A closer look at truth-tables

Let’s look more closely at this use of a truth-table to define a connective.

Here’s the truth-table for ampersand again:

Figure 1. The defining truth-table for ampersand

Let’s highlight some key points about this truth-table.

First, as with every truth-table, each row corresponds to a kind of possible situation—e.g. the first row corresponds to situations in which we have: ‘P’, true; ‘Q’, true. And the table says that ‘(P & Q)’ is true in situations like that.

Secondly, we can take the table to state the meaning of ‘&’: it’s effectively a definition which will determine the correct use of the expression. It expresses what our original English-language attempt at defining ‘&’ was aiming at. (You’ll see later that we can use truth-tables for tasks other than defining sentence connectives, but it’s a job of definition that we’re doing here.)

Thirdly, strictly speaking, we should have used something other than a roman letter in our definitional truth-table. The reason for this (putting it roughly for the time being) is that roman letters are going to stand in for particular sentences, whereas here what we want to say is, effectively: ‘Take any sentence you like, take any sentence you like, and here’s how things will go with the truth-values of a sentence made up from them plugged into ‘&’ … ’

The point here is that this isn’t just about some particular sentences ‘P’ and ‘Q’ but about how ‘&’ works with all sentences. So, we should really have put something like this:

(We’ve said ‘roughly’ here. We’ll come back to this point and sharpen it up later.)

Finally, note that ampersand is what’s called a truth-functional sentence connective: in every case, the truth-value of a sentence or sentential clause made by plugging sentences into an ampersand is settled by the truth-values of the plugged-in sentences.

All of the connectives in our formal logical language will be truth-functional. Not all sentence connectives in natural languages are truth-functional—two examples of connectives which are not are ‘because’ and ‘It is possible that’.

(If you’d like to find out more about the distinction between truth-functional and non-truth-functional sentence connectives, and find out about an example of a non-truth-functional sentence-connective, see the supplementary material in the document linked below. Note that this is optional: you don’t need to know about this material to follow the main thread of the course.)

Now, as we’ve noted, it seems like ‘&’ is close in meaning to the English word ‘and’ (in those cases where ‘and’ is used as a sentence connective). We’ll explore this suggestion in detail shortly, but first we need to get clearer about our broader project and look at how we can test ideas about natural-language connectives.