Tarski on disjunction

Before going off to Florence, I was reworking chapters on the material conditional for IFL2 (in fact I posted a couple of draft chapters here, which I then thought I could improve on,  and so I rapidly took them down again). While away, it occurred to me that it might be prudent/interesting/useful to take a look at how various standard logic texts over the years have handled the conditional in propositional logic. So here I am, starting to work quickly through a pile of some 25 introductory texts, from  Alfred Tarski’s Introduction to Logic and to the Methodology of the Deductive Sciences (1936/1941) to Jan von Plato’s Elements of Logical Reasoning (2013).

I’m writing some telegraphic notes for myself as I go along, but I’ve quickly realized it would take far too much time (and be far too distracting from what I am supposed to be doing) to work these up to give detailed and fair-minded stand-alone reports here. But let me say something about the first book I turned to. For I was surprised and intrigued when reading Tarski to discover  how rocky his arguments are (and indeed how unclear his position is about the relation of ordinary language and the connectives of the formal logician). But let’s not tangle now with what he says about the conditional; his preceding remarks about disjunction already show some of the problems he gets into. So here are some quick notes about those remarks.

Talking of “or” as used to join two sentences, Tarski very confidently asserts that

… in everyday language, the word “or”  has at least two different meanings,

in particular, inclusive and exclusive meanings. But the only supposed illustration given for this once-popular ambiguity claim is hopelessly weak. Tarski writes

… if a child has asked to be taken on a hike in the morning and to a theater in the afternoon, and we reply:

no, we shall go on a hike or we shall go to the theatre,

then our usage of the word “or” is obviously of the second [exclusive] kind, since we intend to comply with only one of the two requests.

But what’s the “no” doing here? It is denying that we will both go for a hike and go to the theatre. So the envisaged reply is arguably just a verbal variant of

 We shall go on a hike or we shall go to the theatre but not both.

But while everyone agrees that something of the whole form P or Q but not both expresses the exclusive disjunction of P and Q, it certainly doesn’t follow that the clause P or Q taken by itself means exclusive disjunction. So the sole given example doesn’t give us any evidence that there is a distinctive exclusive meaning of “or”.

Tarksi continues

In logic and in mathematics the word “or” is used always in the first, non-exclusive meaning.

But is he entitled to this claim? After all, if what he wrote about the hike/theatre example holds good, surely this would too:

… if a child wonders if the number 49 is divisible by both three and seven, we might reply (as a hint):

no, 49 is divisible by three or 49 is divisible by seven: work out which!

And our usage of the word “or” is obviously of the exclusive kind, since we intend to allow only one of the two disjuncts to be true.

So, by Tarski’s lights, we ought to have here a mathematical use of the exclusive or! What he meant, presumably, is that in the formal logician’s regimented usage, when  “or” is symbolised by ““, the disjunction is inclusive: but this isn’t what he actually says at this point.

Tarski then continues,

Even if we confine ourselves to those cases in which the word “or” occurs in its first meaning, we find quite noticeable differences between its usage in everyday language and that in logic. In common language two sentences are joined by the word “or” only when they are in some way connected in form and content …. It is not altogether clear what kinds of connections would be appropriate here, and any attempt at their detailed analysis and description would lead to considerable difficulties. As we shall see, such connections are disregarded in contemporary logic, where consequently one has to allow some strange examples; and indeed, anybody unfamiliar with its language would presumably be little inclined to consider a phrase such as:

2 x 2 = 5 or New York is a large city

as a meaningful expression, and even less so to accept it as a true sentence.

What does “in some way connected in form and content” mean? We are are given no hint. Yet on any natural reading, the claim that the disjuncts of everyday disjunctions will be so connected seems over-strong. To take a contemporary example: an article about fake news lists ten surprising/bizarre claims, and asks us to spot which five claims are true, and which five are fake. We happen to know four of the claims to be true, and we dismiss another four as false. That leaves us with two claims up for grabs, P and Q. And in the circumstances it is now entirely natural to assert P or Q even though these disjuncts need not be in any obvious sense connected in form and content any more than are 2 x 2 = 5 and New York is a large city (it is just that the newspaper article has made P and Q both salient in the context, and we justifiably think one is true).

Further, whatever the first inclination of the untutored, it is odd for anyone to cast serious doubt on the meaningfulness of  2 x 2 = 5 or New York is a large city. (We’d surely now say that here Tarksi is running together questions of meaning, of semantics, and questions of pragmatics, of conversational appropriateness in a given context.) After all, it is precisely because it is a meaningful bit of English that we understand the sentence here perfectly well, and so realize that it takes a bit of work of describe a situation in which this disjunction would be a conversationally natural thing to assert. Yet of course it takes exactly the same amount of work to describe a situation in which 2 x 2 = 5  New York is a large city would be a natural thing to assert. On this count, at any rate, there’s nothing to distinguish between “” and (inclusive, sentential connective) “or”.

Tarski continues

Sometimes we even take the utterance of a disjunction as an admission by the speaker that he or she does not know which member of the disjunction is true, and which is false. And if we later arrive at the conviction that the speaker knew at the time that one—and, specifically, which—of the members was false, we are inclined to look upon the whole disjunction as a false sentence, even though the other member should be undoubtedly true.

Well, no. We might, in those circumstances, conclude that the speaker was being misleading, even culpably misleading, in asserting (only) the disjunction. But that doesn’t make the disjunction false. It is surely a naive common-place — not a bit of high post-Gricean theory — that we can can, in many ways, be misled as to the truth by the utterance of a perfectly true sentence.

It is intriguing, then, to find Tarski (he of the formal semantic account of truth!) seemingly so at sea over what would strike us as elementary distinctions of issues of semantics and literal truth vs issues of pragmatics. As you would now expect, this doesn’t bode well for his discussion of conditionals!