What is "meaningful" for a multi-valued logic?

[nomadreid]

In theories with a two-valued logic, a sentence is "meaningful" if it is (with respect to a given model) either true or false. Does this definition need to be modified for multi-valued logics? If so, how?

 

[Svein]

In theories with a two-valued logic, a sentence is "meaningful" if it is (with respect to a given model) either true or false. Does this definition need to be modified for multi-valued logics? If so, how?

I'm on thin ice here, but I would think that a multivalued logical expression would be "meaningful" if its truth value can be calculated.

 

[nomadreid]

Thanks, Svein, this is also my intuition, but one thing that mathematics (and physics) has taught me is not to trust my intuition.

 

[WWGD]

Isnt 'meaningful" here equivalent to decidable? Usually there is a way of mapping from syntactic to semantic, i.e., like you said, assigning truth value to atoms and predicates so that the truth value of sentences can be computed. I guess you need something of a similar sort, but then again a many-valued logic may not just be assigned vaues in {T,F} , but maybe intermediate values too, I would believe.

 

[nomadreid]

Thanks, WWGD, but "meaningful" and "decidable" are two different things. " "Decidable" is essentially "provable" (with respect to a given theory), whereas a meaningful statement need not be decidable in that theory, although it might be in a more powerful one. The Gödel sentence is meaningful but not decidable in Peano arithmetic. I should have been stricter in my reply to Svein, who wrote that the truth value could be "calculated". Yes, but not necessarily using only the tools of the theory in question . For example, the Gödel sentence can be decided in a new system (more powerful theory, which can end up with a different model), but not in Peano arithmetic. To put it another way: the domain of the interpretation function of a meaningful sentences is a lattice of truth values: for decidable ones one knows which one. Sort of like the difference between a variable and a constant.

 

[ShayanJ]

I'm on thin ice here, but I would think that a multivalued logical expression would be "meaningful" if its truth value can be calculated.

This is the most natural generalization of the two-valued case. I don't see anything wrong with it!

 

[nomadreid]

Svein, your statement "a... logical expression would be 'meaningful' if its truth value can be calculated." runs into the problem with the word "calculated", which I am assuming to mean "calculated on the base of the given theory". "Meaningful" is broader than "computable".Gödel's sentence can be assigned a truth value without causing the system to be inconsistent: you can choose, T or F, as you wish, but there is no way you can calculate that truth value from the given theory.
The generalization of the two-valued case would be that a meaningful proposition can be assigned a truth value in some lattice. I also do not see what would be wrong with the (correctly stated) "natural" generalization, but that is why I posted the question, in case there is something I am overlooking that would require "meaning" to be restricted to the two values.

 

[ShayanJ]

Svein, your statement "a... logical expression would be 'meaningful' if its truth value can be calculated." runs into the problem with the word "calculated", which I am assuming to mean "calculated on the base of the given theory". "Meaningful" is broader than "computable".Gödel's sentence can be assigned a truth value without causing the system to be inconsistent: you can choose, T or F, as you wish, but there is no way you can calculate that truth value from the given theory.
The generalization of the two-valued case would be that a meaningful proposition can be assigned a truth value in some lattice. I also do not see what would be wrong with the (correctly stated) "natural" generalization, but that is why I posted the question, in case there is something I am overlooking that would require "meaning" to be restricted to the two values.

So I guess you have problem with the word "calculated". If that's the case, well you're right but that's a problem in the two-valued case too. Let's use the word "assigned".
But you're actually mentioning a problem with two-valued logic to rule out a definition in multi-valued logic. Just think of a system in a multi-valued logic.Imagine it has some kind of a Gödel sentence which can't be decided. But as you said, we can still assign a truth value to it arbitrarily(and use it as a new axiom I guess!). So it seems to me the situations are similar and if the Gödel sentence can't prevent us in assigning truth values in two-valued logic, it shouldn't cause such troubles in the multi-valued case too.
Of course you're still right to think there may be a problem we're overlooking. But at least we now know that such problem should be non-trivial and needs closer inspections than such general and sloppy discussions.

 

[nomadreid]

Thanks, Shyan. And my apologies for addressing you as Svein in my last post.

So I guess you have problem with the word "calculated". If that's the case, well you're right but that's a problem in the two-valued case too. Let's use the word "assigned".

Indeed, that is good. In the two valued case, it is meaningful if it is possible without contradiction to assign a truth value, and the generalization in question is whether the possibility of assignments of other values besides T or F are still meaningful. The question arises, for example, in quantum physics, where many argue that until a particle's spin is measured, the expression "the electron's spin" is meaningless. That is, the values of 1/2 or -1/2 render it meaningful, but not 1/√2.

But you're actually mentioning a problem with two-valued logic to rule out a definition in multi-valued logic.

Actually, Gödel's Incompleteness Theorems have been extended to multi-valued and fuzzy logics. Of course many multi-valued logics are complete, just as many two-valued ones are, but some incompleteness results are listed in Section 8.5 of "Many Valued Logics 1: Theoretical Foundations" by Bolc and Borowik.

Just think of a system in a multi-valued logic.Imagine it has some kind of a Gödel sentence which can't be decided. But as you said, we can still assign a truth value to it arbitrarily(and use it as a new axiom I guess!).

Yes, and so now we have a new system, which will have yet another undecidable sentence. So adding the new axiom does not get us out of undecidability.

So it seems to me the situations are similar and if the Gödel sentence can't prevent us in assigning truth values in two-valued logic, it shouldn't cause such troubles in the multi-valued case too.

It will cause the same problem in both. But I brought up the Gödel sentence only to point out the difference between "meaningful" and "decidable"; since we have that out of the way, the Gödel sentence is no longer relevant to the question.

at least we now know that such problem should be non-trivial and needs closer inspections than such general and sloppy discussions.

Yes, indeed. Since I am pretty certain that this question has been looked at by those working in the field, ideally someone in Logic would be a contributor to this Forum and enlighten us.