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Defining ampersand

In this video step, York student Lillian shows how our connective ampersand ('&') is defined using a truth-table.
Let’s look at the truth table for one of the connectives in our logical language, ampersand. Remember that we are defining how it works and what it means. Here’s the truth table. We’ll assume we’re working with two basic sentences ‘P’ and ‘Q’ and show how the truth value speaks the truth value of ‘(P ampersand Q)’. Here are the four rows corresponding to the four ways things could be with the truth values of ‘P’ and ‘Q’ covering all the ways things could be with ‘P’ and ‘Q’. ‘(P ampersand Q)’ is true when both are true, and false in all other cases. That seems to correspond closely
to the English connective ‘and’: plausibly, ‘P’ and ‘Q’ is true where both sentences are true and false, where at least one is false.

Here is the truth-table which defines the meaning of our connective ampersand (‘&’):

defining truth-table for ampersand

Figure 1. The defining truth-table for ampersand

Let’s look more closely at this definition and how it fixes the logical powers of ampersand.

Ampersand and ‘and’

The truth-table method gives us a way to define ampersand without using ‘and’. The rows correspond to kinds of possible situation. Each row shows what the truth-value of an ampersand-sentence would be, given the truth-values of the sentences/sentential clauses involved.

Note that the truth-table definition of the meaning of ampersand will make some (very simple) forms of argument come out as valid (on the basis of the meaning given to ‘&’). For instance

  • (α & β)
  • Therefore, α

will be valid. The definition of ‘&’ means that for ‘(α & β)’ to be true, ‘α’ will have to be true; so the truth of the premise will ensure the truth of the conclusion.

(A quick side-comment: it’s possible you’re going a little crazy at this point, thinking ‘This is unbelievably obvious!’ That would be understandable, but it’s worth noting that, in logic, we tend to start with the super-obvious and build up to things that are less obvious. Also, we tend to pick on uncontroversial, hum-drum examples, like ‘It’s raining’, because we want to focus on the meanings of the logical words, rather than getting distracted by the subject-matter.)

The definition of ‘&’ will also ensure that

  • α
  • β
  • Therefore, (α & β)

will be valid.

Notice that, when we write the truth-value for an ampersand-sentence in a particular kind of situation, that truth-value goes in a column directly under the ampersand. (We’ll do the same sort of thing for our other connectives.) This will be important later on when we see how to work out the truth-values of more complex sentences—sentences with clauses that themselves contain connectives.

Now, as noted in the video, the way ampersand works is at least plausibly very similar to the way that ‘and’ works as a sentence connective in English (this, trust us, is no accident).

For instance, it’s intuitively very plausible that if someone says

  • It’s raining and it’s cold

then they’re committed to it being true that it’s raining and also true that it’s cold. (What they’ve said, we think, will be false if it turns out one of those claims is false.)

That should give us some hope that we can use ‘&’ to represent and evaluate arguments expressed in English that involve ‘and’ used as a sentence connective.

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Logic: The Language of Truth

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