I think it's mainly philosophical types who are drawn to them, some to slay them like dragons and some to peep through them like they're doors to somewhere else. — Tate

Don't you love a joke every now and then? Doesn't everybody? Laughter, they say, is the best medicine!

Don't you love a joke every now and then? — Agent Smith

I think
So

Your "solution" to the paradox is standard technique — Agent Smith

Well it's a standard paradox, the sort that @TonesInDeepFreeze showed how to deal with earlier. It posits a set and then asks if the set is a member of itself.

So you have a paradox. But your conclusion is that logic is broken. How do you move from the paradox to that conclusion?

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.

Well it's a standard paradox, the sort that TonesInDeepFreeze showed how to deal with earlier. It posits a set and then asks if the set is a member of itself.

So you have a paradox. But your conclusion is that logic is broken. How do you move from the paradox to that conclusion? — Banno

You know this stuff. I already explained it in my previoud post.

The long and short of it: The LNC is incompatible with paradoxes; one has to go. I know your choice (you deny that there are true paradoxes).

He has some odd notions concerning instantaneous velocity you might find amusing. — Banno

It's simply a matter of bad axioms, useful but not truthful. If you judge good by usefulness, you'll say the axioms are good. If you judge good by truthfulness you'll say the axioms are bad. To choose the latter may seem like having "odd notions" to you.

This is also Hegel's dialectic. In simple terms, all opposites are defined by a shared property. — Jackson

This is the root of the problem, and it's demonstrated well in some of Plato's dialogues, like The Sophist and The Parmenides. If we take two things which are categorically different, and set them up as opposites, then we falsely assign a shared property to them. True opposites, in the absolute sense, cannot share any property, or else they are not absolutely opposite.

Absolute opposition requires a separation of category. So for example, if negative and positive are supposed to be opposite in an absolute sense, they cannot share a common property, or else they are not absolutely opposite. And when we allow what you call the "shared property" to be a property of the of the ideas which are opposite, we make a category mistake because opposite is what is assigned to the properties, not to the object itself. Now we produce a property of the property.

So for example, hot and cold are opposite. We can say these two are possible properties of the same thing. But if we look for a "shared property" of hot and cold, we make the category mistake. Hot and cold are not the type of things which themselves have properties. Hot and cold are defined in different terms, terms of activity (becoming), and becoming is not the type of thing which is described through properties, its described by a change in properties.

It's simply a matter of bad axioms — Metaphysician Undercover

:up:

If a train of logical reasoning ends on a contradiction (paradox), the following possibilities must be considered

1. Fallacies (mistakes in applying the rules of natural deduction)

and/or

2. One/more false premises (axioms/postulates)

If not 1 and/or 2 then and only then

3. The LNC needs to be scrapped + a version of paraconsistent logic needs to be adopted

Arigato gozaimus Metaphysician Undercover san!

:confused:

If we take two things which are categorically different, and set them up as opposites, then we falsely assign a shared property to them. True opposites, in the absolute sense, cannot share any property, or else they are not absolutely opposite. — Metaphysician Undercover

Which is why Hegel's dialectic is different from Plato's.

Hot and cold are not the type of things which themselves have properties. — Metaphysician Undercover

Properties of temperature.

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

Not so fast. The Law of Non-Contradiction is a good rule of thumb for most contexts. But there is one common circumstance where LNC does not apply : Holism. The reductive methods of science are appropriate for things-in-isolation. But when a thing participates in a larger System, it shares qualities of the system, which compromises some of its own properties. To a reductionist observer such holistic behavior may seem inconsistent and paradoxical.

For example : what Einstein called spooky-action-at-a-distance, Schrodinger called "entanglement". Which implies that some quantum particles in a holistic (waveform) system share some properties with other particles. Apparently, in their waveform state, electrons are connected to all other electrons in the universe, in such a way that a measurement of one instantly affects (e.g. flips the spin of) all similar particles. From that perspective, it's not a contradiction, but a feature of Holism : an emergent property. :smile:


Holism ; Holon :
Philosophically, a whole system is a collection of parts (holons) that possesses properties not found in the parts. That something extra is an Emergent quality that was latent (unmanifest) in the parts. For example, when atoms of hydrogen & oxygen gases combine in a specific ratio, the molecule has properties of water, such as wetness, that are not found in the gases. A Holon is something that is simultaneously a whole and a part — A system of entangled things that has a function in a hierarchy of systems.
BothAnd Blog Glossary

"The opposite of a profound truth is also a profound truth"
___Neils Bohr, baffled by apparent violations of LNC

"We are not only observers. We are participators. In some strange sense, this is a participatory universe. Physics is no longer satisfied with insights only into particles, fields of force, into geometry, or even into time and space." ___John A. Wheeler

SHARING IS PARTICIPATING (parts unite with the whole)

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?

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.

If a train of logical reasoning ends on a contradiction (paradox), the following possibilities must be considered

1. Fallacies (mistakes in applying the rules of natural deduction)

and/or

2. One/more false premises (axioms/postulates)

If not 1 and/or 2 then and only then

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

While I don’t purport to be an expert in applied formal logics, this seems worthwhile to mention:

All apparent paradoxes not accounted for via (1) or (2) might also be accounted for in principle by multi-valued logics, with fuzzy logic as one variant. MVLs can, for example, take into consideration things such as partial truths, or else partial falsehoods (e.g., on which a great deal of spin and misinformation is for example dependent, this in real-life applications of truth-values).

Multi-valued logic does not reject the LNC.

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.

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.

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.

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. — TonesInDeepFreeze

From Internet Encyclopedia of Philosophy:

“strong paraconsistency includes ideas like:

Some contradictions may not be errors;
classical logic is wrong in principle;
some true theories may actually be inconsistent.”

Sounds like strong ( as opposed to weak) paraconsistency does see contradictions as ruinous not classical logic.

Yes, so?

Sounds like strong ( as opposed to weak) paraconsistency does see contradictions as ruinous to classical logic.

In the sense you mention a 'truth predicate' we actually say a 'truth function'. Meanwhile, (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. — TonesInDeepFreeze

As someone largely ignorant of formalized logics:

The less technical format of the liar paradox (contrasted to "this sentence is false") is that of “someone who is a member of X claims that all Xs are liars”.

-- This can be true in the sense that all members of X have at some point in their lives told lies, thereby being definable as liars. Non-paradox.

-- It can also be true in the sense that all members of X have a larger than normal propensity to tell lies, thereby again being deemed liars. Also non-paradoxical.

-- The claim can only become paradoxical when interpreting that all members of X can only strictly communicate via lies, without any exception. Yet this interpretation is contradictory to the real-life occurrence of liars.

While I don’t know how to translate the aforementioned into formal logic, I do find that the claim “I am a member of humankind, and all humans I know of (including myself) are liars (on account of having lied at some point in their lives)” to hold a truth value.

Again, as someone ignorant in the formalities of the matter, cannot this latter sentence as expressed be mapped to a truth value in formal logics?

Or would one argue that the sentence as expressed (a variant of “someone who is a member of X claims that all Xs are liars”) does not present a valid format of the liar paradox?

Of course, if we wish to have theorems that are contradictions but without explosion, then classical logic doesn't work, and if one wishes to have contradictions without explosion, then one may say that classical logic is thereby wrong.

But the particular argument given by Agent Smith does not vitiate classical logic itself. Classical logic works just fine for a vast amount of the logic for the sciences. You and I are now communicating with computers built by principles of classical logic. And that does not overlook that for arguably paradoxical statements in a wider scientific context, classical logic may be inadequate.

In particular, Russell's paradox does not ruin classical logic, which gives us an immediate and wonderfully simple solution with the theorem that there does not exist such a contradictory relation.

Classical logic works just fine for a vast amount of the logic for the sciences — TonesInDeepFreeze

And "traditional" mathematics. But with over 26,000 topics in math it's getting harder to pigeonhole.

Rather than get bogged down in whatever vagaries there might be in the Epimenides paradox, I would suggest the clearer, simpler, mathematically "translatable" simpler and more starkly problematic "This sentence is false".

“I am a member of humankind, and all humans I know of (including myself) are liars (on account of having lied at some point in their lives)” to hold a truth value.

Again, as someone ignorant in the formalities of the matter, cannot this latter sentence as expressed be mapped to a truth value in formal logics? — javra

No arithmetically adequate and consistent theory can define a truth predicate by which to then formulate a predicate 'is a liar'.

Keep in mind that Tarski's theorem is a claim only about certain kinds of theories (arithmetically adequate and consistent) formulated in classical logic.

However, given predicates Hx for 'x is a human', and Sxy for 'x states y', and Ly for 'y is a lie', and m for you, then we can write:

Ax(Hx -> Ex(Sxy & Ly)) ... "All humans lie sometimes"
Ey(Smy & Ly) ... you lie sometimes.

And those would have truth values.

But so what? It's not in question that we can formalize a lot of non-problematic things. That we can formalize a lot of non-problematic things doesn't refute that "This statement is false" is problematic.

Yes, I didn't write that correctly. What I meant:

Classical logic with added mathematical axioms works just fine for a vast amount of the mathematics for the sciences.

No arithmetically adequate and consistent theory can define a truth predicate by which to then formulate a predicate 'is a liar'.

Keep in mind that Tarski's theorem is a claim only about certain kinds of theories (arithmetically adequate and consistent) formulated in classical logic. — TonesInDeepFreeze

Thanks for the reply!

Rather than get bogged down in whatever vagaries there might be in the Epimenides paradox, I would suggest the clearer, simpler, mathematically "translatable" simpler and more starkly problematic "This sentence is false". — TonesInDeepFreeze

I know. But to me this derivative of the Epimenides paradox is as much pure gibberish as is the what might be called paradoxical proposition of, "A square is a circle". To each their own, maybe.

gibberish — javra

But it's not gibberish. It's syntactical and it talks about the property of truth as pertaining or not to a given sentence, which is a well understood notion. We don't throw out expressions from the language merely because they present logical problems. The expression is well formed; it is only upon further analysis that we find it is problematic. It would be poor analysis to throw out sentences ad hoc only on the basis that de facto they are problematic.

A square is a circle — javra

That's not paradoxical. Rather, with definitions of 'square' and 'circle' and some theorems of mathematics, it simply, without any paradoxical aspects, implies a contradiction.

A square is a circle — javra

That's not paradoxical. — TonesInDeepFreeze

True. Maybe, more in keeping with "this sentence is false", might be "this square is not a square". But I get the the former has a more of self-referential aspect then the latter.

Still, I'd be grateful to hear of any notion regarding why it should be taken seriously as a proposition. This when something like "this square is not a square" is not. For example, can the proposition be deemed a necessary valid deduction from a grouping of incontestably true premises?

Have you read any of Wittgenstein’s later work, in particular his response to what he called Moore’s paradox? He believed that paradoxes in classical logic were artifacts of a ‘craving for the general, meaning a covering over of all sorts of changes in sense and meaning within what logic held to be self-identical.

"This square is not a square" is seen as a self-contradiction on its face, and its truth value is falsehood, and there is no contradiction in saying its truth value is falsehood.

"This sentence is false" also implies a self-contradiction, but it is not so easy to say its truth value is falsehood, since if its truth value is falsehood then its truth value is truth and if its truth value is truth then its truth value is falsehood.

I am not familiar with Wittgenstein's views on Moore's paradox.