## References for discussion of truth as predication?

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• 24.2k
Again, what you have written shows multiple errors in your understanding of formal logic. And misunderstandings of my post. Experience shows that listing them will only cause you to double down, so I'll leave that aside.

You might benefit from reading Gillian Russell - I've mentioned her to you previously. At issue is if there is a "correct human reasoning" in the way you suppose. She careful shows this to be unlikely, for reasons other than misunderstanding Gödel.

https://gilliankrussell.files.wordpress.com/2018/05/logicalnihilism-philissues-v3.pdf

Your attempt to examine logic without paying due attention to formality is counterproductive.
• 3.3k
we know from Godel that not all true sentences can be shown true in propositional logic.

That is not what Godel said nor what we take from Godel.

To show a sentence is to prove the sentence from a set of axioms and rules of inference.

But if a sentence is contingent, then to show that the sentence is true requires specifying which model or models 'true' pertains to.

Propositional logic:

We have the soundness and completeness theorem:

G |- P if and only if G |= P.

That is, a set of formulas G proves a formula P if and only if every model in which all the formulas of G are true is a model in which P is true.

Soundness (if G |- P then G |= P). Proof is straightforward by induction on length of derivation. I don't know who first proved it.

Completeness (if G |= P then G |- P). It seems this was first proved by Post in 1921.

So:

If a sentence is contingent, then the sentence is not provable by logical axioms alone. (Soundness) Moreover, there is a mechanical method to demonstrate that the sentence is not provable by logical axioms alone (that is, there is mechanical method to adduce a model in which the sentence is false).

If a sentence is contingent, then the negation of the sentence is not provable by logical axioms alone. (Soundness) Moreover, there is a mechanical method to demonstrate that the negation of the sentence is not provable by logical axioms alone (that is, there is mechanical method to adduce a model in which the sentence is true).

If a sentence is logically true, then the sentence is provable by logical axioms alone. (Completeness) Moreover, there is a mechanical method to adduce such a proof.

If a sentence is logically false, then the negation of the sentence is provable by logical axioms alone. (Completeness) Moreover, there is a mechanical method to adduce such a proof.

Also, if a sentence is true in a given model, then it can be demonstrated that it is true in that model. Moreover, there is a mechanical method to demonstrate that it is true in that model.

Also, if a sentence is false in a given model, then it can be demonstrated that it is false in that model. Moreover, there is a mechanical method to demonstrate that it is false in that model.

Predicate logic:

We have the soundness and completeness theorem:

G |- P if and only if G |= P.

That is, a set of formulas G proves a formula P if and only if every model in which all the formulas of G are true is a model in which P is true.

Soundness (if G |- P then G |= P). Proof is straightforward by induction on length of derivation. I don't know who first proved it.

Completeness (if G |= P then G |- P). This was first proved by Godel in 1930, but Henkin's very different proof in 1949 is the one usually referred to.

So:

If a sentence is contingent, then the sentence is not provable by logical axioms alone. (Soundness) But there is no mechanical method to demonstrate that the sentence is not provable by logical axioms alone (that is, there is no mechanical method to adduce a model in which the sentence is false) (follows from Church 1936, corollary of incompleteness theorem).

If a sentence is contingent, then the negation of the sentence is not provable by logical axioms alone. (Soundness) But there is no mechanical method to demonstrate that the negation of the sentence is not provable by logical axioms alone (that is, there is no mechanical method to adduce a model in which the sentence is true) (follows from Church 1936, corollary of incompleteness theorem).

If a sentence is logically true, then the sentence is provable by logical axioms alone. (Completeness) But there is no mechanical method to adduce such a proof (follows from Church 1936, corollary of incompleteness theorem).

If a sentence is logically false, then the negation of the sentence is provable by logical axioms alone. (Completeness) But there is no mechanical method to adduce such a proof (follows from Church 1936, corollary of incompleteness theorem).

Incompleteness (Godel-Rosser in modern form):

If (1) T a formal theory (has a recursive axiomatization and recursive inference rules), and (2) T is consistent, and (3), e.g., Robinson arithmetic is interpretable in T, then T is incomplete (that is, there is a sentence P in the language for T such that neither P nor ~P are theorems of T).

"[...] not all true sentences can be shown true in propositional logic."

The best I can make that a definite thought as to formal logic: If a sentence is contingent, then it is not provable in propositional logic from logical axioms alone. That is true, but it is merely the soundness theorem for propositional logic, and I see no reason to think it is something that comes from Godel.
• 3.3k
From the Wikipedia article on Formalism (Philosophy)
Formalists within a discipline are completely concerned with "the rules of the game," as there is no other external truth that can be achieved beyond those given rules.

An example of Wikipedia promulgating sloppy misinformation.

What is called 'game formalism' or 'extreme formalism' regards mathematics as merely execution of rules for strings of symbols. But formalism in the philosophy of mathematics has variations that are not game formalism. Indeed, it seems that game formalism is not widely accepted while other forms of formalism have more acceptance.

This goes along with the fact that, contrary to the Internet (and even printed) meme, Hilbert did not say that mathematics is just a game played with symbols, as indeed Hilbert did very much view finitistic mathematics as contentual and infinitistic mathematics as applicable even though ideal.
• 262
Again, thanks for the response to my concerns. Two things:

1. My question about whether “on” was a logical connective only meant that I wasn’t aware of a “logic of ‛on’” that had been worked out. For all I know, there is one, but if you start playing with it, you can see why it might be ill-suited for formal functions. “If a is on b, and b is on c, then a is on c” – true or false? Beats me. Depends. But once we disambiguate “on”, what are we going to do with it? All sorts of interesting questions hinge on getting clear about “and”, “or”, “if/IFF”, “can”, “must”, et al. -- well, who knows, maybe we need a better understanding of “on” too.

2. I deliberately didn’t say anything about Kimhi’s use of “syncategorematic” in this thread because it’s so non-standard, and even though he has a go at explaining it about three different ways, his usage isn’t transparent to me. You write:

Therefore, within the proposition "p is true", the expression "is true" is a syncategorematic expression, which adds nothing to the sense of "p".

That’s right, but it’s right for Frege as well. Frege’s assertion (judgment) stroke, indicating “is true”, is syncategorematic.

Where it gets bizarre with Kimhi is his further claim that p itself is syncategorematic. You’re right that he regards p as a fact rather than a Fregean complex, but how then is p used? What is the context we need to provide in order to state a relation involving p? I don’t think that, e.g., joining it with q in ‛p & q’ helps. The problem lies in how facts are asserted – how they’re affirmed or denied. When the ND reviewer writes, “ ‛p’ itself . . . [does] not add anything whatsoever to the sense of ‛p’,” this can only mean that some assertion of p can add nothing to p’s sense, but that’s still orthodox Fregeanism. I think Kimhi wants to say something more radical – that the context needed to make use of (syncategorematic) ‛p’ has to involve a monistic understanding of what it is to assert. He thinks the necessary separation of sense and assertion is all wrong. “From the monist point of view, a simple propositional sign displays a possible act of consciousness.” -- the possibility of affirmation or negation.

And on this mystical note, I’ll stop.
• 24.2k
Soundness (if G |- P then G |= P). Proof is straightforward by induction on length of derivation. I don't know who first proved it.

isn't that Godel's completeness theorem?
• 3.3k

No, that's the converse of the completeness theorem.
• 24.2k

Ok.

Not authoritative, of course, but
David Hilbert and Wilhelm Ackermann are often credited with formalizing the notion of soundness in their work on formal logic systems in the early 20th century. Their 1928 book "Grundzüge der theoretischen Logik" (Principles of Mathematical Logic) laid the groundwork for formal systems, including the notion that derivations in a formal system should correspond to semantic truths.
However, the soundness theorem is usually associated with Kurt Gödel, who in 1930 proved both the soundness and completeness of first-order logic (predicate logic) as part of his doctoral dissertation. This work demonstrated that if a formula is provable, it is also true in all models, and conversely, if it is true in all models, it is provable. The proof of soundness is typically straightforward compared to the proof of completeness, but both are key results in Gödel's work.

So while Hilbert and Ackermann helped define the formal system, it is Gödel's 1930 work that solidified the formal proof of soundness in the context of first-order logic.
— ChatGPT

Have you any thoughts on the OP? Perhaps on "existence is not a predicate"?
• 3.3k
(1) It is possible that Godel first proved the soundness of the predicate calculus. I mentioned in my post that I don't know who first proved it.

(2) Source other than an AI bot?

(3) If G |-P then G |= P is soundness not completeness. Even if Godel was the originator of the soundness proof, still soundness and completeness are different, being converses of each other.

/

I have no comment on discussion of "existence is not a predicate" in this thread. But, whether or not this suits the context of that discussion: In the most basic sense of predicate logic, of course existence is not a predicate, but existence is formulated as a predicate in certain modal logics.
• 3.3k
"Godel’s completeness theorem, which he proved in 1930, shows that if a formula is semantically valid, then it is provable within the system, while the soundness theorem follows as a corollary: if a formula is provable, it is valid." ChatGPT

That's great. ChatGPT caught, in flagrante, lying about the pair of central theorems of mathematical logic.

The soundness theorem does not follow as a corollary of completeness.

Trivially, an inconsistent system is complete and not sound!

It is SO discouraging to look at the Internet.

Not authoritative, of course

Time Magazine's Understatement of the Year
• 3.3k
Followup with the liar bot:

I wrote:

"That's wrong. soundness does not follow from completeness" - TonesInDeepFreeze

ChatGPT replied:

"You're correct. I misspoke. The soundness and completeness theorems for first-order logic are indeed distinct results, and soundness does not follow from completeness.

Here’s a more accurate description:

Soundness Theorem: This theorem states that if a formula can be derived using a formal system (i.e., it is provable), then it is also valid (true in all models). The proof of soundness generally involves showing that the rules of inference in the formal system preserve truth.

Completeness Theorem: Godel's completeness theorem, proven in 1930, states that if a formula is valid (true in all models), then it is provable within the system.

Godel proved the completeness theorem, but the soundness theorem was established in the context of formal logic as well, often attributed to earlier foundational work in formal systems, including work by other logicians and formalists who developed and refined these ideas. Soundness was addressed as part of the broader development of formal systems and logical proofs." - ChatGPT

/

And that is still a quite incorrect statement of the theorems. And slop in the summation paragraph too.
• 24.2k
Cheers. I don't disagree with any of that. Yep, Chat GPT confabulates.
• 2.4k
Again, what you have written shows multiple errors in your understanding of formal logic.

I see you are back to your schtick of non-responses. I will take this as a concession, and move on with the topic of the thread.
• 24.2k

Have a look at the Russell article. Let us know what you think. It takes a more detailed approach to metalogic that we've seen so far, while at the same time being quite broad.

I've made my view pretty clear over the years. There is no "correct" way of thinking in the way supposed by some. Instead we have a range of conversations, growing and spreading in a quite organic fashion. From what little I've understood of your position, you seem to think that Aristotle and a couple of others fathomed the whole of how we ought think, and anything since then is mistaken. I doubt there is a common ground to be found here.
• 3.3k

There was only one (combined) error. I mentioned several facets of the matter just to provide an ample understanding of it.
• 24.2k
• 2.4k
From the Notre Dame review of Irad Khimi's Thinking and Being

Informative review. :up:
• 2.4k
I’ve been working with some ideas in Irad Kimhi’s Thinking and Being. Much of what he talks about concerns the nature of the relationship between predication and truth-assertion. It occurred to me that “Existence is not a predicate” has some obvious parallels with “Truth is not a predication.” That is, neither existence nor truth add anything, conceptually, to what they appear to be predicating ‛existence’ and ‛truth’ of. I can say “A hundred thalers exist” but this adds nothing to the concept ‛a hundred thalers’; I can say “It is true that there are a hundred thalers on the table” but this adds nothing to the proposition ‛There are a hundred thalers on the table’.J

Before looking at affirmation and negation I want to revisit this part of the OP now that I have a better understanding of the context.

Colloquially, I want to say that both predications of truth and existence add something to the thing they are predicated of, for this thing is thought to be truth apt or existence apt. By thinking of such predicables as "apt" they are thought of as logically pre-true or pre-existing. This way of thinking seems to be what we now consider normal. Of course, it is possible to affirm such predications without the words "true" or "exists."

Regarding the idea that existence is not a predicate, I think this is tied to standard compositional syllogistic. It would seem that judgments of existence are atomic in a way that is foreign to the combining and separating that constitutes logical acts, and therefore questions of existence are considered pre-logical (including being prior to logical predication). Formal logic is only concerned with existence qua logical function, as for example is seen by the existential quantifier. Existence, then, is treated as a kind of meta-predicate which is barred from being taken into per se consideration within the object language, given the way that it is not (logically) manipulable.
• 1.7k
All sorts of interesting questions hinge on getting clear about “and”, “or”, “if/IFF”, “can”, “must”, et al. -- well, who knows, maybe we need a better understanding of “on” too.J

Yes, there are many logical systems other than Frege's First-Order Logic, such as noted in the Wikipedia article on Non-classical logic. Some, I am sure, not invented yet.
• 1.7k
Where it gets bizarre with Kimhi is his further claim that p itself is syncategorematic. You’re right that he regards p as a fact rather than a Fregean complex, but how then is p used? What is the context we need to provide in order to state a relation involving p? I don’t think that, e.g., joining it with q in ‛p & q’ helps. The problem lies in how facts are asserted – how they’re affirmed or denied.J

I think Kimhi wants to say something more radical – that the context needed to make use of (syncategorematic) ‛p’ has to involve a monistic understanding of what it is to assert.J

Following on from @Leontiskos, it seems that truth and existence are meta-predicates rather than predicates.
Not "the apple is on the table is true" but "the apple is on the table" is true.
Not "the apple exists" but "the apple" exists.

For Khimi, the proposition "p" is a syncategorematic expression, where "p" is a fact.

Khimi agrees with Frege that a proposition can be both asserted (reference) and unasserted (sense), but whereas Frege thinks that sense and reference can be disassociated, Khimi disagrees and believes that sense and reference are two parts of a single unity

The fact that "I saw the Morning Star" is both asserted, in that it refers to the planet Venus,
and unasserted, in that its sense is of a star the rises in the morning.

For Frege, "unicorns are mythical creatures" can have a sense, a horse-like animal with a single projecting horn from its forehead, even though it doesn't have a reference.

For Khimi, if a proposition has a sense, then it must also have a reference, as sense and reference cannot be disassociated.

Khimi agrees with Wittgenstein that there are no negative facts,

It comes back to Wittgenstein's puzzle, how can not-p negate p, when p may not be the case.

Khimi holds that the idea of a judgement without a contrary is incoherent, in that if I judge the postbox to be red, then I must also be judging whether the postbox is not-red.

This makes sense in that I only know what something is when I know what it isn't.

I only know what it means for there to be rain if I also know what it means for there not to be rain.

To know something, I must know not only "p" but also "not-p".

This concept is mirrored in Wittgenstein's truth tables in the Tractatus. Wittgenstein did not invent truth tables, but their use in modern logic is usually traced back to the Tractatus. Because an exhaustive list of the truth-possibilities of a proposition tells us everything we need to know about that proposition, the truth table then shows us what we need to know about that proposition.

For example, the proposition "It will rain" can be understood by all its possibilities: "it will rain and I will get wet", "it will rain and I won't get wet", "it won't rain and I will get wet" and ""it won't rain and I won't get wet".

A fact such as "p" has meaning only when it can be judged to be "not-p".

A fact such as "the postbox is red" has meaning to me when I can judge whether it is "not-red"

For Khimi, the proposition "the postbox is red" has a sense, as well as a reference, but also "the postbox is not-red" must have a sense as well as a reference, as sense and reference cannot be disassociated.

The question is, what does "the postbox is not-red" refer to?

The colour purple is not-red, as well as the colours orange, brown, turquoise and violet.

Therefore, the colour not-red could be the colour violet.,

Therefore "the postbox is not-red" could be referring to "the postbox is violet".

This introduces possible modal worlds. Wittgenstein is important for his introduction of modality.

From the SEP article Possible Worlds

However, the predominant version of combinatorialism finds its origins in Russell's (1918/1919) theory of logical atomism and Wittgenstein's (1921, 1922, 1974) short but enormously influential Tractatus Logico-Philosophicus.

IE, "p", although syncategorematic, gets its meaning from its contradictory pair "not-p".
• 262
Yes, I had recourse to it several times in drafting my various posts about Kimhi. Much better than the Hanna piece.
• 2.4k
“From the monist point of view, a simple propositional sign displays a possible act of consciousness.” -- the possibility of affirmation or negation.J

Reading more of Kimhi's book, I am appreciating it, especially the way he explodes the Fregean paradigm over and over. I think Kimhi could help clear up the truth-functional confusions overflowing in <this thread>, which are all essentially based on the Fregean form-substitutability between (A & ¬B) and (B & ¬B).

But another overlap between the child thread and Kimhi is as follows:

In English we can deny in a manner that does not affirm the negation of any proposition, and this violates the way that propositional logic conceives of the LEM. In fact, going back to flannel’s thread, this shows that a contradiction in English need not take the form (A ^ ~A). In English one can contradict or deny A without affirming ~A.

(Propositional logic seems to assume, prima facie, not only the commonsensical idea that C is neither A nor B, but also the deeply counterintuitive idea that C is neither ¬A nor ¬B. Usually if C is neither A nor B then it must be both ¬A and ¬B.)

Compare Kimhi:

[For the non-Fregean] The truth-connector is therefore seen as an expression of an operation. In fact, we can speak of truth and falsity operations, which are performed by . . . is true and . . . is false respectively. The assertion “p is true” is the same as “I truly think p.” There is no logical gap between these assertions. By contrast, the assertion “p is false” is not the same as “I truly think not-p.” Thus, truth and falsity operations are not symmetrical. However, they both apply to p and “A thinks p.” It is only in judgments about others that the use of . . . is false is required in addition to negation. — Kimhi, Thinking and Being, 93

I would want to say that calling something false is to deny, not to negate, and that the asymmetry of affirmation and denial is well represented by Kimhi's final sentence here. Denial requires an interlocutor in a way that affirmation does not (and this interlocutor could also be merely represented). The corollary here seems to be that saying "p is false" is not the same as saying "not-p".

At this point I would want to see denial and negation as distinguished according to what Kimhi says on 87 about Aristotle, where denial is an assertion "away from" and a negation is a "separation." But I would have to look into this more, and I know Kimhi will go on to speak about whether and in what way these are truly distinct.

(I also really like Kimhi's work in unifying the various formulations of the PNC, a point that I have often found difficult to convey to those whose paradigm precludes it.)
• 262
I'm glad to hear you're reading Kimhi -- not for the faint of heart! In fact you may find parts of it easier going than I did, due to your background in Aristotle.

I didn't participate in the thread you refer us to, and I'm not prepared right now to try to take it all in. But your quoted comment about negation versus denial is definitely apropos. It may come down to the difference between 'not-X' (negation as an operation within a proposition) and 'It is not the case that (p)' (denial of a proposition), though I'm not sure about this. What Kimhi adds to this, in a manner I'm still grappling with, is the unity part: the claim that "the assertion 'p is true' is the same as 'I truly think p'." In general, the role of an act of consciousness in Kimhi's philosophy is what allows him to take a thoroughly monist stance on these matters, but as I've said before, I think he could have done a much clearer job explaining it.

I also agree that he's good on the PNC. One of the most appealing and lucid sections of the book.
• 2.4k
I'm glad to hear you're reading Kimhi -- not for the faint of heart! In fact you may find parts of it easier going than I did, due to your background in Aristotle.J

A lot of it is on point for me, even though he is going deeper than I have seen others go. It is also bringing together a number of disparate interests of mine, which is great.

It may come down to the difference between 'not-X' (negation as an operation within a proposition) and 'It is not the case that (p)' (denial of a proposition)J

I think that's right, but I think it can be elaborated further.

What Kimhi adds to this, in a manner I'm still grappling with, is the unity part: the claim that "the assertion 'p is true' is the same as 'I truly think p'." In general, the role of an act of consciousness in Kimhi's philosophy is what allows him to take a thoroughly monist stance on these matters, but as I've said before, I think he could have done a much clearer job explaining it.J

Yes, but in his defense I think it is very hard to elucidate the manner in which the intellect knows truly, and how truth is both psychological and ontological. This is also related to the table that the ND reviewer gives, which would also be a good jumping off point:

$\begin{array} {|c|c|}\hline A\,believes\,p. & A\,believes\,p. & A\,believes\,p. \\ \hline p. & Not-p. & A's\,belief\,is\,correct. \\ \hline So\,A's\,belief\,is\,correct. & So\,A's\,belief\,is\,incorrect. & So\,\,p. \\ \hline \end{array}$

Reveal
$\begin{array} {|c|c|}\hline A\,believes\,p. & A\,believes\,p. & A\,believes\,p. \\ \hline p. & Not-p. & A's\,belief\,is\,correct. \\ \hline So\,A's\,belief\,is\,incorrect. & So\,A's\,belief\,is\,correct. & So\,\,p. \\ \hline \end{array}$


I didn't participate in the thread you refer us to, and I'm not prepared right now to try to take it all in.J

I wouldn't really recommend reading it, but it is an interesting test case of what would happen if we ignore Kimhi's points, such as the point that (b∧¬b) is not a genuine proposition. In large part that thread is just people assuming that it is a genuine proposition, and also some devoted Fregeans being adamant about this. Obviously that assumption leads to problems at every turn.

One of those "disparate interests" that Kimhi brings together is the strange situation I find myself in on TPF, where there are some who have become very proficient at the manipulation of logical symbols—which they are quick to lord over others—and yet they seem to have no idea of the purpose of logic. For example, the problem with viewing (b∧¬b) as a proposition is that it does not fit with the final cause of logic, and these people have no understanding of the question, "What is the final cause of logic?" They have no reference point outside the internal machine of logic.

A similar point can be made with respect to Frege’s use of semantic notions such as “the sense of . . .” and “the reference of . . . ,” namely, insofar as we come to see them as pointing to similarities and differences displayed by the notation, we recognize that they are not predicates, since all predicates are expressible using the notation; hence we come to see them as mere means deployed to communicate the use of the notation to a learner. In the end, the success of this communication requires this realization. One can say, therefore, that Frege’s universalist conception is eliminativist with respect to both semantical and formal-categorical discourse. Note that the very construction of the Begriffsschrift is not in any obvious way internal to the fact-stating discourse, namely, we cannot describe it as the actualization of the capacity of linguistic self-consciousness which is internal to the activity of assertion as such. — Kimhi, Thinking and Being, 91-2

Picking on Banno again, a stark example of this sort of thing can be seen in a thread trying to figure out what logic even is:

. . .You are appealing to usage, but the etymology and the historical usage point very clearly to logic as an art of reasoning.

They say that one of the best ways to learn something is to teach it. A few weeks ago I looked at your thread which is intended to teach propositional calculus (link). It's no coincidence that in your third substantial post you were already into truth tables. But even in your first substantial post you said, "What we want to do is to examine the relations between these propositions, rather than their contents." It seems to me that it would have been more apt to say that we want to examine the relations that obtain between these propositions based on their content. Relations hold or fail to hold in light of the content of the relata, and this has everything to do with truth.

Now a pedagogue might choose to introduce the rules of logic before introducing the purpose of logic, much like you could teach a child to kick a ball before introducing them to the game of soccer. Of course I am not convinced that this is sound pedagogy.

The posts I was responding to are defending what would be labeled by Kimhi "the modern schematic conception of logic" (pp. 89-90). It is fascinating to me that Banno would teach logic without starting by telling his pupil what logic is for. When is the pupil ever to learn what the tool of logic is used for? Or the teacher? This is something like the epitome of a functionalist approach which attempts to prescind from all ontological questions.

Thus one reason I am interested to read Kimhi is to understand what pitfalls Frege was attempting to avoid in constructing a system that has led to such oddities ("psycho-logicism").
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