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
Your "solution" to the paradox is standard technique — Agent Smith
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
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
He has some odd notions concerning instantaneous velocity you might find amusing. — Banno
This is also Hegel's dialectic. In simple terms, all opposites are defined by a shared property. — Jackson
It's simply a matter of bad axioms — Metaphysician Undercover
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
Hot and cold are not the type of things which themselves have properties. — Metaphysician Undercover
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.3. The LNC needs to be scrapped + a version of paraconsistent logic needs to be adopted — Agent Smith
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
homological relationship to the empirical-material world it's modeling? — ucarr
signifier (math model) & its referent ( material object) — ucarr
can the math model successfully model a self-contradictory material object without containing within itself any contradictory math expressions? — ucarr
[...] then foundational logic of math needs reexamination. — ucarr
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
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
The LNC is incompatible with paradoxes — Agent Smith
The LNC needs to be scrapped + a version of paraconsistent logic needs to be adopted — Agent Smith
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
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
Classical logic works just fine for a vast amount of the logic for the sciences — TonesInDeepFreeze
“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. — TonesInDeepFreeze
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
gibberish — javra
A square is a circle — javra
A square is a circle — javra
That's not paradoxical. — TonesInDeepFreeze
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