Chapter 10 Truth-functional connectives
10.1 The idea of truth-functionality
Let’s introduce an important idea.
A connective is truth-functional iff the truth value of a sentence with that connective as its main logical operator is uniquely determined by the truth value(s) of the constituent sentence(s).
Every connective in TFL is truth-functional. The truth value of a negation is uniquely determined by the truth value of the unnegated sentence. The truth value of a conjunction is uniquely determined by the truth value of both conjuncts. The truth value of a disjunction is uniquely determined by the truth value of both disjuncts, and so on. To determine the truth value of some TFL sentence, we only need to know the truth value of its components.
This is what gives TFL its name: it is truth-functional logic.
Many languages use connectives that are not truth-functional. In English, for example, we can form a new sentence from any simpler sentence by prefixing it with ‘It is necessarily the case that…’. The truth value of this new sentence is not fixed solely by the truth value of the original sentence. For consider two true sentences:
Shostakovich wrote fifteen string quartets.
Whereas it is necessarily the case that , it is not necessarily the case that Shostakovich wrote fifteen string quartets. If Shostakovich had died earlier, he would have failed to finish Quartet no. 15; if he had lived longer, he might have written a few more. So ‘It is necessarily the case that…’ is not truth-functional.
10.2 Symbolizing versus translating
All of the connectives of TFL are truth-functional, but more than that: they really do nothing but map us between truth values.
When we symbolize a sentence or an argument in TFL, we ignore everything besides the contribution that the truth values of a component might make to the truth value of the whole. There are subtleties to our ordinary claims that far outstrip their mere truth values. Sarcasm; poetry; snide implicature; emphasis; these are important parts of everyday discourse, but none of this is retained in TFL. As remarked in chapter 5, TFL cannot capture the subtle differences between the following English sentences:
Dana is a logician and Dana is a nice person
Although Dana is a logician, Dana is a nice person
Dana is a logician despite being a nice person
Dana is a nice person, but also a logician
Dana’s being a logician notwithstanding, he is a nice person
All of the above sentences will be symbolized with the same TFL sentence, perhaps ‘’.
Now, we keep saying that we use TFL sentences to symbolize English sentences. Many other textbooks talk about translating English sentences into TFL. However, a good translation should preserve certain facets of meaning, and—as we just saw—TFL just cannot do that. This is why we will speak of symbolizing English sentences, rather than of translating them.
This affects how we should understand our symbolization keys. Consider a key like:
Dana is a logician.
Dana is a nice person.
Other textbooks will understand this as a stipulation that the TFL sentence ‘’ should mean that Dana is a logician, and that the TFL sentence ‘’ should mean that Dana is a nice person. But TFL just is totally unequipped to deal with meaning. The preceding symbolization key is doing no more and no less than stipulating that the TFL sentence ‘’ should take the same truth value as the English sentence ‘Dana is a logician’ (whatever that might be), and that the TFL sentence ‘’ should take the same truth value as the English sentence ‘Dana is a nice person’ (whatever that might be).
When we treat a TFL sentence as symbolizing an English sentence, we are stipulating that the TFL sentence is to take the same truth value as that English sentence.
10.3 Indicative versus subjunctive conditionals
We want to bring home the point that TFL can only deal with truth functions by considering the case of the conditional. When we introduced the characteristic truth table for the material conditional in chapter 9, we did not say anything to justify it. Let’s now offer a justification, which follows Dorothy Edgington.11 1 Dorothy Edgington, “Conditionals”, Stanford Encyclopedia of Philosophy (Fall 2020) (https://plato.stanford.edu/archives/fall2020/entries/conditionals/).
Suppose that Lara has drawn some shapes on a piece of paper, and coloured some of them in. We have not seen them, but nevertheless claim:
If any shape is grey, then that shape is also circular.
As it happens, Lara has drawn the following:
In this case, our claim is surely true. Shapes C and D are not grey, and so can hardly present counterexamples to our claim. Shape A is grey, but fortunately it is also circular. So our claim has no counterexamples. It must be true. That means that each of the following instances of our claim must be true too:
If A is grey, then it is circular
(true antecedent, true consequent)
If C is grey, then it is circular
(false antecedent, true consequent)
If D is grey, then it is circular
(false antecedent, false consequent)
However, if Lara had drawn a fourth shape, thus:
then our claim would have been false. So this claim must also be false:
If B is grey, then it is circular
(true antecedent, false consequent)
Now, recall that every connective of TFL has to be truth-functional. This means that the truth values of the antecedent and consequent alone must uniquely determine the truth value of the conditional as a whole. Thus, from the truth values of our four claims—which provide us with all possible combinations of truth and falsity in antecedent and consequent—we can read off the truth table for the material conditional.
What this argument shows is that ‘’ is the best candidate for a truth-functional conditional. Otherwise put, it is the best conditional that TFL can provide. But is it any good, as a surrogate for the conditionals we use in everyday language? Consider two sentences:
If Hillary Clinton had won the 2016 election, then she would have been the first woman president of the USA.
If Hillary Clinton had won the 2016 election, then she would have turned into a helium-filled balloon and floated away into the night sky.
Sentence 1 is true; sentence 2 is false, but both have false antecedents and false consequents. (Hillary did not win; she did not become the first woman president of the US; and she did not fill with helium and float away.) So the truth value of the whole sentence is not uniquely determined by the truth value of the parts.
The crucial point is that sentences 1 and 2 employ subjunctive conditionals, rather than indicative conditionals. They ask us to imagine something contrary to fact—after all, Hillary Clinton lost the 2016 election—and then ask us to evaluate what would have happened in that case. Such considerations simply cannot be tackled using ‘’.
We will say more about the difficulties with conditionals in chapter 13. For now, we will content ourselves with the observation that ‘’ is the only candidate for a truth-functional conditional for TFL, but that many English conditionals cannot be represented adequately using ‘’. TFL is an intrinsically limited language.