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Testing our ideas about natural-language connectives

Here we look at how we can investigate whether natural-language sentence connectives have the same truth-tables as connectives in our formal language.
Netting, showing the connections between the cords
© Clint Adair @ Unsplash

We’ve said that it looks like ampersand is close in meaning to the sentence connective ‘and’ in English. But how can we test this idea?

Before we look further at this, we need to clear something up. So far we’ve been speaking in terms of the meanings of ampersand and ‘and’. Strictly speaking, what we are really interested in is whether ‘and’ sentences have the same truth-conditions as the corresponding ampersand-sentence. This is a fairly precise question and it’s what matters to questions of validity. The idea of the (whole) meaning of a word or sentence is much harder to pin down (because the idea of meaning is less clear). From now on, in looking at ‘and’ and at other sentence connectives, we will be primarily concerned with this issue about truth-conditions.

There’s one fairly straightforward way we can test ideas about the truth-conditions associated with particular expressions. As speakers of natural languages, we’re pretty good sources of evidence about the meanings of expressions in the languages we speak. We can exploit this in testing our ideas about the meanings of expressions, including connectives.

It’s important to be clear about how our abilities as speakers of natural languages are relevant. As ordinary speakers, we’re typically not very good at what you might call direct explication of meaning. For example, it’s unlikely that any of us would immediately come up with the sort of account of meaning we’ve given for ampersand if we were asked to explain the meaning of ‘and’. If we were asked to explain the meaning of ‘and’ before being introduced to the kinds of ideas and methods that logicians have developed over many years, we’d probably be stumped: we know how to use ‘and’, but spelling out its meaning in explicit terms is a different, challenging task.

So, what sort of evidence can we provide that will help us decide if the meaning and logical powers of ‘and’ (specifically, the truth-conditions with which it is associated) can be represented using ampersand?

Well, we know how to use ‘and’, and part of that is at least having an idea about whether a sentence containing it would be true or false in some particular situation. This gives us a way of testing the match between a connective in our formal language and a natural language connective: we can think of situations in which sentential clauses have particular truth-values, think about whether we’d say a sentence involving the natural language connective is true or false in that situation, and see whether that matches the truth-value determined for the corresponding sentence of our formal language. If we find a case or cases in which there’s a mismatch, then the meanings differ (at least in some uses). If we don’t find mismatch cases, that supports the idea that the expressions are close in meaning or in fact have the same meaning in the specific sense of expressing the same truth-condition/having the same truth-table.

So, are there any mismatch cases for ampersand and ‘and’? Are there any cases in which ‘(A & B)’ would be true but ‘A and B’ would be false, or any cases in which ‘(A & B)’ would be false but ‘A and B’ would be true? If there are, then the idea that ‘and’ has the same truth-table as ampersand is weakened.

(A quick side comment: Note that, even if we did find some mismatch cases, it might not be that everything was lost. For instance, it might be that ‘and’ is ambiguous. That is, it might be that ‘and’ has one meaning in some cases and contexts and another meaning in other cases and contexts. We’ll say more about ambiguity later.)

© University of York
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Logic: The Language of Truth

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