Three laws of logic apply to all propositions
¬(p ∧ ¬p) Law of non-contradiction
(p ∨ ¬p) Law of excluded middle
p = p Law of identity

Because of Quine's paper: https://www.ditext.com/quine/quine.html most philosophers have been confused into believing that there is no such thing as expressions of language that are {true on the basis of their meaning}.

The unique contribution I have made to this is that the semantic meaning of these expressions is always specified by other expressions. When we can derive x or ~x by applying truth preserving operations to a set of semantic meanings then this perfectly aligns with Wittgenstein's concise critique of Gödel: https://www.liarparadox.org/Wittgenstein.pdf

Unless P or ~P has been proved in Russell's system P has no truth value and thus cannot be a proposition according to the law of the excluded middle. Sometimes this "proof" requires an infinite sequence of steps.

Unless P or ~P has been proved in Russell's system P has no truth value and thus cannot be a proposition according to the law of the excluded middle. Sometimes this "proof" requires an infinite sequence of steps. — PL Olcott

Simplest is this: if P is undecidable, then neither P nor ~P are provable in R. I don't know what R is, but let's assume it stands for the kinds of systems that are actually relevant to this discussion, and which include arithmetic as described by Godel. As such, for clarity let R = G, and let us refer to P as unprovable in G.

P can be and is a proposition in G. It says, using Godel's methodology, that P is not provable in G. The law of excluded middle is misapplied here. To say that it has no truth value qua is a misstatement because it is true. In G itself ,it is undecidable, thus not provable in G.

Now, you have been repeatedly evasive and substantively non-responsive to many posts and questions through at least several threads. My own criticism of your claims, which themselves may not go far enough, is that while you can devise whatever limited system you like, your claim is that you can generalize it where it does not apply. And that is wrong, ignorant, and stupid.

Simplest is this: if P is undecidable, then neither P nor ~P are provable in R. I don't know what R is — tim wood

You simply ignored most of what I said.
I didn't read beyond the point where you proved that you ignored my definition of R.
(expressions of language specifying semantic meanings)
I am simultaneously carrying many other conversations so I must stop reading as
soon as I hit the first big mistake.

And now we see you do not know what a sentence is, else you'd have finished reading it.

Simplest is this: if P is undecidable, then neither P nor ~P are provable in R. I don't know what R is, but let's assume it stands for the kinds of systems that are actually relevant to this discussion, and which include arithmetic as described by Godel. As such, for clarity let R = G, and let us refer to P as unprovable in G. — tim wood

I already specified that the R I am referring to is the set of semantic meanings specified as expressions of language. This is the key foundation of my whole point and cannot be ignored. This R is the ultimate foundation of the truth of all expressions of language that are {true on the basis of their meaning}.

Truth preserving operations applied to these expressions that fail to derive P or ~P prove that P is not a proposition because it violates the law of excluded middle.

The context I am using is ordinary mathematical logic applied to classical logic:


For a given language, we have different models. A model is an interpretation of the meaning of the symbols of the language. Per a given model, every sentence receives exactly one of the two truth values. That is, per a given model, no sentence is both true and false, and every sentence is either true or it is false. Moreover, no proof has an infinite number of steps, since we cannot mechanically check an infinite number of steps.

Some sentences are true in every model (these are called 'logical truths')

Some sentences are true in some models and false in other models (these are called contingent sentences')

Some sentences are false in all models (these are called ''logical falsehoods').


If P is a sentence and t is a closed term term, then

~(P & ~P) is a theorem in every theory and it is true in every model (non contradiction)

P v ~P is a theorem in every theory and it is true in every model (excluded middle)

t = t is a theorem in every theory and it is true in every model (identity).


Moreover, we have the meta-theorem that a sentence is true in every theory if and only if it is provable in every theory.

/

With formal theories, it is required to have a mechanical method to check whether a given sequence of formulas is a proof, and for that we need a mechanical method to check whether a given sequence of symbols is a formula of a certain kind, and for that we need a mechanical method to check whether a given sequence of symbols is a formula. And we need a mechanical method to check whether a given formula is a sentence.

It would be circular if, to know whether a given formula is a sentence, we needed first to know whether a given formula is such that either it or its negation is provable. To know whether it is a sentence we would need to know whether it is provable, but to know whether it is provable, we would need whether either it or its negation is provable.

Moreover, we have the meta-theorem that there are theories such that there are sentences such that neither the sentence nor its negation are provable in the theory. This does not contradict the law of excluded middle (P v ~P), since the law of excluded middle semantically is that either P is true or ~P is true, and the law of excluded middle syntactically is that P v ~P is provable in all theories, but the law of excluded middle is not that in all theories either P is provable of ~P is provable.

For a given language, we have different models. A model is an interpretation of the meaning of the symbols of the language. Per a given model, every sentence receives exactly one of the two truth values. That is, per a given model, no sentence is both true and false, and every sentence is either true or it is false. — TonesInDeepFreeze

That may make conventional sense. In my system semantic meaning is fully integrated directly into the language. This makes things such as the principle of explosion impossible. (A & ~A) semantically entail FALSE.

On may reasonably propose an alternative formalized logic, but a formalized logic requires that we have a purely mechanical method by which to determine whether a given finite sequence of sentences is or is not a proof, which requires a mechanical method by which to determine whether a given sequence of symbols is or is not a sentence.

I already specified that the R I am referring to is the set of semantic meanings specified as expressions of language. This is the key foundation of my whole point and cannot be ignored. This R is the ultimate foundation of the truth of all expressions of language that are {true on the basis of their meaning}.

Truth preserving operations applied to these expressions that fail to derive P or ~P prove that P is not a proposition because it violates the law of excluded middle. — PL Olcott

And I've granted R all day long. But you're not talking just about R, but generalizing your claims beyond R, and as you persist beyond reason, so with reason I call you out and warn against engaging with you. And not to be forgot, you have been asked about R itself and given no answer. That is, R does not exist and I suspect cannot exist, either way, how is R an "ultimate foundation" of anything? By contrast, Godel et al were exactly rigorously clear about what their system(s) are.

And I've granted R all day long. But you're not talking just about R, but generalizing your claims beyond R, and as you persist beyond reason, so with reason I call you out and warn against engaging with you. And not to be forgot, you have been asked about R itself and given no answer. That is, R does not exist and I suspect cannot exist, either way, how is R an "ultimate foundation" of anything? By contrast, Godel et al were exactly rigorously clear about what their system(s) are. — tim wood

When any expression P or ~P has no connection through truth preserving operations to elements of the set of expressions of specified semantic meanings then P is not a proposition. This is almost the same thing that Wittgenstein says. I generalized what Wittgenstein said to apply to every expression that is {true (or false) on the basis of its meaning}. No connection to any meaning: then meaningless.

On may reasonably propose an alternative formalized logic, but a formalized logic requires that we have a purely mechanical method by which to determine whether a given finite sequence of sentences is or is not a proof, which requires a mechanical method by which to determine whether a given sequence of symbols is or is not a sentence. — TonesInDeepFreeze

One expression of formal language or formalized natural P or ~P can either be connected to a set of
semantic meanings specified as formal language or formalized natural language through a set of truth preserving operations or not. Some aspects of classical logic (such as the Principle of Explosion) are not truth preserving. If neither P nor ~P can be connected to elements of the set of semantic meanings then P and ~P are meaningless.

The meanings of sentences are given by the method of models. The truth or falsehood of sentences is determined by rules operating on the truth and falsehood of the components of the sentences, down to the truth or falsehood of the atomic sentences.

The principle of explosion adheres to the principle of truth preservation.

The principle of truth preservation is: All cases in which the premises are true are cases in which the conclusion is true. Put another way: There are no cases in which the premises are true but the conclusion is false.

Since there are no cases in which a contradiction is true, there are no cases in which both a contradiction is true and the conclusion is false.

the set of expressions of specified semantic meanings — PL Olcott

Please define this. If it is a constructed set, please show how it is constructed.

The meanings of sentences are given by the method of models — TonesInDeepFreeze

I am NOT doing it that way. The meanings of terms are specified in a knowledge ontology type hierarchy. The compositional meaning of expressions is derived through something like Montague grammar.

The principle of explosion adheres to the principle of truth preservation. — TonesInDeepFreeze

Nothing can be semantically derived from the expression that "cats are not cats"

The principle of truth preservation is: All cases in which the premises are true are cases in which the conclusion is true. Put another way: There are no cases in which the premises are true but the conclusion is false. — TonesInDeepFreeze

That is simply not good enough. X is semantically entailed by a set of premises if and only if X is a necessary consequence of all of the premises.

the set of expressions of specified semantic meanings
— PL Olcott
Please define this. If it is a constructed set, please show how it is constructed. — tim wood

It has taken me twenty years to derive the architectural overview that I just provided. A key aspect of this is defining expressions that are {true on the basis of their meaning} where meaning is expressed using other expressions.

I started with absolute truth and found the most people believe that absolute truth only comes from God and they don't believe in God. The ten years after that I started talking about analytical truth only to find that Quine successfully convinced most people that it does not exist.

Recently I came up with the above expressions that are {true on the basis of their meaning} where meaning is expressed using other expressions.

My whole system is just like expanding the syllogism so that it applies to every expression that is {true on the basis of its meaning}.

It is anchored in a type hierarchy knowledge ontology to specify the semantic meaning of terms of a formal or formalized natural language. https://en.wikipedia.org/wiki/Ontology_(information_science)

The compositional meaning of expressions of this language are derived from something like Montague grammar.

* 'entailment' and 'consequence' are usually taken as specifying the same relation. That is the relation between a sets of sentences G and a sentences P that holds when there are no models in which all the members of G are true but P is false.

* If C is any contradiction and P is any sentence, then we have C -> P, but that does not allow inferring P. Rather, we would infer P from (C -> P) & C. But since we never have C, don't have (C -> P) & C so we still don't have P.

* Montague semantics is based on compositionality as with the method of models (though with extended aspects such as types, modality, intensionality and possible world models).

As with any subject, before purporting to critique mathematical logic and model theory, one should know something about it. The number of decades one has been floundering in ignorance and confusion on the subject is not a positive index of the cogency of one's critique.

I asked you a specific and clear question.

Please define this. If it is a constructed set, please show how it is constructed. — tim wood

You ether cannot or will not answer it. You describe what you call a set and make certain claims about it. You have not shown that it exists or can exist, or how it's built, and you certainly have not shown how it can satisfy the claims you make for it.

Let me start you out. You adduce a set of what you call truths; you then iterate some kind of syllogistic reasoning on the and produce new conclusions. And done. How does this meet any of your claims?

Rather than merely bandying Richard Montague, the poster would do well to start at the beginning with symbolic logic as presented in his textbook:

Logic: Techniques Of Formal Reasoning, 2nd ed. - Kalish, Montague and Mar

Start there, with the basics of the subject before pretending to speak meaningfully about more advanced topics.

* 'entailment' and 'consequence' are usually taken as specifying the same relation. — TonesInDeepFreeze

I am talking about semantic entailment that has nothing to do with model theory.
The reason the error of the Principle of Explosion has slipped through the cracks
is that semantics was divorced from logic.

When we try to answer what is it about {A cat is not a cat} that makes the
{Moon is made from green cheese} true and we come up with nothing
then the ruse of the POE is exposed.

* If C is any contradiction and P is any sentence, then we have C -> P, but that does not allow inferring P. Rather, we would infer P from (C -> P) & C. But since we never have C, don't have (C -> P) & C so we still don't have P. — TonesInDeepFreeze

https://en.wikipedia.org/wiki/Principle_of_explosion Disagrees.

* Montague semantics is based on compositionality as with the method of models (though with extended aspects such as types, modality, intensionality and possible world models). — TonesInDeepFreeze

I am talking about how to fully integrate semantics directly in the language and have no need for model theory. If we don't do this then we will not understand that the POE is simply wrong. There is nothing about {cats are not cats} that proves {the Moon is made from green cheese}.

You ether cannot or will not answer it. You describe what you call a set and make certain claims about it. You have not shown that it exists or can exist, or how it's built, and you certainly have not shown how it can satisfy the claims you make for it. — tim wood

It took me twenty years to unequivocally prove that something just like the analytic side of the analytic synthetic distinction really exists. Most philosophers remain convinced by Quine there is no such thing as {true on the basis of meaning}.

When I anchor this in {the meanings must be specified as expressions} of language and X or ~X must be derived by applying truth preserving operations to these expressions of semantic meaning then this {true on the basis of meaning} is proven to exist. Also undecidable sentences are rejected as failing the Law of excluded middle.

Rather than merely bandying Richard Montague, the poster would do well to start at the beginning with symbolic logic as presented in his textbook: — TonesInDeepFreeze

In other words you did not understand that I just provided the essence of the foundation of expressions that are {true on the basis of their meaning} thus establishing that something just like the analytic side on the analytic/synthetic distinction has been proven to exist.

Do you understand that in terms of these discussions and your replies to me you're talking crazy - most of your replies being either or both nonsense and non-sequiturs?

↪PL Olcott Do you understand that in terms of these discussions and your replies to me you're talking crazy - most of your replies being either or both nonsense and non-sequiturs? — tim wood

Let just talk about the POE. (A & ~A) prove B no matter what A and B are.

(1) We know that "Not all lemons are yellow", as it has been assumed to be true.
(2) We know that "All lemons are yellow", as it has been assumed to be true.
(3) Therefore, the two-part statement "All lemons are yellow or unicorns exist" must also be true, since the first part of the statement ("All lemons are yellow") has already been assumed, and the use of "or" means that if even one part of the statement is true, the statement as a whole must be true as well.

You disagreed with the above.

You're incoherent, here. And it looks like you do not understand the distinction between valid and true.

↪PL Olcott You're incoherent, here. And it looks like you do not understand the distinction between valid and true. — tim wood

I do know the distinction between valid and true. https://iep.utm.edu/val-snd/

So see how you can use this distinction to explain how what you said
diverges from what Wikipedia said.

You disagreed with the above. — PL Olcott

So see how you can use this distinction to explain how what you said
diverges from what Wikipedia said. — PL Olcott

What are you referring to?

* If C is any contradiction and P is any sentence, then we have C -> P, but that does not allow inferring P. Rather, we would infer P from (C -> P) & C. But since we never have C, don't have (C -> P) & C so we still don't have P. — TonesInDeepFreeze

Disagrees with this:

(1) We know that "Not all lemons are yellow", as it has been assumed to be true.
(2) We know that "All lemons are yellow", as it has been assumed to be true.
(3) Therefore, the two-part statement "All lemons are yellow or unicorns exist" must also be true, since the first part of the statement ("All lemons are yellow") has already been assumed, and the use of "or" means that if even one part of the statement is true, the statement as a whole must be true as well. — PL Olcott

Given C, a contradiction, the expression C => P is true. That is because C is false, and whenever the antecedent is false, the implication is true - them's the rules. But it is an elementary and serious error to suppose this shows that P is true. For P to be true, C must first be affirmed. That is, C ^ (C => P) => P, C being true, affirms P. And this is exactly - or should be - what Tones said.

But that is not at all what you have said. You have stipulated that both L and ~L are true, and then created the expression (L v X), noting that it is true. And it is, but it says nothing about X.

By the way, Tones is Tones, I am not, I am tim wood.

The principle of entailment goes very far back in the history of logic. It is in model theory that the principle is given mathematical exactness. The model theoretic version adheres to the general principle: A set of premises entails a conclusion if and only if there are no circumstances in which the premises are all true but the conclusion is false.

/

If a proposal for a logic does not include models than it is fundamentally different from Montague grammar/semantics.

/

"If C is any contradiction and P is any sentence, then we have C -> P, but that does not allow inferring P. Rather, we would infer P from (C -> P) & C. But since we never have C, don't have (C -> P) & C so we still don't have P."

(1) That is exactly correct. (2) Wikipedia is not a reliable source on the subject of logic. (3) I highly doubt that Wikipedia disagrees with what I wrote anyway.

/

I have not opined on the analytic-synthetic distinction.

The ordinary distinction between truth and validity is:

A sentence is true or not per a given model.

A sentence is valid if and only if it is true in every model.