4/03/2006

Tarski, Quine, and logical truth

The following must have been addressed already in the literature, but I'm going to mention it anyway--perhaps a better-informed reader can point me in the direction of the relevant research. W.V.O. Quine offers the following characterization of logical truth:
The logical truths are those true sentences which involve only logical words essentially. What this means is that any other words, though they may also occur in a logical truth (as witness 'Brutus,' 'kill,' and 'Caesar' in 'Brutus killed or did not kill Caesar'), can be varied at will without engendering falsity. ("Carnap and Logical Truth," ยง2)
All I wanted to mention here is that Alfred Tarski had already shown, in 1936's "On the Concept of Logical Consequence," that Quine's characterization is a necessary condition for a sentence to be a logical truth, but not a sufficient one. For example, if one is using an impoverished language that (i) only has proper names for things over five feet tall, and (ii) only has predicates applying only to things over five feet tall, then the sentence 'George W. Bush is over five feet tall' will be a logical truth -- because no matter what name from this impoverished language we substitute for 'George W. Bush' or what predicate we substitute for 'over 5 feet tall' in this sentence, the resulting sentence will be true.

Now some historical questions: did Quine think his condition was sufficient, or just necessary? (I quickly checked "Truth by Convention," and I didn't find any conclusive evidence that he considered it sufficient.) If Quine does consider this a proper definition of logical truth, how does he/ would he answer Tarski's objection? -- and/or why doesn't Quine simply adopt Tarski's definition of logical truth, viz. 'truth in all models'?

(You might think Tarski's objection shouldn't count for much, since I used a very contrived language to make the point against Quine. In Tarski's defense, however, (a) assuming that every object has a name in our language also seems somewhat artificial, and (b) Tarski proved (elsewhere) that a single language cannot contain names for all (sets of) real numbers (See "On Definable Sets of Real Numbers," reprinted in Logic, Semantics, Metamathematics).)

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3 Comments:

At 6/5/06, Blogger Ole Thomassen Hjortland said...

I believe Quine was aware of the distinction pointed out by Tarski (Tarski actually writes about this in his 1933 monograph on truth as well). In his 'Philosophy of Logic' (1970), Quine discusses the difference between objectual (referential) and substitutional quantification precisely with respect to the size of the language.

This point is closely tied to the definition of truth and logical consequence (Etchemendy (1990) shows that the type of quantification is essential for these definition), so it would be surprising if Quine was not aware of the analogy.

However, these remarks by Quine is made many decades after he wrote 'Truth by convention'. The latter paper was written in the early thirties, so it's quite possible that Quine wasn't aware of the problem at that point.

 
At 8/5/06, Blogger GF-A said...

hi Ole --

You are of course right that the place to look for Quine's mature view on the relation between his characterization of logical truth and Tarski's is in Q's Philosophy of Logic. I realized this shortly after posting this entry, so I pulled out the book, and there's even a section directly comparing Quine's account to Tarski's. However, some of what Quine says there confused me, and I'm actually now still trying to understand it -- if I manage to figure it out, I'm probably going to post something about it.

But there was one thing about your comment that I wondered about -- Quine is committed to objectual quantification being the "real" or "fundamental" tpe of quantification, and that carries over to his discussion of logical truth as well. That is, from what I could tell in Philosophy of Logic, the distinction [objectual quantification/ substitutional quantification] does not match up with the distinction [Tarskian logical truth/ Quinean logical truth].

 
At 9/5/06, Blogger Ole Thomassen Hjortland said...

Hi,

I must admit that Quine's position is a bit puzzling. I had to look up some of it to remind myself. It appears that Quine is arguing that substitutional logical truth (Bolzanos logical truth) and model-theoretic logical truth (Tarski logical truth) are equivalent, but he adds that the language must be "sufficiently rich", i.e., rich enough for elementary number theory (see p. 53 in Phil. of Logic). But I can't see that he anywhere explicitly points to the fact that substitutional logical truth will overgenerate in cases where the non-logical vocabulary is sufficiently sparse.

Regarding the analogy to quantification, I realise that my comment was not very clear. My point was that it would be strange if Quine endorsed substitutional logical truth unrestricted (that is, without the reminder about size of the non-logical vocabulary) since he discusses the related problem with substitutional quantification (namely that it will not capture entities we are ontologically committed to, but which are not named, e.g., some real numbers).

I am looking forward to your next post on the topic.

 

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