Our first connective: ampersand ('&')
The first connective we’ll define for our logical language is ‘&’ — called ampersand.
We’ll stipulate (fix by definition) that ‘&’ is what’s called a two-place sentence connective. That is, to make a grammatical sentence or grammatical sentential clause with ‘&’, we need to plug two sentences/sentential clauses into it, one on either side.
We’ll then fix the meaning of ‘&’ by stipulating (fixing by definition) what the truth-conditions of an ampersand-sentence are in terms of the truth-values of the plugged in sentences.
The truth-value of a claim (or sentence) is just whether it is true or false. For example, the truth-value of the claim that two plus two is four is (always) true; the truth-value of the claim that two plus two is five is (always) false. The term ‘truth-value’ is useful in allowing us to talk the particular way things are with a claim (or sentence) in relation to it being true or false without having to use the (very long and awkward) phrase ‘the particular way things are with a claim (or sentence) in relation to it being true or false’.
Here is a first try at specifying the meaning of ampersand. (We’ll see shortly that this first try will need some further work in order to give us what we need.)
- A sentence ‘(α & β)’ is true if and only if α is true and β is true
This means that if we make a larger sentence by putting one sentence, say, ‘It is raining’ in front of ‘&’, and putting another sentence, say, ‘It is cold’ after ‘&’, then the resulting sentence, ‘It is raining & it is cold’, will be true if and only if ‘It is raining’ is true and ‘It is cold’ is true. And it follows that if one or both of α, β is false, then ‘(α & β)’ is false. (If you’re wondering why the ampersand-sentence includes brackets, we’ll come to that in a little while.)
So far, so good, it would seem. But you might have a worry here. Look again at what we just said to try to stipulate the meaning of ‘&’. There’s something that might appear a bit dodgy about it, if we’re hoping to use ‘&’ to clarify the meaning and logic of ‘and’ in English. Can you see what it is?
The problem is that the word ‘and’ crops up (twice) in our attempted definition: once in ‘if and only if’; and once in ‘α is true and β is true’. If we’re trying to get clear about ‘and’, you might think it’s a bit dubious to go round in quite such a tight little circle: we wanted to understand ‘and’ better, and so introduced ‘&’, but then we used ‘and’ itself to define ‘&’.
Can we solve this problem?
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