Philosophy Now Forum

godelian wrote: ↑Sun Nov 17, 2024 5:28 pm The truth of a theory is the collection of sentences that are true in its standard model.

That's just more circular reasoning.

Given a bag of sentences what identifies any given sentence as "true in the standard model"?

What makes the sentence "1+1=2" true in the standard model?

Skepdick wrote: ↑Sun Nov 17, 2024 5:55 pm

godelian wrote: ↑Sun Nov 17, 2024 5:21 pm You originally wrote that your views are in line with Tarski's work, and not in violation of it, but now you write that "Tarski's theory of truth is circular". You keep moving the goalpost.

Those two sentences are not mutually incompatible. What is it that you don't understand?

"1+1=2" is true is 1+1=2 is true.

This fails to address WHY it's true.

For some sentences in PA's standard model, we know why they are true, as we can prove them from PA. For the vast majority, however, it is impossible to say why they are true, because they are true but unprovable:

http://www.sci.brooklyn.cuny.edu/~noson ... ovable.pdf

True But Unprovable
by Noson S. Yanofsky

The world that Gödel described can be extended. Of all the mathematical facts, only some are expressible with language. Any mathematical fact that is provable is automatically expressible because proofs must be done in a language. This figure is not drawn to scale because the expressible facts are miniscule within the collection of all facts.

We have come a long way since Gödel. A true but unprovable statement is not some strange, rare phenomenon. In fact, the opposite is correct. A fact that is true and provable is a rare phenomenon. The collection of mathematical facts is very large and what is expressible and true is a small part of it. Furthermore, what is provable is only a small part of those.

godelian wrote: ↑Sun Nov 17, 2024 6:00 pm For some sentences in PA's standard model, we know why they are true, as we can prove them from PA.

Skepdick wrote: ↑Sun Nov 17, 2024 3:38 pm axiomatization on its own is worthless, because it trivially follows from Godel's completeness theorem which equates truth with provability in 1st order logic. So provable = true?!?! But then that's proof theory, not model theory.

godelian wrote: ↑Sun Nov 17, 2024 6:00 pm For the vast majority, however, it is impossible to say why they are true, because they are true but unprovable:

You can't even explain why the provable ones are true!

They are a direct consequence of the axioms - yes.
IF the axioms are true
THEN the sentence is true.

But without soundness you don't know that the axioms are true.

Without soundness proof != truth. Proof only justifies validity.

Skepdick wrote: ↑Sun Nov 17, 2024 6:05 pm You can't even explain why the provable ones are true!

The model is constructed like that.
It is not a model unless all sentences provable from the theory are true in the model.

godelian wrote: ↑Sun Nov 17, 2024 6:27 pm The model is constructed like that.
It is not a model unless all sentences provable from the theory are true in the model.

Why do you keep talking in circles?

What does it mean for "1+1=2" to be "true in the model"?

YOU insisted on drawing a distinction between proof theory (syntax) and model theory (semantics).
YOU insisted that there is a distinction between syntactic provability and semantic truth!

I know that 1+1=2 is provable in PA.
I am asking you if it's true in PA.

Skepdick wrote: ↑Sun Nov 17, 2024 6:30 pm What does it mean for "1+1=2" to be "true in the model"?

Since "1+1=2" is provable from PA, it will be true in all its models. Any arbitrary set (along with the symbols + and =) in which this sentence is true -- as well as all other sentences provable from PA -- is a model of PA. The natural numbers { 0, 1, 2, ... } is such set.

godelian wrote: ↑Sun Nov 17, 2024 6:34 pm Since "1+1=2" is provable from PA, it will be true in all its models.

You said there's a difference between proof theory and model theory.
A distinction between syntax and semantics!

You said there's a distinction between truth and provability.

I know that 1+1=2 is provable in PA.
I am asking you what makes it true in PA.

godelian wrote: ↑Sun Nov 17, 2024 6:34 pm Any arbitrary set (along with the symbols + and =) in which this sentence is true

What makes the sentence true?!?!?

godelian wrote: ↑Sun Nov 17, 2024 6:34 pm -- as well as all other sentences provable from PA -- is a model of PA. The natural numbers { 0, 1, 2, ... } is such set.

That makes it provable in PA!

Why is that true in PA?

Skepdick wrote: ↑Sun Nov 17, 2024 6:36 pm Why is that true in PA?

That sentence is provable in PA. It is true in the natural numbers and all other models that interpret PA. Not only the natural numbers interpret PA. There are many more sets that do.

godelian wrote: ↑Sun Nov 17, 2024 6:38 pm

Skepdick wrote: ↑Sun Nov 17, 2024 6:36 pm Why is that true in PA?

That sentence is provable in PA. It is true in the natural numbers and all other models that interpret PA. Not only the natural numbers interpret PA. There are many more sets that do.

Skepdick wrote: ↑Sun Nov 17, 2024 6:36 pm You said there's a difference between proof theory and model theory.
A distinction between syntax and semantics!

You said there's a distinction between truth and provability.

I know that 1+1=2 is provable in PA.
I am asking you what makes it true?

Skepdick wrote: ↑Sun Nov 17, 2024 6:40 pm I know that 1+1=2 is provable in PA.
I am asking you what makes it true?

You need to hunt down all the sets in which it is true. It happens to be true in the set {0,1,2, ... }.
You can check this by evaluating the arithmetic expression by using the sum algorithm.

godelian wrote: ↑Sun Nov 17, 2024 6:52 pm

Skepdick wrote: ↑Sun Nov 17, 2024 6:40 pm I know that 1+1=2 is provable in PA.
I am asking you what makes it true?

You need to hunt down all the sets in which it is true.

What would make it "true" in any set?

godelian wrote: ↑Sun Nov 17, 2024 6:52 pm It happens to be true in the set {0,1,2, ... }.

What makes it true in the set {0,1,2, ... }?

godelian wrote: ↑Sun Nov 17, 2024 6:52 pm You can check this by evaluating the arithmetic expression by using the sum algorithm.

What makes an evaluation "true"?

Belinda wrote: ↑Sun Nov 17, 2024 1:33 pm True. Also true, religious people and also non-religious may be indoctrinated and both religious and non-religious frequently are indoctrinated.

Yes, of course. One can be indoctrinated by anything, if one has no sufficient reason to believe in that thing.

Religious doctrines must be combined with liberal education

Why? If any ideology can be indoctrinatory, what makes you think "liberal education" is magically exempt from that?

There must be as much autocracy as befits a liberal legal system.

Autocracy is not liberal.

Religions must be democratic...

Again, why?

Why "must" Islam, Hinduism or Catholicism, for examples, be "democratic"? They certainly aren't, by nature.

When religions support castes the religions have become cultish.

So you would regard Hinduism as a "cult," then? Because that's where you find "castes."

But hierarchy is a feature of reality. Things automatically rank themselves by quality. The only way to eliminate all hierarchies is to eliminate all quality, all standards, a basis of excellence, and make everything reduce to the lowest possible level of function.

Why would that be a good thing?

...the saints and the seers of all great religious traditions are held to be saints and seers, because they have singlehandedly taught pagans more civilised ways to live together. I include of course, not only Jesus but also Muhammad , Mandela, and Confucius.

Well, Mandela wasn't a religious figure, so far as I know. And Mo married a six-year-old, then sexually invaded her at age 9, as any imam can proudly tell you. That in addition to his habit of killing anybody he regarded as an "infidel." I'm not sure just how much more "civilized" we ought to regard that to be...And even in the case of Jesus Christ, I wouldn't say that teaching "civilization" was His primary mission.

Jesus ,Confucius, the Buddha, and Muhammad for instance merited absolute trust...

Hmmm...see above.

Superstition! Yes that's a problem.

Indeed. But who gets to say what it a "superstition," and what is a legit "religious" belief, and what is truth? It's certainly not self-evident. For example, is errant or poorly-theorized science "superstitition"?

Skepdick wrote: ↑Sun Nov 17, 2024 7:24 pm What makes it true in the set {0,1,2, ... }?

In the case of 1+1=2, it is true in the natural numbers because it is provable from PA.

In the case of unprovable but true statements, we cannot easily discover them and we usually don't know why they are true in the natural numbers. We just know that the unprovable statements outnumber the provable ones.

"What makes an evaluation "true"?"

OMG, what makes it true is what meets the agreed-on criteria established between two or more conventional language users. Functions like LEM and LOI in logic are 'true' because they derive from our particular psychologistic physiology... the logic gate structure of nerve activity in the brain makes experienced reality correspond to certain principles a priori, i think. Likewise, a fly, even though it cannot reason, directs its behavior as if either the french fries exist or they do not (LEM). And, it does this so consistently as to suggest that the fly can not be doing it accidentally or arbitrarily. In which case, insofar as we call the fly a goal oriented intentional machine, we know by its similar nervous system that it too must be presented with conceptual certainty of its environment by the inner workings of its brain, and that, like Aristole, the fly asserts with a certainty it can't deny that the french fries are the french fries and are not anything else. Its brain makes it believe this.

'True' evaluations are, as N kinda once put it, ones that have worked for a protracted period and have not on account of the danger they would create if untrue, jeopardize the animal making the judgements by being thought to be true.

Logic and math are experential nuances created by ionized particles traveling membranes and crossing clefts. If a fly had more of a brain and a language, he'd be able to make a symbolic representation of the objects and relations in his world and, in doing so, would be employing the same rules (as us) in those representations. Indeed, I'm saying that flies are mathematicians.

promethean75 wrote: ↑Mon Nov 18, 2024 3:40 am "What makes an evaluation "true"?"

OMG, what makes it true is what meets the agreed-on criteria established between two or more conventional language users. Functions like LEM and LOI in logic are 'true' because they derive from our particular psychologistic physiology... the logic gate structure of nerve activity in the brain makes experienced reality correspond to certain principles a priori, i think.

Psychologism is indeed an alternative ontology in mathematics. However, I was using Tarski's semantic theory of truth to elucidate what truth means in model theory. Tarskian truth of a sentence means that it is a particular fact in particular model-theoretical structure, i.e. the model or universe for the theory. The term "structure" is defined in universal algebra:

https://en.wikipedia.org/wiki/Structure ... cal_logic)
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it.

From the model-theoretic point of view, structures are the objects used to define the semantics of first-order logic, cf. also Tarski's theory of truth or Tarskian semantics.

For example, the structure [ {0,1,2, ...}, {0,1,+,x} ] contains the natural numbers {0,1,2, ...} and fixes the interpretation of the non-logical symbols {0,1,+,x}. This approach allows us to define the truth according to a correspondence theory: A sentence S in the language L of theory T is true, if S is true in model M and model M interprets theory T. This correspondence theory for truth is called Tarski's Semantic Theory of Truth:

https://iep.utm.edu/s-truth

The Semantic Theory of Truth

The semantic theory of truth (STT, hereafter) was developed by Alfred Tarski in the 1930s. The theory has two separate, although interconnected, aspects. First, it is a formal mathematical theory of truth as a central concept of model theory, one of the most important branches of mathematical logic. Second, it is also a philosophical doctrine which elaborates the notion of truth investigated by philosophers since antiquity. In this respect, STT is one of the most influential ideas in contemporary analytic philosophy. This article discusses both aspects.

Tarski's semantic theory of truth is the dominant theory for truth in mathematics.