# 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 “$\lor$“, 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 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 $\lor$ 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 “$\lor$” 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!

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### 12 Responses to Tarski on disjunction

1. Rowsety Moid says:

It seems to me that if “or” is ambiguous, then context might often indicate which of inclusive or exclusive “or” was meant; and the “no” in “no, we shall go on a hike or we shall go to the theatre” contributes to such a context; so what’s the problem? I’d say the example does illustrate that there is a distinct exclusive meaning. That the ambiguity could also be resolved by adding “but not both” doesn’t stop that from being so.

The rest of what he says also seems quite reasonable to me, since he’s talking about “everyday language” and people “unfamiliar” with “contemporary logic”.

And though it may be “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”, that doesn’t, it seems to me, entail that people wouldn’t be inclined to see the whole disjunction as false.

BTW, have you encountered the sort of people (quite common in computing) who like to respond as A in what follows:

A: Let’s go out for lunch.
B: Ok. Mexican or Chinese?
A: Yes.

• Peter Smith says:

On the first point, I of course wasn’t offering an argument that “or” isn’t ambiguous. I was only claiming that Tarski’s one example doesn’t show that it is — in other words, I was just noting that the particular exchange he describes can, pretty obviously, be construed as sensible equally well by someone who holds that “or” is (in its strictly literal meaning) univocal. It is interesting that Tarski apparently didn’t see this or feel the (strong?) pull of the modern theorist’s methodological maxim: do not multiply senses beyond necessity.

On the second point I agree that some people might be inclined, at first blush, “to see the whole disjunction as false”. But surely the question — if we are to find a distinction between everyday “or” and the logicians’ vel — is whether this judgement would be stable under a little probing and reflection And that isn’t at all clear.

• Rowsety Moid says:

I’m having trouble understanding your argument. If someone thinks, along with Tarski, that “or” is ambiguous, then they can agree that “it certainly doesn’t follow that the clause P or Q taken by itself means exclusive disjunction”, because something beyond the mere “P or Q” has to establish which sense is meant. That follows from “or” being ambiguous. But then you say “So the sole given example doesn’t give us any evidence that there is a distinctive exclusive meaning of “or”.” Why doesn’t it give any evidence?

You seem to be thinking that the evidence would have to be a case where “P or Q” on its own was clearly exclusive, but that won’t happen if “or” is ambiguous. The only evidence will be cases where an “or”, taken in context, is naturally understood as exclusive (plus such things as native speakers saying they at least sometimes understand “or” as exclusive.)

(I think it may even be that most people who haven’t become familiar with inclusive “or” in maths, logic or programming think of “or” as exclusive more often than not. The inclusive sense can seem quite strange.)

“We shall go on a hike or we shall go to the theatre but not both” is a different sentence that puts the “or” in a different context; if that “or” can be read as inclusive, that doesn’t mean the “or” in Tarski’s example is inclusive.

And though you don’t offer an explicit argument that “or” isn’t ambiguous, that seems to be what you have in mind when you disparage the ambiguity claim as “once-popular”, when you call it a “claim” rather than (say) an observation, and when you mention “the modern theorist’s methodological maxim” of not multiplying senses beyond necessity.

On the other point, I can’t agree that “the question … is whether this judgement would be stable under a little probing and reflection”. The internet gives numerous examples of someone sticking even to questionable views no matter what probing and opportunities for reflection they are given — and of people being “probed” into questionable views when they’d started somewhere better.

2. Daniel Nagase says:

A quick comment: it may be worth remembering that Tarski sometimes seems very close to Ricketts’s Carnap, in that he apparently considers natural language to lack a well-defined logical structure. In that sense, there just isn’t a well-defined semantics for natural language and it may only admit of a pragmatic analysis. That may explain some of his carelessness regarding semantics vs. pragmatics.

3. M.V. says:

A: Let’s go out for lunch.
B: Ok. Mexican or Chinese?
A: Yes.

Small fix:

A: Let’s go out for lunch.
B: Mexican or Chinese. Ok?
A: Yes.

:)

• Rowsety Moid says:

I thought the example might be enough to remind someone of the phenomenon if they’d encountered it; it probably isn’t clear enough on its own to show what the phenomenon is.

A wants B to express a preference. B knows this but deliberately misunderstands the question. B would do this even if A had asked “Would you prefer Mexican or Chinese?” or “Would you rather go to ___ or ___?” (naming two restaurants), because B is fond of this form of “wit”.

• Rowsety Moid says:

Sorry — I have “A” and “B” backwards there — it should be B who wants A to express a preference, and A who’s fond of that form of “wit” — and it’s not possible for me to edit a comment after I’ve posted it.

4. Dawid Milczarek says:

>>What he meant, presumably, is that in the formal logician’s regimented usage, when “or” is symbolised by “\lor“, the disjunction is inclusive (…)<>to use the word “or” by itself only in the first meaning, and to replace it by the compound expression “either… or…”
whenever the second meaning is intended.<<

5. Rowsety Moid says:

Further defence of Tarski.

1. Re “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” — Could we not also say “while everyone agrees that something of the whole form P or Q or both expresses the inclusive disjunction of P and Q, it certainly doesn’t follow that the clause P or Q taken by itself means inclusive disjunction”?

If not, what makes one of them acceptable but not the other?

2. Re “no, 49 is divisible by three or 49 is divisible by seven: work out which!” — I don’t think that shows that, by Tarski’s lights, there’s a mathematical use of xor; and so I also don’t think that Tarski must have meant that disjunction is inclusive in the formal logician’s regimented usage. The example is of ordinary English that happens to be about numbers and division. If ordinary English has both inclusive and exclusive or, that won’t suddenly change as soon as numbers or arithmetical operations are involved.

3. Re “2 x 2 = 5 or New York is a large city” — I don’t think it matters that it would take the same amount of work to describe a situation in which it would be conversationally natural to assert that sentence when it’s written with “\lor” as when it’s written with “or”. I don’t even think Tarski’s talking about a difference between “or” and “\lor” there, at least not primarily.

One problem is that we aren’t unfamiliar with logic, and so we’re not the best people to judge how such sentences would seem to people who are. We don’t have any trouble seeing a meaning for that sentence; however, I don’t think we understand the sentence just because it’s a meaningful bit of English: instead, we’re helped to see it as meaningful by our exposure to logic.

Also, we can see a meaning even without coming up with a situation in which it would be a conversationally natural thing to assert. Perhaps someone unfamiliar with logic could be led to see it as meaningful by being presented with a description of a suitable situation, but I don’t think that’s contrary to what Tarski said. He was talking about how people were inclined, not about what they could be led to think.

In any case, people don’t normally start by determining a sentence’s meaning and then come up with a suitable situation; instead, a sentence appears already in a situation or context, and that helps them decide what meaning was intended. The approach that starts with the meaning assumes something rather questionable, namely that there’s only one meaning the sentence could have. Instead, there might be a number of different possible meanings, each calling for a different sort of situation to make it conversationally natural to assert it.

6. Nate says:

This is fascinating. I might be able to offer some of that ‘outsider’ view: I’m a programmer who’s fascinated with Prolog but has been struggling for a while to grasp the fundamentals of formal logic, as much of the field is so different from the basic assumptions of programming and makes very little intuitive sense to me. Particularly the material conditional strikes me as ‘just wrong’ – and I find my naive intuition in strong agreement with what Tarski seems to be saying here in this quote! So I already think I might like Tarski.

Here’s what I see, as an outsider to the field of formal logic – and leaving aside the difference between exclusive and inclusive or and just focusing on that phrase “some way connected in form or content”:

1. Asserting ‘A or B’, and putting such a statement into a logical world or database, to a human, is a MUCH stronger statement than simply evaluating the material conditional truth function ‘A or B’. So *much* stronger that that the assertion and the truth function seem to be almost two different things!

An ‘A or B’ statement like, say, ‘I assert that Santa Claus exists OR the sky is blue’ is a very odd thing for a human to say. It’s also a very odd thing to put into a computer database. We are saying that we know for sure that one or more very specific facts are true — yet that we DON’T know for sure which ones. We’ve somehow lost data without losing faith in that data. But how would we get a situation like this? In logic (or at least in data processing), don’t we usually deal in discrete, observed facts and then work forward toward higher-level abstractions? How would you personally observe a disjunction? Couldn’t it only come from a process of inference, or summarisation (some kind of managed filing and forgetting)? This suggests that what an OR statement is describing is something at a deeper level than observed facts.

2. One could go further and say that, from a programming perspective, there is a fundamental difference between AND assertions and OR assertions. A set of ANDs constitutes a database or a logical world: ‘A and B and C…’. ORs, however, constitute rules for inferring further information about that world.

We can get to this idea by looking at logical implications (not-A or B) and how they seen to encode almost everything about ‘function evaluation’… and that the heart of a logical implication is the disjunction. There’s something very ‘big’ about a disjunction, in other words. It contains infinities within it. We should therefore probably be careful about inferring or asserting something like this.

Another way of looking at this is to say that evaluating the truth function ‘A or B’ means looking at the state of the database or world right now, and yes we can answer this with ‘true’ or ‘false’ just by checking the state of the world right now.

However, *asserting* the truth function ‘A or B’ means saying that it *remains* true regardless of how the state of the world changes around it. That means it’s saying something about *the state of all possible worlds*, and we surely can’t check the truth of such an *assertion* just by checking the state of the world *right now*!

So ‘A and B’ is a simple fact, but ‘A or B’ must be some kind of rule, or meta-fact, describing an infinity of possible situations. And that suggests that there are two levels of ‘truth’ associated with a disjunction: evaluating it as a truth-function can tell us ‘*can* this rule be possible’, but it doesn’t tell us ‘*is* this rule possible?’

3. The Prolog language also points us in this direction. A Prolog database contains ANDs (sets of data) and rules (OR statements). Prolog’s Robinson resolution proof procedure, however, disallows us from inferring ORs arbitrarily; we can only add new facts to the database by following existing rules. Following this fairly restrictive procedure – rather than using the usual rules of classical logic which allow much more flexible inferences – turns out to be quite a simple way of managing paradoxes even in the face of contradictory data. I suspect there is a very deep reason for this, and that’s that ORs functionally act like rules and ANDs act like data; they are statements at two very different levels of abstraction.

In summary, I think ‘AND’ statements are statements about what is observed – but ‘OR’ statements are statements about *rules that govern the world*. As humans, I think we naturally grasp this level of abstraction associated with ‘OR’, that it’s saying something at a higher level of complexity than just ‘AND’… and we see this very clearly in the sense of unease the untrained logician gets when looking at the material conditional. The naive human idea of ‘if… then’ or implication is much more along the idea of ‘permanently true rule’ than ‘temporarily true-or-not-true truth function’. And the truth of a rule is something at a deeper level than the truth of a fact.

I think this is maybe what Tarski is getting at, and I think there’s something worth investigating there.

Regards, Nate

7. Stefan Koller says:

Short answer: I find your invocation of Grice highly compelling.

Long Answer: Regarding your: “But what’s the “no” doing here? It is denying that we will both go for a hike and go to the theatre.”
Indeed. You get a similar effect in German and Ancient Greek if you don’t use ‘or’ simpliciter but ‘either… or’ (the Greek actually uses the same word, ^etoi, for ‘either’ and ‘or’ in such constructions; cf. Aristotle’s Categories). In that case, the second use of ‘or’ takes on an exclusive meaning, but only because of the rest of the sentence (sentential context). I’d think English works similarly when the impatient parent goes,

“Look here, we’ll EITHER go to the movies OR climb that mountain.” (caps added for emphasis)

But if that’s right, then your idea that ‘or’ only has 1 meaning, namely inclusive, doesn’t strike me as correct. It rather looks to me that ‘or’ has (at least) two meanings, one standard (inclusive), and one that’s only triggered in the right sort of sentential context. Pretty much Grice’s point regarding ‘and’.

8. Rowsety Moid says:

One problem for the Gricean view is that words in natural languages have whatever senses they actually have. Natural languages don’t have to minimise the number of senses or have the number of senses that’s most convenient for a linguistic theory. (Even some people who think “or” doesn’t have an exclusive sense think it has more than one, btw. R. E. Jennings may be an example.)

So I ask this: since many people believe that “or” has an exclusive sense, and sometimes use “or” intending it to have that meaning, and can even be correctly understood as meaning exclusive-or, how can it make sense to say there’s no such meaning in English? Is it supposed to be impossible for any language to have a disjunction word that has both an inclusive and an exclusive meaning? Or if it is possible for a language, why can’t English be such a language?