Logic of truth

Banno

I’m re-starting the thread that was locked by the mods because it was thought to overlap with another thread.

I’m doing this in good faith, after due consideration and in recognition that the mods may well decide to lock it again. I maintain that the content here is parallel to but distinct from the material in the other thread, that it is substantial enough to constitute a complete thread, that it is better positioned here in the logic sub-forum in the hope of attracting attention from those around here who have a more substantial understanding of logic, so they can point out where I go wrong.

Basically this thread is exegesis, while the other thread is a general discussion. So if you want to add your own theory of truth, do it on the other thread.

And besides, it will keep my posts together in a way that allows me to track my own reasoning.

In my enthusiasm, which might well evaporate, I anticipate discussing Tarski's T-sentences, Kripke's three-valued logic and revision theory.

Tarski's strategy
There's a strategy that Tarski used in his original paper, and that Davidson later mirrored. I'll have a go at paraphrasing it.

We want a theory of truth.

We can ask what such a theory might look like. If it is adequate to its task, it will deliver, for every sentence, something that tells us if that sentence is true.

So it will have the following form, which is not yet a T-sentence:

For any sentence p, p is true if and only if ϕ

Further, to avoid circularity, the notion of truth cannot occur in ϕ.

And finally, this will not work for a language strong enough to talk about its own sentences, because directly it will be able to generate a sentence of the form

This sentence is false

Putting these together, if we have as one of our sentences

Snow is white

then our theory will produce a sentence in the metalanguage that looks like

"snow is white " is true iff s

where s is a sentence in the metalanguage.

You should be able to see where this is going. All we need to do now is work out what s might be. So far Tarski is setting out what is needed for any theory of truth. A bare minimum is that it generate for each sentence in our language something that is true exactly when that sentence is true.

Since this assumes that these sentences are either true or false, it assumes realism. That is, so far, nothing Is needed along the lines of proof, knowledge, belief, justification or whatever for our sentences to be true.

Kripke’s theory doesn’t make this presumption.

Banno

Designation and Satisfaction
So we have, as a general form for any theory of truth, what Tarski called "Material adequacy",

For any sentence p, p is true if and only if ϕ

And we want to understand what ϕ is.

And we have that in order to avoid the Liar Paradox, we avoid having a language that can talk about itself. Instead, we employ a second language, and use it to talk about the truth of our sentences. We call this the metalanguage, and it talks about the object language. Our sentence "For any sentence p, p is true if and only if ϕ" is a part of the metalanguage, referring to any sentence p of the object language and ϕ is a sentence in the metalanguage

So what is ϕ?

The obvious solution is that ϕ and p are the same. ϕ=p.

But the problem here is that ϕ and p are in different languages. In the metalanguage, p is effectively a name for a sentence in the object language.

Tarski worked around this by introducing terms in his metalanguage that refer to the same thing as terms in the object language; the notion of designation; and then using this to define truth in terms of satisfaction.

Suppose we restrict the object language to being about a group of people, Adam, Bob and Carol...

And in the metalanguage we can have a definition of "designates":

A name n designates an object o if and only if (( n = "Adam" and o = Adam) or ( n = "Bob" and o = Bob) or( n = "Carol" and o = Carol)...

Doubtless this looks cumbersome, despite my having skipped several steps, but it gives us
a metalanguage and and object language both talking about the same objects, Adam, Bob and Carol..., and a way to use the same name in both languages.

We want to add predication. To do this, Tarski developed satisfaction. Suppose we have two nationalities in our object language, English and French. We need a way of talking aobut those nationalities in the metalanguage. We can define "satisfaction":

An object o satisfies a predicate f if and only if ((f="is english" and o is English) or (f="is french" and o is french)

And so, in a cumbersome way, we have the object language and the metalanguage talking about the same predicates and objects.

Here I've used finite lists, but it is possible to construct similar definitions for designation and satisfaction for infinite objects and predicates, and for n-tuple predicates. I'm just not going to do it here.

Banno

Defining truth

So now we have the material adequacy condition for a theory of truth, together with definitions of designation and satisfaction that serve to allow us to talk about the object language using the metalanguage.

Putting these together we get, in the metalanguage, a definition of the truth of any sentence in the object language.

We have

For any sentence p, p is true if and only if ϕ

and given designation and satisfaction, we can take any sentence p in the object language and develop a sentence in the metalanguage that both designates the same things and is satisfied in the same conditions.

So we take any sentence p of the object language, apply to it our definitions of designation and satisfaction, and produce a new sentence in the metalanguage that means the very same. We might call this new sentence in the metalanguage S, and write

For any sentence p, p is true if and only if S

Tarski started with a general requirement for any theory of truth, then used satisfaction to show that what was needed for ϕ is another sentence.

The more casual way of setting this out is to treat p as a sentence in the metalanguage that has the same conditions of satisfaction as a sentence in the object language, and then to name that sentence in the object language "p". p is some sentence in the metalanguage, "p" a sentence in the object language that satisfies the same conditions, and so

"p" is true if and only if p

Banno

An implication for any language.
Stepping up on level, what Tarski has done is to set out material adequacy as a condition for any theory of truth, then analyses meaning in terms of satisfaction, and use satisfaction to define truth. He was able to tie meaning down using the notion of satisfaction, and use it to define truth.

We have material adequacy:

For any sentence p, p is true if and only if ϕ

and we tie meaning down by sticking to one sentence, so that the meaning cannot be ambiguous. We name the sentence on one side, and use it on the other.

"p" is true if and only if p

...and hey, presto, we have a definition of truth.

A diversion, which I am allowed to do because no one has read this far. In Truth and Meaning and elsewhere, Davidson flips this around. Where Tarski held down meaning and defined truth, Davidson holds down truth to get at meaning.

So Davidson takes

For any sentence p, p is true if and only if ϕ

and points out that if we take a sentence p and produce anther sentence ϕ that satisfies the condition of material adequacy, then ϕ gives the meaning of p.

Brilliant!

Banno

Model Theory

This stuff:

A name n designates an object o if and only if (( n = "Adam" and o = Adam) or ( n = "Bob" and o = Bob) or( n = "Carol" and o = Carol)...

An object o satisfies a predicate f if and only if ((f="is english" and o is English) or (f="is french" and o is french) — Banno

...gives an interpretation to the terms of our metalanguage by setting out which objects those terms designate. Doing this is constructing a model, and here is perhaps the main contribution of Tarski to logic: Model Theory.

Banno

Tarski's indefinability theorem.

Another side issue, but with relevance. And again I will jump all over it without too much formal consistency, but if you want more detail, use google.

We know from Gödel that any logic sufficiently powerful to include arithmetic will be incomplete, or inconsistent.

So suppose we have a language, with a set of axioms, from which we can deduce bits and pieces of arithmetic. Gödel showed that if it is consistent, then there will aways be some bits of arithmetic that remain outside of that deductive sequence. Things that are unprovable, but nevertheless true.

Tarski took that notion and applied it to truth, and showed that, just as there are always theorems that cannot be proved, there cannot be a definition of truth within that language. Another language is needed, or at least an extension of the language.

The proof takes a first-order language with "+" and "=", and assigns a Gödel number to every deduction, as in the incompleteness proofs. It then finds a Gödel number for a definition of truth, and shows that it is not amongst the list of Gödel numbers of the deductions. Hence, that definition is not amongst the deductions of the language.

In plain language, an arithmetic system cannot define arithmetic truth, for itself.

Hence it was apparent to Tarski that in order to talk about truth, one needed an object language and a metalanguage. This is what he developed in his definition of truth.

Banno

Kripke's theory of truth

Tarski's ideas lead to a hierarchy of languages that, like Russian Dolls, each give the truth of the language that they enclose.

Can a language contain its own truth predicate? Various theories do manage this trick. The one I'd like to bowdlerise next derives from a paper by Kripke. The trick, as mentioned earlier, is avoiding the liar paradox: "This sentence is false".

Again, suppose we restrict the language to being about a group of people, Adam, Bob and Carol... and their respective nationalities, English, French... We can construct any number of sentences from these: Adam is English", "Bob is English", "Adam and Bob are french"...

We start by adopting three truth values instead of two. So as well as assigning "true" and "false" to the statements of our language, we add a third value, pictured as sitting in between - not true and not false. (a Kleen evaluation)

Let's call this third value "meh"

We assign "meh" to all the statements of our language.

Then we can give an interpretation to the language, and assign "true" or "false" to these as appropriate; so "Adam is English" is true, and "Adam is French" is false, and so on.

Notice that so far any sentence that contains the term "true" will still have the truth value "meh". So "'Adam is English' is true" is neither truth nor false.

We then start to permit sentences that contain "true" or "false" to be assigned values other than "meh", but under strict conditions. So:

If "Adam is English" is true, then we allow that "'Adam is English' is true" is also true.
If "Adam is French" is false, then we allow that "'Adam is French"' is false" is true.
And so on. Generally, if p is true, then "p is true" is true, and '"p is true" is true' is true, and so on; if p is false, then "p is false" is true, and '"p is false" is true' is true, and so on.

But notice that in this construction, we never get to assigning a truth value to the sentence "this sentence is false". So it remains with the truth value "meh" - neither true nor false.

Banno

The Kripke sort of construction given above has an antirealist bent.

So suppose our language were the whole of mathematics, and we adopted a constructivist position, such that a mathematical theorem is true only if there is a proof that it is true. We can adopt the antirealist position that the Goldberg Conjecture, since it is unproven, has the truth value "meh" - is neither truth nor false.

creativesoul

Truth value gaps...

Agent Smith

This sort of construction has an antirealist bent.

So suppose our language were the whole of mathematics, and we adopted a constructivist position, such that a mathematical theorem is true only if there is a proof that it is true. We can adopt the antirealist position that the Goldberg Conjecture, since it is unproven, has the truth value "meh" - is neither truth nor false. — Banno

Something not quite right there. Did you mean (the Goldbach conjecture is) true XOR false? Any proposition is either true or false (principle of bivalence).

sime

I don't think it helps to introduce "meh" as a truth value for undecided arithmetical propositions, because that would distort the existent meaning of arithmetical truth values for both the constructive and classical senses of arithmetic.

In the constructive case, the truth value of an arithmetic proposition is considered a 'Win' or 'True' if there exists a proof of the proposition, and is considered a 'Loss' or 'False' if there is a proof of it's refutation. But introducing a truth value for the status of undecided arithmetic formulas is tantamount to calling a failure to prove or refute them a 'Draw', which distorts the concept of mathematical truth by muddying the distinction between a mathematician's abilities and his subject matter.

IMO, in constructive logic it is better to resist assigning a truth value to undecided propositions so that truth values always refer to what has been proved, rather than to what hasn't been proved. Draws should only be considered a third truth value in cases where there is a constructive definition of drawn games such as in Chess, unlike arithmetic that doesn't possess a natural concept of a draw

As for the classical case, the Law of Excluded Middle suffices to denote the truth value of undecided propositions; unlike in the constructive case, the classical meaning of A OR B doesn't entail either a proof of A or a proof of B, therefore A OR ~A interpreted as meaning TRUE OR FALSE suffices as the truth 'value' for undecided propositions of classical arithmetic.

bongo fury

metalanguage — Banno

I'm not sure, but: you mean object language? The interpretation is that fragment of the metalanguage that interprets terms of the object language?

ssu

One thought, hopefully adequately understandable:

What if our premisses are wrong when we try to make a theory of truth as we try to do it?

That we just assume it's a straight forward logic "if...then" as we ordinarily do. But self-reference, or generalizing to all possibilities and we end up in problems where we need a metalanguage or some hierarchical system to avoid a paradox.

And the wrong premiss is that when we add self-reference, or the infinite, our finite logic simply cut it? And this is what the paradoxes tell us.

Moliere

Something not quite right there. Did you mean (the Goldbach conjecture is) true XOR false? Any proposition is either true or false (principle of bivalence). — Agent Smith

Accepting a third truth value basically rejects the principle of bivalence.

A good read @Banno - you simplified it enough that I think I followed along :)

Something I'm not following -- if we designate our meta-language to refer to the same objects, does it still, at the same time, function as a meta-language? Sort of like having L1 and L2, with the same strings, but slightly different meanings?

I have a hard time thinking in terms of a meta-language. Like, clearly formal and defined -- but I'm not sure I understand how the meta-language performs both the meta-language's function of talking about L1 and the function of the object language which talks about the objects. At this part:

Suppose we restrict the object language to being about a group of people, Adam, Bob and Carol...

And in the metalanguage we can have a definition of "designates":

A name n designates an object o if and only if (( n = "Adam" and o = Adam) or ( n = "Bob" and o = Bob) or( n = "Carol" and o = Carol)...

Doubtless this looks cumbersome, despite my having skipped several steps, but it gives us
a metalanguage and and object language both talking about the same objects, Adam, Bob and Carol..., and a way to use the same name in both languages. — Banno

I might just have to crack open the paper again to follow these steps, and that's fine, but I thought I'd note something I'm not following.

Banno

I don't think it helps to introduce "meh" as a truth value for undecided arithmetical propositions, because that would distort the existent meaning of arithmetical truth values for both the constructive and classical senses of arithmetic. — sime

Doubtless you are right. But that's how I understand Kleen evaluations; albeit using "meh" for "undefined" - it's easier to type.

So the truth table for conjunction is

$\begin{array} {|r|r|}\hline \&= & true & meh & false \\ \hline true & true & meh & false \\ \hline meh & meh & meh & meh \\ \hline false & false & meh & false \\ \hline \end{array}$

(What's the simplest way to construct a table here? Neither HTML nor MathJax seem to work... doubtless my coding skills are not up to it.)

Ah... Thanks, Whichever mod that was. IS there a better way? Can we do HTML tables?

Banno

Something not quite right there. Did you mean (the Goldbach conjecture is) true XOR false? Any proposition is either true or false (principle of bivalence). — Agent Smith

No and yes. In Kripke's system the truth value of an unproven conjecture would be neither true nor false. Hence, meh. It's a non-classical logic, so the principle of bivalence is dropped.

Banno

I'm not sure, but: you mean object language? The interpretation is that fragment of the metalanguage that interprets terms of the object language? — bongo fury

No, although perhaps those sentences are in a metametalanguage.

I'm not real fussed, since the point is not to set out the exact formal logic - What I've written is a long way from that, but to explain roughly what is going on, mostly to myself.

Banno

↪ssu

Not sure. The logics here are an attempt to make these issues clear.

Banno

A good read Banno - you simplified it enough that I think I followed along :) — Moliere

Thanks - that's a bonus, since the aim was to simplify it enough that I seem to understand it.

if we designate our meta-language to refer to the same objects, does it still, at the same time, function as a meta-language? — Moliere

Yep, because the object languagecan talk about Adam and Bob, but can't talk about itself, however the metalanguage can talk about Adam and Bob, and about the sentences of the object language.

So we have Adam, Bob, Carol,...

And in the object language we can write about them: (Adam is English).

And in the metalanguage we can write about them : (Adam is English), and add sentences from the object language: ("Adam is English" is true)

Agent Smith

No and yes. In Kripke's system the truth value of an unproven conjecture would be neither true nor false. Hence, meh. It's a non-classical logic, so the principle of bivalence is dropped. — Banno

Doesn't it make more sense to say the truth value of unproven statements is unknown (it is true/false, we just don't know which) instead of neither true nor false.

Agent Smith

Accepting a third truth value basically rejects the principle of bivalence. — Moliere

I see but wouldn't it be better to say we don't know if p or ~p instead of we know neither p nor ~p. There's a difference, no?

ssu

Not sure. The logics here are an attempt to make these issues clear. — Banno

By logic, do you mean first order logic?

To me it seems like one takes the natural numbers and then assumes you can deduce from the natural numbers things like infinity or irrational/transcendental numbers. Doesn't go like that.

But perhaps first a question that someone hopefully could answer:

Tarski's indefinability theorem and the Incompleteness theorems of Gödel are in some literature called as incompleteness results. Are they equivalent or just how they link to each other? For me it seems like talking about the same issue just from a bit different viewpoint. Am I wrong?

RussellA

We want to add predication. To do this, Tarski developed satisfaction. — Banno

Quick question.

I perceive something in the world that is cold, white and frozen, and I name it "snow".
Therefore, "snow" means something in the world that is cold, white and frozen.

For Tarski, Convention T is "p" is true IFF p. Therefore, "snow is white" is true IFF snow is white.

Tarski's T convention assumes that in the object language the subject is satisfied by its predicate, in other words, the subject "snow" has the property "is white".

I perceive something in the world that is the ground and name it "the ground".

Therefore, Tarski's Convention T may be written as "snow is on the ground" is true IFF snow is on the ground.

But Tarski's T convention assumes that in the object language the subject is satisfied by its predicate. This would mean that "snow" has the property "is on the ground".

But is it true that "snow" has the property "is on the ground", as for snow, being on the ground is a contingent rather than a necessary fact ?

What am I missing ?

Moliere

Yep, because the object languagecan talk about Adam and Bob, but can't talk about itself, however the metalanguage can talk about Adam and Bob, and about the sentences of the object language.

So we have Adam, Bob, Carol,...

And in the object language we can write about them: (Adam is English).

And in the metalanguage we can write about them : (Adam is English), and add sentences from the object language: ("Adam is English" is true) — Banno

OK, that helps me understand "object language" a lot better. It's a literal moniker - a language for objects and objects only, and especially not its own sentences.

So a thought -- I balked at the meta-language because of its artificiality, however this makes me wonder -- could the meta-language just be a natural language? Like, the meta-language is for our object language, but it can have other functions too. So really it's just its role and relationship to the object language that makes it the meta-language.

Or no?

Banno

Quick question. — RussellA

There's no such think in philosophy.

I perceive something in the world that is cold, white and frozen, and I name it "snow".
Therefore, "snow" means something in the world that is cold, white and frozen. — RussellA

That's pretty much the theory of descriptions. Although intuitively appealing, it's fraught with issues and generally held to be incorrect. It's outside the scope of this thread.

Similarly, arguably, neither being on the ground nor being white are necessary properties of snow, since in some possible world snow is black and floats in a layer at head height. That is, there is no logical contradiction in snow behaving in this way. But again, modality is outside the scope of this thread.

But

"Snow is white" is true IFF snow is white, and
"Snow is on the ground" is true IFF snow is on the ground, and indeed
"Snow is turquoise with purple polkadots" is true IFF snow is turquoise with purple polkadots

are all true.

Banno

↪Moliere

I had taken it that the object language was amongst the objects talked about in the metalanguage. Interesting.

The image I have of the place of natural language in the metalanguages is a bit like the place of $\omega$ in Cantor's ordinal numbers. So we can number the object language as language 1. We can number the metalanguage which allows us to talk about language 1, as language 2. We can number the meta-metalanguage, which allows us to talk about both language 1 and language 2, as language 3, and so on. Each language allows us to talk about truth in the languages with a lower cardinality. So every metalanguage has a cardinality.

But a natural language allows us to talk about the truth values of the whole sequence of metalanguages, more or less as $\omega$ is larger than any whole number.

This is just my own musing, not a part of this exegesis.

Banno

By logic, do you mean first order logic? — ssu

More the first order logic with the additional items that make the stuff discussed here possible.

So you asked:

What if our premisses are wrong when we try to make a theory of truth as we try to do it? — ssu

Premises might have a few different meaning here. So there are a bunch of rules that set out the game of first order logic - I found this neat summary. There are also axiomatisations, systems in which a specified set of tautologies is assumed. See rules 1 through 8 in this axiomatisation.. This system is both consistent and complete. Only and every true tautology can be deduced from the axioms. The proof of this is called "Gödel's completeness theorem", mostly in order to cause utter confusion.

There's a potted history on SEP.

Tarski and Kripke proceed by adding stuff to this system.

creativesoul

"Snow is white" is true IFF snow is white, and
"Snow is on the ground" is true IFF snow is on the ground, and indeed
"Snow is turquoise with purple polkadots" is true IFF snow is turquoise with purple polkadots

are all true. — Banno

Not true in the same sense that any of the left side utterances are though.

Interesting.

RussellA

"Snow is turquoise with purple polkadots" is true IFF snow is turquoise with purple polkadots — Banno

I am attempting to understand Tarski's logic of of truth.

Tarski's T-Schema states "S" is true IFF S

Tarski's Semantic Definition of Truth establishes the T-Schema, whereby "S" is true IFF S, where "S" is in an Object Language, and S is in a Metalanguage.

The following T-Schema are all true:
"Snow is turquoise with purple polkadots" is true IFF snow is turquoise with purple polkadots.
"Snow is white" is true IFF snow is white
"Snow is a volcano" is true IFF snow is a volcano.

If S was not limited in some way, the T-Schema would be "S" is true

For every possible statement "S" in an object language, an S may be found in a metalanguage. For example, given the proposition "snow is a volcano", there is a true T-Schema such that "snow is a volcano" is true IFF snow is a volcano.

It follows that for every possible "S", a true T-Schema may be found, meaning that every possible "S" will be true.

If every possible "S" is true, the term IFF becomes redundant, and the T-Schema may be reduced to "S" is true.

What limits possible values of S ?

However, there is a term IFF in the T-schema, meaning that not all propositions "S" in an object language are true. It follows that there are limitations as to what S can be in the metalanguage.

My belief is that the S in the metalanguage is limited by correspondence with the world, in that I perceive something in the world that is cold, white and frozen, but I don't perceive something in the world that is cold, a volcano and frozen.

However, if S is not limited by correspondence with the world, yet S must be limited by something (otherwise the T-Schema would be "S" is true), then what does limit S ?

What prevents some values of S from being a possibility in the metalanguage ?

Moliere

↪Banno

Fair, I'm distracting you. :)

I think, for me at least, the next step would be -- if you accept that a natural language can be a meta-language -- to actually say that that's the end of the infinite regress.

The object-language kind of does function along the lines of conversations about objects. We just accept the object language as its being used, and even if people are actually using English they do use it in such a way that "passes over" the liar's paradox

Metaphysician Undercover

But
"Snow is white" is true IFF snow is white, and
"Snow is on the ground" is true IFF snow is on the ground, and indeed
"Snow is turquoise with purple polkadots" is true IFF snow is turquoise with purple polkadots

are all true. — Banno

As implied here

↪creativesoul

, this is a fine example of equivocation. The final phrase, "are all true", uses "true" in a different way from the other three.

So, which form of "true" are you talking about in this thread? Or is it the case that "the logic of truth" is itself just deceptive sophistry?

Logic of truth

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