an implication with a false antecedent (or a conditional with a false conditional clause) is ipso facto true.

In this context, the notion of vacuous truth is just a sop to convey the message that the case of the false antecedent is no good reason for anyone to feel aggravated because, it is alleged, this has no consequence whatsoever, hence, "vacuous" truth.

So, please, stop feeling aggravated.