# What does it mean, intuitively and then also precisely, that a particular English word is not truth functional?

What does it mean, intuitively and then also precisely, when we say that a particular English word is not truth functional? Let me present some examples.

Example 1 As far as I can tell from a book I am reading about propositional logic, the truth value at a particular time of the sentence Max is home whenever Claire is at the library (call this S) is not determined by the truth values, at that same time, of the atomic sentences Max is home (call this A) and Claire is at the library (call this B).

S appears to be similar to the sentence B -> A. However, it is possible that Claire is in fact at the library (B is true), Max is in fact home (A is true), but the sentence as a whole is false. My interpretation is that there is the possibility of the sentence being false in this case because whenever does not refer only to one moment or instant in time, but instead refers to all instants in time in which Claire is at the library. The atomic sentences on the other hand refer only to one instant in time.

If we had the sentence D being Max is home if Claire is at the library, apparently if refers to one instant in time. Therefore, the truth values of the atomic sentences relate to one instant in time, and the sentence D with the word if also has a truth value that is related only to that same instant in time. So the truth value of D depends solely on the truth values of the atomic sentences, and it is therefore truth functional.

Example 2 Consider sentence S If Max had been home, then Carl would have been there too.

Looks like a sentence with a material conditional: B -> A, where B is Max had been home and A is Carl would have been there too.

From what I read in my book, it is possible that B is true and A is true, but the sentence as a whole is false. First of all, it is confusing to think of a truth value for Max had been home, because of the tense of the verb.

Intuitively, if it is possible that Max had been home is false, but the sentence as a whole is also false, then this means that the sentence does not follow the truth table of a material conditional, and hence does not depend solely on the truth values of the constituent atomic sentences.

For example, say Carl is a dog, and the person who says S thinks Max is taking Carl with him everywhere he goes. But what if instead of this circumstance being true, someone else altogether has Carl. Even if Max were home, Carl would not be with him, because Carl is with this other person.

Example 3 If the book is here then the mailman delivered it.

A is The book is here, B is the mailman delivered it, and S is the sentence A -> B.

If the book is here, and the mailman delivered it, then the sentence is true. If the book is here, and the mailman did not deliver it (say someone else brought it over), the sentence is false. If the book is not here, whether the mailman delivered it or not, the sentence is true.

Questions

It seems like the concept of whether a sentence is truth-functional or not is not super sharply defined. Intuitively, in example 3, it is truth-functional because when I assign truth values to the atomic sentences, my imagination cannot come up with a scenario wherein the sentence as a whole has a truth value not equal to the truth value we would expect from a truth table analysis. In examples 1 and 2, I did come up with such a scenario.

Is it correct to say that if I can come up with a scenario wherein the truth value of the sentence differs from the truth-table expected value, then the particular English connective is not truth-functional?

I would imagine there is no airtight criterion for determining whether a word in English is or is not truth-functional.

• Your 1st example sentence includes hidden quantification over time instants (states) so it's not simply truth functional of its atomic constituents A & B at a same instant, FOL could describe it, PL is not enough. Your 2nd example sentence is the typical counterfactual conditionals and were first discussed as a problem for the material conditional which treats them all as trivially true. Modal logic is usually used to address these conditionals such as Lewis's strict conditional and Stalnaker's variably strict conditional... Jan 13 at 23:11
• To expand on Double Knot's answer: this is an old topic in logic. There are contexts in which the normal rules of predicate logic break down. These are called modal contexts. See plato.stanford.edu/entries/logic-modal Jan 13 at 23:40
• See e.g. Indicative Conditionals Jan 14 at 7:29
• For most English words to be "truth-functional" or not is non-sensical, take "red" or "apple", for example. Even logic related words, and, or, if-then, are not meant to be fully truth functional. This is an idealization that applies, strictly speaking, only to formal languages. The basic idea is right, a connective is truth functional if the truth value of the composite is determined by the truth values of its constituents. But words of natural language only, at best, approximate this ideal, so one can typically find contradicting "scenarios" for all of them. Jan 15 at 2:01

The short answer is that no English word is truth functional, unless it is given an artificially specified meaning. Truth functions are a feature of formal logic, and formal logic only approximates the meanings of words in natural languages.

Take, for example, the word 'and'. This is the connective that tends to be the least argued about by logicians. If A is true, also B is true, then "A and B" is true, and vice versa. So, 'and' looks like a truth function, but even here there are exceptions. We can use 'and' to conjoin things rather than propositions, e.g. "Alice is sitting between Bob and Carol", or "Dave and Ellen make an odd couple". 'And' also often carries the conversational implicature of sequence, i.e. it suggests "and then".

Or consider negation. One might think that 'not' would be the easiest logical particle to understand, but many papers and even entire books have been written about negation, e.g. Laurence Horn's "A Natural History of Negation".

What about the word 'or'? Classical propostional logic treats it as a truth function, but this is controversial. Intuitionism and relevance logic treat it differently and have different rules of implication. Also, in ordinary usage it often carries the conversational implicature of "this or that but I don't know which".

Your examples are all concerned with conditionals, where the gap between formal representations of conditionals and their natural language sense is notoriously great. The truth functional conditional is called the material conditional, or material implication. It has a special role in classical logic, because it is the connective that is introduced by the rule of conditional proof (or the deduction theorem) and eliminated by modus ponens. But the gap between material implication and the ordinary sense of 'if' in English is a gulf a mile wide.

In your first example, 'whenever' functions as a conditional with a quantifier covering all times, or all situations. It can be glossed as, "at any time t, if Claire is in the library at t, then Max is at home at t". Whether this counts as truth functional depends on how you propose to understand whether the conditional is true when Claire is not in the library. Material implication treats this as trivially true, but this gives odd results. If Claire never goes to the library, then the material implication interpretation would make it true that Max is on the Moon whenever Claire is in the library. We might be better off understanding this conditional as a relevant implication, or in terms of what holds in possible worlds close to the actual world.

The second example is a counterfactual. The antecedent is known (or assumed) to be false. This is more obviously not a truth function, since counterfactual conditionals are not all trivially true. "If someone had given me a million pounds last month I would have bought a new car" is true, but "If someone had given me a million pounds last month I would have bought a gun and killed myself" is not. There are many different proposals for understanding counterfactuals. A popular account, proposed in slightly different versions by David Lewis and Robert Stalnaker, is that we are making a statement about what holds in a possible world close to the actual world. So, "If Max had been home, Carl would have been there too" is glossed as: "Max was not home in the actual world, but the closest possible world in which Max was home and Carl was there too is closer to the actual world than the closest possible world in which Max was home and Carl was absent". What counts as 'close' is fleshed out by Lewis in terms of a number of rules of thumb.

Your third example could possibly be interpreted as you suggest, as a truth function. But again, it is rather odd. If the book is not here, then presumably the mailman did not deliver it, or else it would be here. Unless you are allowing for other unstated possibilities, such as somebody stole the book after it was delivered. If we don't know whether the book is here or not, then it makes sense to say that if the book is here the mailman delivered it. But if we know the book is not here, then we have no use for the conditional. A case like this, especially if there are unstated possibilities, might better be understood as a conditional probability: "it is highly likely that the mailman delivered the book, supposing it is here". If the book is known to be absent, this has a false antecedent, and so its probability is undefined.

The upshot seems to be that there is no single good way to formalise conditionals. There is certainly an immense literature on the subject: thousands of papers and scores of books. Some of the better books are Jonathan Bennett's "A Philosophical Guide to Conditionals", David Sanford's "If P then Q", Ernest Adams' "The Logic of Conditionals", David Lewis' "Counterfactuals", and Robert Stalnaker's "Knowledge and Conditionals".