The LNC needs to be scrapped + a version of paraconsistent logic needs to be adopted — Agent Smith

There are systems with all three: LNC, contradictions, and non-explosiveness. You can look it up yourself; you can educate yourself about this subject on which you are so opinionated yet so ill-informed.

The LNC is incompatible with paradoxes — Agent Smith

With a paraconsitent logic, one can have both LNC and non-explosiveness. In such a logic, we may have LNC as a theorem (or theorem schema) and also have each conjunct of a contradiction as theorems and also have non-explosiveness. You can look it up yourself.

Paradoxes such as you have mentioned are informal. For purposes of formal classical mathematics we are more careful in formulation so that the paradoxes don't occur
— TonesInDeepFreeze

Right. And part of that formality is rules for the use of the truth predicate that are artificial. This is why the value of the solution you point out does not extend to the realm of ordinary language, where if a statement can't be asserted, it can't be true or false. — Tate

In the sense you mention a 'truth predicate', we actually say a 'truth function'. On the other hand, as to truth predicates, (Tarksi) for an adequately arithmetic theory, there is no truth predicate definable in the theory.

For a language, per a model for that language, in a meta-theory (not in any object theory in the language) a function is induced that maps sentences to truth values. It's a function, so it maps a statement to only one truth value, and the domain of the function is the set of sentences, so any sentence is mapped to a truth value.

And, (same Tarksi result said another way) for a semantic paradox such as the liar paradox, the statement can't be asserted in any arithmetically adequate consistent theory, so it is not mapped to any truth value.

But let's go back to the earlier context in which a poster claimed that the paradoxes ruin classical logic for mathematics:

We must distinguish between informal paradoxes and formal contradictions. We know (the soundness theorem) that classical first order logic alone does not prove contradictions. In the case of the Russell set, first order logic itself proves that there is no such relation such that for all objects x there is an object y such that x bears the relation to y if and only if y does not bear the relation to itself. In the case of the liar paradox, we have the result that the liar sentence is not formalizable in an adequately arithmetic consistent theory. And, as far as we know, and by certain arguments, the mathematical axioms of set theory are consistent.

So if the informal paradoxes motivate us to view them as needing to be allowed formally, then we do wish to allow contradictions in theories but not have them explosive, and then we adopt a paraconsistent logic instead of classical logic. But that is not the ruination of classical logic.

homological relationship to the empirical-material world it's modeling? — ucarr

(1) I don't know your meaning of 'homological' applied to relationships between a mathematical theory and empirical observation.

(2) There are two senses of 'model'. The first is the formal notion of a model for a language: a state-of-affairs is a model for a mathematical language, and the state-of-affairs is a model of a mathematical theory if and only if the theory is true in the state-of-affairs. The second sense is the exact reverse of the first sense: a mathematical theory "modeling" states-of-affairs such as those observed empirically. Personally, in order to be clear at all times, I prefer to use only the first sense.

signifier (math model) & its referent ( material object) — ucarr

If we must use the word 'signifier' here, I would say that the signifier is not a model but rather a theory.

can the math model successfully model a self-contradictory material object without containing within itself any contradictory math expressions? — ucarr

I put it this way: There is no model of a contradictory theory. (That's for classical logic. We may find other things pertain in other kinds of logic.)

[...] then foundational logic of math needs reexamination. — ucarr

It is widely viewed in the study of logic that classical first order logic does not exhaust all the forms of reasoning about many kinds of subject matter.

How many points do I have to throw in the bag to fill it?
— Hillary

'throw in', 'bag', and 'fill' (in your context) are not mathematical terms, so I can't give you a mathematical answer to your question.

However, the mathematical question "how many 3D-points are in a non-empty volume?" does have the mathematical answer: the cardinality of the set of real numbers. — TonesInDeepFreeze

So there you have my reply. What was your point in asking your question?

how the liar sentence was banished from the kingdom of propositions — Agent Smith

Agent Smith is ignorant of how it actually works in formal mathematics.

The choices are clear. — Agent Smith

Yes, and Agent Smith ignores the most obvious choice.

I believe the culprit wished to point out flaws in my reasoning. — Agent Smith

For there to be a culprit there needs to be a misdeed. It's a bizarre view that the culprit is not the one irresponsibly spreading misinformation but rather the one who corrects that misinformation.

Do we need to do an overhaul of the logic we're using in this forum and in philosophy as a whole? — Agent Smith

You claimed that classical logic, thereby classical mathematics, is devastated by the paradoxes. I gave you fulsome explanation that the paradoxes do not occur in the ordinary mathematical theories. Paradoxes such as you have mentioned are informal. For purposes of formal classical mathematics we are more careful in formulation so that the paradoxes don't occur. I explained specifically, in detail, for the case of Russell's paradox. But you choose to utterly ignore information that is given to you. I said I wondered why someone would prefer to remain ignorant rather than avail themself of information given to them even at no cost. One explanation though is that the person finds themself more charming or fascinating to fancy themself as some kind of of novel thinker rather than to exercise the common humility in recognizing that there are brilliant and wise thinkers in the past who have come up with entire fields of study, such as mathematical logic, in which we find rigorous and brilliant solutions

My understanding of paraconsistent logic, from Graham Priest, is that things can contradict each other and still be true — Jackson

That sounds like dialetheism. Paraconsistent logic is characterized by the absence of the explosion principle. What Graham Priest text do you refer to?

unless we do something to halt the principle of explosion, we're doomed! — Agent Smith

I gave you copious explanation why you are wrong about that. You are blatantly wrong about it. I cannot fathom what reward you find in posting blatant misinformation over and over again.

The Grelling-Nelson paradox — Agent Smith

Your example is not the Grelling-Nelson paradox.

I'm going back over concepts in notes I already gave you.

As I mentioned for other examples, your question depends on whether we are working with predicate logic or modal predicate logic. With predicate logic, there is no existence predicate. With modal predicate logic, there is an existence predicate but it's complicated to explain. I'll address the question with predicate logic and say that modal predicate logic is better suited.

In either case, the first crucial distinction you need to be very clear upon is a distinction I've mentioned: names vs. predicates.

'Joe Biden' and 'Peter Parker' are names.

'is a dog' and 'is an apple' are predicates.

In predicate logic (at least in certain airtight treatments, such with the Fregean method) every term (variable, constant or compound naming term) evaluates to some member of the domain of discourse.

Let 'b' stand for Joe Biden. 'b' is a constant (a name).

Let 'k' stand for Peter Parker. 'k' is a constant (a name).

Let 'D' stand for 'is a dog. 'D' is a predicate.

Let 'L' stand for 'is an apple'. 'L' is a predicate.

Let 'E' stand for 'there exists'. 'E' is a quantifier, and it is not a predicate.

/

Eb (intending to mean "there exists Joe Biden" or "Joe Biden exists") is not well-formed. It is not a formula. It is not considered.

So "Joe Biden exists" is not really translatable into predicate logic at face value in way that you might wish. Sure, you might say Ex x=b. But that's just a logical truth, a theorem of identity theory. It doesn't say much. It has nothing to do with the fact that there exists a person who is Joe Biden. No matter what name we choose and symbolize with 'b', we still have the trivial theorem Ex x=b.

Perhaps the best workaround would be to have a predicate 'B' that stands for the conjunction of all the required properties for being Joe Biden. Then we would say:

ExBx.

And that is well-formed and is meaningful.

/

But the Peter Parker example is trickier (when, for sake of discussion, we dismiss that the name 'Peter Parker' is taken as otherwise referring to a fictional character, and we are only interested in the sense of names being for objects in the real world). It's tricky because I can't say "the conjunction of all the required properties for being Peter Parker" since we agreed that there is no such conjunction of properties.

The best I can offer is to have a predicate 'K' that stands for the conjunction of all the required properties of being Peter Parker if there were a real person just like the fictional character. Then maybe (I'm not sure because of that subjunctive 'if there were' bit):

~ExKx

/

"Some dogs exist" is not problematic:

ExDx

/

"All apples exist". It's not clear what that is supposed to mean. Is it supposed to mean that everything that is an apple is something that exists? But see my remarks earlier in this post about existence as a predicate. I don't think I have a satisfactory symbolization other than:

Ax(Lx -> Ey y=x)

But, if I recall, that is, mutatis mutandis, what you came up with one or your own formulas. And my point stands that it doesn't really say much.

/

So for three of your examples, modal predicate logic is much better suited than mere predicate logic.

It could be that I'm getting mixed up between the principle of bivalence and the law of noncontradiction. — Agent Smith

It is not difficult.

excluded middle: P or not-P

non-contradiction: not(P and not-P)

bivalence: (P or not-P) and not(P and not-P)

So bivalence is just the conjunction of excluded middle and non-contradiction.

You propose that there are closed well formed formulas that are meaningless (have no interpretation or the valuation function also has meaningless in its range). Mathematical logic does not have that presupposition, so it is not, in its own terms, taking meaninglessness as falsehood. Indeed, the notion that contradictions are false is the ordinary notion through the centuries of the subject of logic. And it facilitates the ordinary notion of entailment.

Of course, one is free to develop a logic in which contradictions are not false but instead valuations include meaningless in addition to true and false, or whatever multi-valued logic one wants to have. I'm only telling you how it happens to work in ordinary predicate logic in which the domain of the valuation function is the set of sentences and the range is {true false}. So every sentence is assigned truth or falsehood and not both per any given model. This is desirable in ordinary predicate logic, especially, as I mentioned, for facilitating the ordinary notions including the ordinary notion of entailment.

Anyway, the context of discussion was the mathematical paradoxes, especially Russell's paradox, and especially the other poster's wildly mistaken notion that classical logic is trivial (proves every formula) because it is inconsistent, unless it eschews the principle of explosion*. In that context, one would ordinarily take it that contradictions are false.

* Contrary to the other poster's foolishness, classical logic is not trivial (it does not prove every formula) and it is not inconsistent, and therefore it does not need to eschew the principle of explosion.

Also, aside from providing semantical interpretation, and myriad other result in model theory, we use models for consistency proofs, relative consistency proofs, and independence proofs (the independence of the axiom of choice and the independence of the continuum hypothesis most famously).

And Robinson's non-standard analysis was developed using model theory.

Anyway, languages (not just formal languages) have both syntax and semantics. Models are the ordinary semantics for languages in predicate logic. And logic itself is not just the study of proof but perhaps even more basically the study of entailment. And entailment is semantical in the sense that 'truth' is determined by the model theoretic semantics for a language.

Also, I overlooked that you said "not adopting". So I added more response accordingly.

This means that if we adopt the method of models, Russell's Paradox is impossible. — Tate

That is incorrect. No matter about models, if you have inconsistent axioms, then you derive Russell's paradox. Then, it is merely an additional note, not confined to Russell's paradox or unrestricted comprehension, that any inconsistent axiom is perforce a non-logical axiom.

What are the consequences of not adopting that method? — Tate

The method of models is ubiquitous in mathematical logic. There are many theorems about models (the subject is called 'model theory') so I don't know how to say simply what the consequences are, since there are many consequences.

Three of the most famous consequences are the completeness (and soundness) theorem, the compactness theorem, and Lowenheim-Skolem. Those are sometimes considered to be the "Big Three Pillars" (my term) of first order logic.

[Edit: I overlooked that you said not adopting.]

The crucial consequence of not having a method of models is that we would need to find some other means of providing semantical interpretations for theories. The method of models is the way we say what the formal sentences mean.

I've never understood why this is so. — Tate

Per the valuation function for truth in models (the Tarski definition by recursion on formulas), every sentence is either true in the model or false in the model but not both. And per that function, the negation ~P of a sentence P is true in the model if and only if P is false in the model; and P is true in the model if and only if ~P is false in the model.

Now, suppose a contradiction P & ~P were true in a model. A conjunction is true in a model if and only if both conjuncts are true the model. So both P and ~P would be true in the model. But then P would be true in the model and false in the model, which is impossible.

You know the derivation of Russell's paradox, right?

Assume EbAy(yeb <-> ~yey)
"There exists a set b such that for all sets y, y is a member of b if and only if y is not a member of y".

So beb <-> ~beb
"b is a member of b if and only if b is not a member of b".

So EbAy(yeb <-> ~yey) yields a contradiction.

But any sentence that yields a contradiction is false in every model, so EbAy(yeb <-> ~yey) is false in every model.

See post above that I added to.

EbAy(yeb <-> ~yey) is false in not just some models but it is false in every model.

What would that sentence be? — Tate

The schema of unrestricted comprehension specifies an infinite number of axioms:

If F is a formula and b is a variable that does not occur free in F, then all closures of

EbAy(yeb <-> F)

are axioms.

Every instance of that schema is a non-logical axiom. [Edit: I have to think more closely whether every instance is non-logical. But still, the point prevails that unrestricted comprehension has non-logical instances, and in particular an instance used to derive Russell's paradox is not just non-logical but it is also inconsistent.]

Moreover, the particular instance:

EbAy(yeb <-> ~yey) is not just false in some models but it is false in every model, as we know from Russell's paradox.

Note also that this is not specific to set theory. To derive Russell's paradox from unrestricted comprehension, we don't rely on anything specific about 'e', the membership relation. Rather for any 2-place predicate R whatsoever (whether 'R' stands for 'is a member or' or 'shaves' or 'is a parent of' ...) we derive a contradiction from:

EbAy(Ryb <-> ~Ryy)

In the sense that these principles are untrue in some models? That doesn't make any sense to me. How can a principle be false? — Tate

It would be false in some models if it were formalized as a first order sentence, or, for a schema, it would have false instances if the schema were formalized.

The important point for the prior discussion here is that Cantor did not confine himself to merely logical principles. The Cantorian paradoxes are not a failing of logic.

Does this mean you're sort of stretching the idea of non-logical axioms to address the problems associated with naive set theory? — Tate

Sure, where the set theory is not formalized with axioms, we can at least point out that the pre-formal principles it uses are non-logical, at least in the sense that they would be non-logical axioms if they were formalized.

It appears that axioms were created specifically to block the path to Russell's Paradox. — Tate

It is common for people to put it that way, but it could be misleading to put that way.

Adding axioms cannot block the derivation of statements that are already derivable without the added axioms (i.e. the logic is monotonic). What axiomatic set theory does to avoid Russell's paradox is not add axioms but to refrain from adding the axiom schema of unrestricted comprehension that results in Russell's paradox. Then other axioms are added that permit derivation of the desired mathematical theorems.

Specifically, Zermelo refrained from adding the schema of unrestricted comprehension, so that schema is not available to use to derive Russell's paradox. Then Zermelo does add the schema of separation instead of unrestricted comprehension and the other axioms of Z set theory.

And Fraenkel did the same except he adds the schema of replacement (which is strong than separation but weaker than unrestricted comprehension) rather than the schema of separation. (I think the schema of replacement is important mainly for deriving transfinite recursion, which can't be derived in mere Zermelo set theory.)

Did Cantor's original set theory have non-logical axioms? — Tate

Cantor didn't have axioms. But of course he did use non-logical principles even if not formalized as axioms.

Except for the pure predicate calculus itself, any mathematical theory (such as formalized in predicate logic) has non-logical axioms.

How many points do I have to throw in the bag to fill it? — Hillary

'throw in', 'bag', and 'fill' (in your context) are not mathematical terms, so I can't give you a mathematical answer to your question.

However, the mathematical question "how many 3D-points are in a non-empty volume?" does have the mathematical answer: the cardinality of the set of real numbers.

glue two points together — Hillary

throw as many points in the bag — Hillary

Those are your personal, impressionistic locutions. Real analysis doesn't have such terminology.

How can you construct a continuum with points as building blocks? — Hillary

The real continuum is constructed in formal axiomatic set theory. We prove that there exists a set and an ordering on that set (which are unique up to isomorphism) such that the ordering has the completeness property.

Your question suggests that you are not familiar with the basics of the subject.

Sherlock Holmes doesn't exist. Where s = Sherlock Holmes, (∀x)¬(x=s) = ¬(∃x)(x=s) — Agent Smith

That's wrong. This is correct:

Hx for "there is an x such that x is a Scotland Yard detective named 'Sherlock Holmes'".

~ExHx for "there does not exist a Scotland Yard detective named 'Sherlock Holmes'".

Also, the '=' sign is a 2-place predicate symbol that goes between terms, not between formulas. Using '=' between formulas is not syntactical. For equivalence of formulas, use the biconditional '<->' that is a 2-place conncective.

The diagonal argument is constructive and intuitionistically valid
— TonesInDeepFreeze

The argument only goes to show that the continuum cannot be broken up in points. — Hillary

The proof shows that there is no enumeration of the set of real numbers. As formalized, the proof uses only first order logic from the axioms of Z set theory, and it could even be Z without the axiom of infinity by couching the proof with an hypothesis, rather than a given, that there exists an infinite set. Moreover, as I mentioned, the proof can even be couched in constructive mathematics with a notion of a "potential infinity" as a computational process without an upper bound of actions.

And the proof shows no such thing that the set of real numbers is not made of points. The continuum is <R less-than-on_R>. By definition, every real number is a point. It's a definition and does not require proof, and the diagonal argument does not contradict it.

Leading to confused notions of infinitesimals or differentials. — Hillary

Ordinary real analysis does not use infinitesimals. And the non-enumerability of the set of real numbers does not lead to use of infinitesimals.

Smith is not ad confused as you suggest — Hillary

I demonstrated exactly the manner in which he is confused and misinformational.

A maximally great being exists: (∀x)(Mx→(∃y)(y=x)) — Agent Smith

That's wrong. This is correct:

Mx for "x is maximally great"

ExMx for "there is an x such that x is maximally great", which is to say "a maximally great being exists".

If god is the maximally great being then god exists = Mg→(∃y)(y=g) — Agent Smith

That's wrong. This is correct:

g for "god"

Meanwhile, it is merely a logical truth that Ey y=g. Whatever 'g' stands for, it is something that exists. If 'g' stands for god, then it is not required to argue that god exists, since existence is given by the mere fact that 'g' stands for some existent which we have stipulated to be god. But a more meaningful assertion is:

Gx for "x is a god", or with monotheism, "x is a god and nothing else is a god". Then ExGx for "there is a god", or with monotheism, "there is a god and only one god".

That is, if you have a constant symbol such as 'g', then it is a given that 'g' refers to something. So Ey y = g

(In modal logic, we can have an existence predicate, but that is another subject, more complicated, and requires specification of the modal logic.)

What's the difference between (∃x)(Gx) (God exists) and (∃x)(x=g) (there exists something and that something is god) where g is God? — Agent Smith

ExGx does not say "god exits". Rather, 'G' is a predicate symbol, so that Gx may be taken as saying "x has the property of being a god", or, for monotheism, "x has the property of being the one and only god". Then ExGx says "there is an x such that x has the property of being a god", or with monotheism, "there is an x such that x as the property of being a god and nothing else has that property'.

Again, Ex x = g is merely a logical truth. 'g' is a constant' and so there is an x such that x = g.

In sum, there is a difference between a name 'g' for some object and a predicate 'G' for a property. When we use a name 'g' then it names something. When we state 'Gx' we assert that x has the property G.

A formalized version of "paraconsistent logic" (logic of paradox) is the Fuzzy Logic — Gnomon

If I'm not mistaken, there is work in combining formal paraconsistent logic with formal fuzzy logic. But fuzzy logic itself is not a formalization of paraconsistent logic.

Could you give an example of a non-logical axiom — Tate

Each axiom of group theory is a non-logical axiom.

what makes it non-logical? — Tate

A non-logical principle is one that is not true in at least one model. — TonesInDeepFreeze

Good. You are at your most eloquent with emojis.

silly games — Agent Smith

Explaining that ZFC does not have unrestricted comprehension that yields the existence of the Russell set is not silly game playing.

What exactly is wrong with what I said? — Agent Smith

I said exactly what is wrong with what you said just a few posts ago! And I explained also in posts yesterday. And from months ago I've explained why various of your posts are confusion and outright misinformation.

infinitesimals — Hillary

Historically, infinitesimals were not given a rigorous treatment. However eventually non-standard analysis was devised in which infinitesimals are constructed with the ordinary set theoretic axioms.

A thousand apologies. — Agent Smith

As your posting history suggests, your thousand apologies will be followed by a thousand more of your egregiously misinformational posts. One only has to sit back, have a cup of tea, and wait for the next one.

Cantor's diagonal argument uses negative self-reference, proves by reductio ad absurdum — ssu

The diagonal argument does not require reductio ad absurdum. And 'negative self-reference' needs a definition.

The diagonal argument is constructive and intuitionistically valid. Though it makes use of an ingenious technique, that technique relies on no logical or mathematical principles that are not common elsewhere in mathematics. Indeed, even without the assumption of infinity, the diagonal argument would still go through as couched in terms of "potentially infinite" processes rather than infinite sets.

axioms don't have to be explained — ssu

Sure, but it is famously the case that certain mathematicians and philosophers do explain the axioms and give justifications for them.

Yet as we don't take as an axiom that all numbers can be well-ordered — ssu

The axiom of choice implies that every set has a well ordering, thus, in particular, the set of real numbers has a well ordering.

the set theory based axioms of mathematics allows Russell's set to, well, exist — Agent Smith

Let's select that quote in particular. It is pure misinformation. ZFC, which is the common set theory for mathematics, is formulated so that it does not allow the usual proof that the Russell set exists, and no one has shown that ZFC does prove that the Russell set exists.

The existence of the Russell set was proven using unrestricted comprehension. But ZFC does not have unrestricted comprehension.

You should inform yourself and stop egregiously posting misinformation.

I explained why you are incorrect. You are terribly mixed up and you don't know what you're talking about. And you add additional confusions and misinformation with each post.

He's defining U in terms of T. — L'éléphant

First, I see a purported definition of U that mentions U in the definiens, which is circular.

Second, I see T mentioned in the definiens of a definition of P8, but I don't see a prior definition of T.

Third, I don't see T mentioned in his purported definition of U.

Take his post one step at a time. When he purports to define U, he is circular. So we have to stop there and fix that.

I read the post. Then I went back to the first place that, as far as I can tell, he doesn't make sense. His theory U is not defined; his proposed definition is circular, so such questions that mention it are nugatory unless we first fix that definition.