I "Theory" in multi-valued logic?

[yossell]

A theory in mathematical logic is often understood to be a set of sentences closed under logical implication.

So understood, it doesn't matter whether the sentence was proved from axioms or using rules of inference. You can formalise the same theory either way. So the details about whether you're using a system of natural deduction or one with many axioms don't really matter.

So understood, the notion of a theory is indeed wider than 'the set of sentences true in a model M'. We can (and should) talk about theories without having to talk about the truth of the sentences of the theory at all. An inconsistent theory (classically understood) is the set of all sentences (of the formal language under study).

'The set of sentences true in model M' (assuming classical model theory) does pick out a theory, since 'true in M' is defined to be closed under classical logical implication.

This definition of a theory still works in non-classical logic -- for instance, an intuitionistic theory will be a set of sentences closed under intuitionistic implication.

How non-classical models treat truth in a model, and how they relate this notion to the non-classical logic they are models for, will probably vary in different theories. But I would avoid talking about the truth of a sentence of a theory independently of a given model for that theory (as some here seem to come close to doing) -- that discussion seems to go beyond what mathematical logic is about.

 

[SSequence]

'The set of sentences true in model M' (assuming classical model theory) does pick out a theory, ...

I don't quite understand your post but this sentence is interesting.

I don't understand how this would be the case? Consider the set of sentence true when we are quantifying on $\mathbb{N}$ (as usual). This set of sentences define what can be roughly called "truths of (true) arithmetic" [the (true) in bracket to imply that with "different natural-numbers/finite-ordinals than $\mathbb{N}$", we will have some "truths" that wouldn't be true in $\mathbb{N}$?]. Call this set $A$. Yet different concrete theories seem to prove different subsets of $A$ [1]? So how does the set of truths for (standard) natural numbers single out a unique theory? I suppose am missing something?

[1] OK I suppose one objection could be that a model for a stronger theory would have many more "objects" than just natural numbers. But still, on the very least, the stronger theory would still pose all the questions that can be posed (in weaker theory such as PA).

P.S.
This also got me thinking that does model theory "guarantee" that set-theory has a model with same "natural numbers" as $\mathbb{N}$? And if the answer to prev. question is "no", would the (possible) non-existence of such a model mean that the common-truths/theorems for all the models satisfying set-theory with "different" natural numbers [assuming that set-theory only proves "true things" (otherwise it seems to become hopelessly complicated for me if we only assume consistency)?] wouldn't agree with $A$? Or there could still be agreement with $A$?

But anyway, I don't even have enough superficial understanding of the complicated model theory related issues underlying this. More specifically, the idea of (non-standard) in natural numbers (and more generally ordinals) is quite unclear for me.

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Regarding the specific topic at hand, I don't have much to add.

 

[yossell]

Consider the set of sentence true when we are quantifying on $\mathbb{N}$ (as usual). This set of sentences define what can be roughly called "truths of (true) arithmetic" [the (true) in bracket to imply that with "different natural numbers than $\mathbb{N}$", we will have some "truths" that wouldn't be true in $\mathbb{N}$?]. Call this set $A$. Yet different concrete theories seem to prove different subsets of $A$ [1]? So how does the truths for (standard) natural numbers single out a unique theory? I suppose am missing something?

I can think of two things that may be worrying you.

1.
'Different concrete theories seem to prove different subsets.'
Yes -- but why is this a problem? Different theories and different subsets go hand in hand.

Begin with a first order axiomatisation of arithmetic and let A be the smallest set containing the axioms and closed under logical consequence. By Godel's incompleteness theorem, we know this theory is incomplete.

Contrast with the theory B: the set of sentences true in the intended model of PA. This theory is complete -- for every sentence of first order arithmetic, either A or ¬A is contained in B. However, this theory cannot be recursively axiomatised.

Accordingly, A and B are different theories -- but the set of sentences true in the standard model is a unique set.

2.
Different concrete theories can contain different predicates -- one theory might contain '+' and 'x'; another might contain '!', or other predicates. Yet, as '!' can be defined in standard PA, in terms of + and x, we do not think of the theory formulated with ! as a genuinely different theory -- even though the set of sentences is, strictly speaking, different.

I reply: Fair point! -- I agree it could be good to treat theories which are mere definitional extensions as not genuinely distinct theories at all.

[1] OK I suppose one objection could be that a model for a stronger theory would have many more "objects" than just natural numbers. But still, on the very least, the stronger theory would still pose all the questions that can be posed (in weaker theory such as PA).

Stronger in what sense? The trouble with the standard first order formulation of arithmetic is that it has non-standard models -- models that contain 'numbers' which are greater than every natural number. 'Stronger' formulations -- such as second order formulations -- rule out these 'extra' objects.

P.S.
This also got me thinking that does model theory "guarantee" that set-theory has a model with same "natural numbers" as $\mathbb{N}$? And if the answer to prev. question is "no", would the (possible) non-existence of such a model mean that the common-truths/theorems for all the models satisfying set-theory with "different" natural numbers [assuming that set-theory only proves "true things" (otherwise it seems to become hopelessly complicated for me if we only assume consistency)?] wouldn't agree with $A$? Or there could still be agreement with $A$?

I'm not sure where you're going with this -- in model theory, we don't expect and can't get one theory's models to have the 'very same' domain as another theory's models. If T has a model M, and if M' is isomorphic to M, then M' is a model of T also, even if its domain is different.

 

[SSequence]

1.
'Different concrete theories seem to prove different subsets.'
Yes -- but why is this a problem? Different theories and different subsets go hand in hand.

Begin with a first order axiomatisation of arithmetic and let A be the smallest set containing the axioms and closed under logical consequence. By Godel's incompleteness theorem, we know this theory is incomplete.

Contrast with the theory B: the set of sentences true in the intended model of PA. This theory is complete -- for every sentence of first order arithmetic, either A or ¬A is contained in B. However, this theory cannot be recursively axiomatised.

Accordingly, A and B are different theories -- but the set of sentences true in the standard model is a unique set.

Yes, I guess I understand what you are saying here. I was "assuming" that something like (or similar to): "recursive enumerability of theorems" was a necessary pre-requisite condition when you wrote theory.

Stronger in what sense? The trouble with the standard first order formulation of arithmetic is that it has non-standard models -- models that contain 'numbers' which are greater than every natural number. 'Stronger' formulations -- such as second order formulations -- rule out these 'extra' objects.

I just mean stronger in the sense that it proves "more" facts about the (real) natural numbers. Just like a "stronger theory" might prove con(PA) [alongside with everything that PA proves] but PA itself wouldn't.

I'm not sure where you're going with this -- in model theory, we don't expect and can't get one theory's models to have the 'very same' domain as another theory's models. If T has a model M, and if M' is isomorphic to M, then M' is a model of T also, even if its domain is different.

What I was asking was whether model theory proves whether set-theory must have a model with same "finite ordinals" as $\mathbb{N}$ (the actual natural numbers).

And the further (secondary) question was that if the answer is no, then does it (necessarily) imply [assuming set-theory to prove only "true things"] that set-theory proves facts about "finite ordinals" which do not match with "truths of arithmetic" [i.e. truths of questions posed by quantification over $\mathbb{N}$ (actual natural numbers)]?

OK I did a quick search. This question (it comes out as very first link) seems to be close to what I am asking:
https://math.stackexchange.com/questions/647480

Point-1 by the user CarlMummert seems to answer the first-part perhaps? (but I need to read carefully what $\omega$-model means actually...)
"First, three caveats:
1. Nothing that we can prove within ZFC can justify this. It would be naively possible that ZFC is consistent but not ω-consistent (and thus has no ω-model), and in that case we could still prove all the same things in ZFC."

 

[Stephen Tashi]

It is an interesting question as to the difference between an axiom and a rule of inference,

It's indeed an interesting question!

The study of formal languages (and mathematical exposition in general) assumes the student has certain basic perceptive abilities for symbols and patterns of symbols and that these symbols and patterns have basic permanent properties similar to permanent properties of physical objects.

For example, the student must be able to perceive whether two symbols at different locations "are the same" - i.e. are the same with respect to some properties but different with respect to the property of location - however location is imagined - different location in a sequence of printed symbols from left to right or different location on an imaginary tape in a Turing machine etc.

We use "ordinary" logic to reason about strings of symbols and think of the symbols as having properties than correspond to properties of physically implemented symbols (e.g. printed symbols).

There is a clear separation between the metalanguage and metalogic versus the content of the language until we begin to use symbols to denote things that happen in the metalanguage.

For example if the rules in the metalanguage allow us to begin with the sequence of symbols "$P \land Q$" and write down other sequences that conclude with "$Q$" we can summarize this fact by saying "$Q$" is "derivable from "$P \land Q$" and abbreviate it as "$"P \land Q" \ \vdash \ "Q"$.

Do we have terminology to distinguish statements that refer only to manipulations of symbols in the language versus statements that refer to some intepretation of the symbols? (e.g. the distinction between $(P \land Q) \implies Q$ versus $"P \land Q"\ \vdash\ "Q"$). $\$ Is an axiom one type of statement and rule of inference the other type?

 

[nomadreid]

It has always seemed to me that a rule of inference occupied a funny place somewhere between semantics and higher-level syntax , between model and theory.

On one hand, it refers to the truth values of statements, hence attached to the semantics of the theory.

On the other hand, it is equivalent to a higher-order statement quantifying over the sentences of the theory, with the truth values now being constants in a corresponding higher-order theory.

So, clearly not belonging to the theory but also not belonging to an interpretation in the sense of sets which satisfy sentences, it seems to inhabit a third level somewhere.

I haven't come to terms with where that third level belongs. Just saying that it is metalogical seems to be hand-waving.