Tea or coffee?

I find logic and argumentation interesting, so when I saw, when browsing on the Internet, that recently a book titled Bad Arguments. 100 of the Most Important Fallacies in Western Philosophy had been published, I couldn’t resist buying it. Already when I read the first chapter, it was clear that it was a good choice, for the chapter is about one of the most confusing words in language: Or.

Consider this example: I have invited my friend John and I ask him what he wants to drink: “Do you like tea or coffee?” Then John will say, for instance, “Tea”. He might also have said “Coffee”, but not “Tea and coffee”, for making a choice is what you are supposed to do in such situations. It’s not polite to ask both for tea and for coffee. However, John might also have said that he wants to drink nothing at all.

Take now another situation. John and I are talking about drinks and then I ask him: “Do you like tea or coffee?” John replies “Tea”, although he might also have replied “Coffee” or even that he doesn’t like tea nor coffee or just that he likes both tea and coffee. That he might have replied the latter is remarkable, for in both examples I asked the same question and nevertheless in the first example John can’t choose both coffee and tea, and in the second example he can. How is this possible?

In order to make this clear, let me write the examples with the help of symbols. I start with the first case. Since logicians prefer the letters p and q for indicating objects, sentences etc, let me use the letter p for “I like tea” and q for “I like coffee”. Moreover, if such a statement is the case logicians call it “true” (or T for short), and if it is not the case they call it “false” (F). Then we can describe my question as “p or q?” and John’s answer is “p ”, which means that p is true and q is false and “p or q” is true. However, if John would have preferred coffee, q would have been true and p would have been false, but “p or q” would also have been true. If he didn’t want a drink at all, both p and q would have been false and so “p or q” would be. Moreover, John knows that it is not polite to ask both for tea and for coffee (p is true and q is true), so for this possibility “p or q” is also false. We can put these results in a table (a so-called “truth table”) in this order:

                                   p          q          p or q

                                   –––––––––––––––––

                                   T         F              T

                                   F          T             T

                                   F          F             F

                                   T         T             F

In the same way the second example can be described with symbols and reproduced in a truth table. Then my question to John is again “p or q?”. John answered “p” as we have seen so p is true and q is false and “p or q” is true. If John had answered “q” instead, q would have been true and p false, but still “p or q” would have been true. However, now it is also possible that John likes both tea (p) and coffee (q), as we have seen, so now – as both p and q are true – “p or q” is true! Including also the possibility that John doesn’t like tea and coffee at all (p, q and “p or q” are false), we get this truth table:

                                   p          q          p or q

                                   –––––––––––––––––

                                   T         F              T

                                   F          T             T

                                   T         T              T

                                   F          F              F

What has this analysis brought to us? I started with two examples in which, as I supposed, the same question was asked, albeit in different situations. Then I clarified the question and the possible answers by constructing truth tables for the two situations. What we see then, however, is that the supposedly same question led to different truth tables in different situations. This is only possible if I actually have asked two different questions, despite the same wording. Although the difference might be in the word “like” (which has slightly different meanings in the examples, indeed), it’s not the “like” that determines the contents of the truth tables, but the choices we are allowed to make do. In each case the choice can be reduced to the question “tea or coffee”. Since tea is tea and coffee is coffee, the upshot is that the word “or” has two different meanings in my examples. In the first example “or” excludes the possibility to choose both coffee and tea, but the second example allows this possibility. Cases like these have logicians made to distinguish two kinds of “or”. They have called them “exclusive or” and “inclusive or” respectively. If we ask a question with the first or, they call it an “exclusive disjunction” (p ⊕ q in symbols), while they talk of a “logical disjunction” (p v q) if we ask a question with the inclusive or. So, when your host asks you whether you would like tea or coffee, it might be possible that he asks you whether you would like to have both.

Reference

Arp, Robert; Steven Barbone; Michael Bruce (eds.), Bad arguments. 100 of the most important fallacies in Western philosophy. Oxford, etc.: Wiley Blackwell, 2019.

See especially Jason Iuliano, “Affirming a Disjunct”, pp. 39-41.