A system for use with a multi-valued semantics can be paraconsistent or not.

However, as far as I know, a paraconsistent system can't have a classical 2-value semantics. — TonesInDeepFreeze

:ok: An important point!

Also, don't forget that a paraconsistent logic, depending on its formulation, can have both LNC and non-explosiveness.

Regarding the mathematical handling of Russell's paradox as opposed to dealing with the problem informally, Russell gives a non-mathematical analogue:

Suppose there is a person who shaves all and only those who do not shave themselves.

There 'shaves' is in place of 'is a member of'. And a solution works analogous to the mathematical approach:

There is no relation 'x shaves y' such that for all x there is a y such that y is shaved by x if and only if y does not shave y (there is no person such that that person shaves all and only those who do not shave themselves). Just as, in mathematics, there is no membership relation such that for all x there is a y such that y is a member of x if and only if y is not a member of y.

For "This sentence is false" we have something similar: In a consistent theory, there is no sentence that says of itself that it is false.

There should be a subdiscipline of logic that studies paradoxes; they're an existential threat to the framework of knowledge we've built for ourselves.

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

I'm trying to use homological in a parallel with onomatopoeia as it's used pertaining to verbal language.

Onomatopoeia - a word that sounds like the noise it describes. Examples - boing, gargle, clap, zap, and pitter-patter

Thus a math expression homological to a state-of-affairs, as specified in our example here, expresses contradictory conclusions that are both valid.

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

You're telling me that a math expression that asserts a claim is nonetheless considered theoretical?

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.) — TonesInDeepFreeze

You're telling me that all legal permutations of classical logic expressions are devoid of contradictions?

If so, it must be the case that classical logic parameters categorically exclude contradiction.

If so, this is an example of a mathematician modulating axioms to fit a metaphysical principle (LNC).

If so, then, in the wake of QM, a mathematician can re-jigger axioms to admit contradictions, which action, you suggest, has already been taken.

Thus a math expression homological to a state-of-affairs, as specified in our example here, expresses contradictory conclusions that are both valid. — ucarr

'valid' has a technical meaning. I wonder whether you are using 'valid' in some other sense.

Anyway, a formula is a contradiction if and only if it is of the form "P & ~P" (or sometimes we say it is a contradiction if and only if it proves a formula of the form "P & ~P").

I'm taking classical logic as the context throughout this discussion unless mentioned otherwise:

A contradiction is not true in any model, so, a fortiori, it is not the case that there is a contradiction that is true in every model (i.e. there is no such thing as a contradiction that is valid).

You're telling me that a math expression that asserts a claim is nonetheless considered theoretical? — ucarr

No. 'Theory' in such contexts is has not the same sense as 'theoretical'. A theory is a set of sentences closed under deduction. An interpretation of the language for a theory may be of any kind of entities or states of affairs we may wish to stipulate, not just "theoretical" ones.

classical logic parameters categorically exclude contradiction — ucarr

'parameters' has technical meanings in mathematical logic and mathematics. I don't know what you mean personally by "logic parameters". And I don't know what you mean by "categorically exclude contradictions". I can tell you that in classical logic:

We can write contradictions. We can put any contradiction as a line in a proof.

There is no model in which a contradiction is true. So, a contradiction entails every sentence (i.e. the semantic version of the principle of explosion ).

A consistent theory never has a contradiction as a member (i.e. no contradiction is a theorem of a consistent theory).

An inconsistent theory has every sentence as a member (i.e. with an inconsistent theory, every sentence is a theorem of that theory, i.e. the principle of explosion), so, a fortiori, an inconsistent theory has every every contradiction as a member (i.e. with an inconsistent theory, every contradiction is a theorem of that theory).

this is an example of a mathematician modulating axioms to fit a metaphysical principle (LNC) — ucarr

I guess by "moduating axioms" you mean "choosing axioms".

Yes, the axioms of classical logic are chosen so that with them we achieve soundness and completeness:

A sentence P is provable from a set of sentences G if and only if there is no model in which all the members of G are true but P is false.

The logical axioms are true in every model, and there is no model in which a sentence and its negation are both true. So, it is not the case that there is a sentence and its negation that are both provable from the logical axioms alone.

a mathematician can re-jigger axioms to admit contradictions — ucarr

One can depart from classical logic and allow that a system can prove contradictions but without explosion. However, the semantics for such a paraconsistent framework may be more complicated than the 2-value semantics for classical logic (which for the portion that is the sentential calculus alone, are quite simple). So I don't recall for paraconsistent systems, how, if at all, one specifies the difference between a logical axiom and a non-logical axiom or non-logical premise.

paradoxes; they're an existential threat — Agent Smith

That's true! I cannot understand why more people don't lie awake at night about it! It requires a global response. We need the World Bank, the World Heath Organization, the International Court of Justice, Interpol, the United Nations Security Council, and the entire cast of 'Glee' on this!

Either that, or Agent Smith just doesn't know the meaning of 'existential threat'.

paradoxes; they're an existential threat — Agent Smith

That's a paradox itself. :cool:

That's true! I cannot understand why more people don't lie awake at night about it! It requires a global response. We need the World Bank, the World Heath Organization, the International Court of Justice, Interpol, the United Nations Security Council, and the entire cast of 'Glee' on this!

Either that, or Agent Smith just doesn't know the meaning of 'existential threat'. — TonesInDeepFreeze

:snicker:

Paradoxes are an existential threat to epistemology (truth) & logic. When these two are assaulted (successfully), our world comes crashing down around our ears!

That's a paradox itself. :cool: — jgill

:cool:

paradoxes; they're an existential threat
— Agent Smith

That's a paradox itself. — jgill

How so?

Paradoxes are an existential threat to epistemology (truth) & logic. When these two are assaulted (successfully), our world comes crashing down around our ears! — Agent Smith

There have been controversial puzzles in epistemology for centuries. I don't see any crashing down of the world related to this. What do you think will happen, the media will announce that philosophers still haven't reached agreement on solutions to the logical and linguistic paradoxes and then the financial markets will all collapse, followed by all the populations lapsing into chaos and war?

financial markets will all collapse, followed by all the populations lapsing into chaos and war? — TonesInDeepFreeze

Like the tv show Lost in Space. The robot flashing lights, that does not compute, that does not compute!

There have been controversial puzzles in epistemology for centuries. I don't see any crashing down of the world related to this. What do you think will happen, the media will announce that philosophers still haven't reached agreement on solutions to the logical and linguistic paradoxes and then the financial markets will all collapse, followed by all the populations lapsing into chaos and war? — TonesInDeepFreeze

It's hard to say what'll actually happen - chaos is inherently unpredictable! All I can say, with a fair amount of certainty, is we would be utterly baffled by everything, aren't we already?