Can a sentence without a truth-value ever be approximately true?

I am curious to hear people's thoughts on the question in the title. There has been a lot of philosophical work done on the idea that a sentence can be strictly speaking false, yet nonetheless approximately true (or 'truthlike' or 'verisimilar'). For example: I am 5'11", but if someone said 'Greg is 6 feet tall,' we want to say that that claim is approximately true or something like that. But what if the claim was (strictly speaking) neither true nor false? (Readers may insert their own favorite truth-valueless sentence here.)

I ask because, as I mentioned in an earlier post, I'm toying with the idea that the Pessimistic Induction over the history of science plus something like Kuhnian incommensurability (esp. untranslatability) will lead us not to the conclusion that current science is likely to be false, but rather is likely to lack a truth-value. For if we cannot translate the claims of a pre-revolutionary language into the post-revolutionary one, then the pre-revolutionary language (from our current point of view) is truth-valueless, not false.

I ask the question in the title because one common realist response to the Pessimistic induction is: "Well, yes, our current scientific theories are probably not exactly true, but they are approximately true." If truth-valueless sentences cannot be approximately true, then this response is not available to the realist.


N. N. said...

There's a line from Aristotle that I can't quite put my finger on concerning approximate truth. It says something to the effect that no statement is completely false, i.e., there's some truth in any statement. I'll see if I can't dig it up.

It seems to me that one of the things that could be meant by saying a position is approximately true is that it's partially true. That is, parts of it are true. I'm thinking of a conjunction of statements, most of which are true, but some of which are false. The few false one's make the position false, but there is some sense to saying it's mostly true.

Greg Frost-Arnold said...

Hi N.N.,

I can go along completely with the idea in your 2nd & final paragraph, but it doesn't appear to get at the question I was asking: in the big conjunction, every single atomic sentence is either true or false -- that is, every sentence has a truth-value.

Perhaps though your idea can be modified slightly to answer my original question: instead of having a false belief in a big conjunction with true atomic sentences, switch that false one to a sentence without a truth value. Then the whole conjunction would be 'mostly true' or something like that. (Or, if we adopted the strong Kleene scheme for 3-valued logic, we could say sentences of the form '[T] or [truth-valueless]' are true simpliciter, and sentences of the form '[F] and [truth-valueless]' are false period, without the 'approximate' or 'partial' business.)

Martin Cooke said...

Is the sense of approximation for the realist something like when a mathematical model is a good model of something that is in itself unknown, i.e. its predictions are mathematically close to observations? Surely such a model might not claim to be a true representation of reality, i.e it might lack truth-values?

Anonymous said...

Inconsistent sentences might provide an interesting example. To use a silly example - consider the claim 'everything I say is false.' This cannot be assigned a truth value. But it surely might be approximately true (perhaps, after all -most- things I say are false.)

Cecil said...

Sorry, I ought to add more detail to the above. The scenario I am imagining is something like this: I have just uttered a long list of falsehoods. Realizing this, I say 'everything I say is false'. (Or perhaps - everything I say today is false.) If this sentence is true, then it is false. So it cannot be true. If it is false, then something I say/have said is true. But everything in the long list of sentences previously uttered is false, and we just showed that 'everything I say is false' is also false. So 'everything I say is false' can't be false.

Greg Frost-Arnold said...

Hi Kevin --

Thanks for stopping by. After your first comment, I thought you might have that example from Kripke's "Notes on a Theory of Truth" in mind. And that is probably as good an example of 'approximately true but truth-valueless' as we can hope for.

But 2 more questions:
(1) Are there any examples that are nearly as good that do not involve semantic (or other linguistic) concepts?

(2) I haven't looked at Kripke's "Notes..." in several years, but wouldn't his theory of truth account for this example? (I'm just thinking: Why else would he mention this example, if it weren't a case where his theory outperforms Tarski's?)

Greg Frost-Arnold said...

oops -- now I need an addendum. Kripke's paper was "Outline of a Theory of Truth"...

Robbie Williams said...

I guess it might depend on how you think of truth-value gaps.

Suppose you're a supervaluationist about the relevant truth-value gaps. Then you'll have at your disposal a bunch of "precisifications" of the language, each of which will classify each of your sentences as either true or false. And then you could apply your standard notion of verisimilitude over the precisifications.

I guess it'll be a bit delicate, how exactly to extend the verisimilitude talk to the original (gappy) sentence. But one option might to supervaluate again.

So if S has two precisifications, p1 and p2, with p1 false and p2 true, I guess p2 will have maximal verisimilitude, and say p1 has verisimilitude of degree d, then one might say that the sentence S has verisimilitude of indeterminate degree, but determinately at least of degree d.

(Supervaluationists are sometimes prepared to talk of sentences being "nearly" true if they're true on most precisifications (where truth=truth on all precisifications), which is another complicated factor in the vicinity. At a guess I'd say it's orthogonal to the verisimilitude issue).

Greg Frost-Arnold said...

Hi Robbie --

Thanks for that - it strikes me as eminently reasonable. 2 further questions:

1. Do you have a reference for someone saying (something in the neighborhood of) 'true in almost all precisifications' = 'nearly true'?

2. I'm curious: what's the rationale for your "guess" at the end of your comment that 'nearly true' (in the 'true in almost all precisifications' sense) is orthogonal to approximate truth? I certainly agree that they're distinguishable phenomena (since truth-valuelessness is certainly different from falsehood), but it seems like they might be something like species of a single genus... or at least analogous. (Near truth in a setting that allows truth-value gaps is analogous to approximate truth in a setting without gaps or gluts.)

Robbie Williams said...

Hi there,

I can't think of a reference off the top of my head (though I'm pretty sure I've seen it around). Supervaluational degree theorists (cf. Williamson 1994 5.5; Lewis 1970, Kamp 1975) will say that the more precisifications you're true on, the truer you are. But that's not exactly the idea here. If I think of the reference, I'll post it.

The reason I was thinking that approximately true and nearly true were orthogonal, is that it seems to me that it makes sense to talk about precise sentences being apprxoimately true (e.g. a precise statement of newtonian physics) whereas those thinks aren't in the running to be nearly true. And conversely, paradigm instances of indeterminacy might well be classified as nearly true, while not looking like cases of approximate truth. E.g. suppose we stipulate that all numbers less than 6 are nice, and numbers greater than 6 are not nice. Is 6 nice? Fine et al think this is indeterminate: true on one precisification, false on another. I don't think this is in the running for being "approximately true" in anything like the sense in play in the science case.

But maybe something clever could be done to unify the notions: really all I was concerned with is not to commit myself to any sort of unification in the comment.