"Quasi-set theory Q is a first-order ZFU-style set theory. Its underlying logic is just like first-order classical logic without identity (the system L mentioned before), with a significant difference: the semantics for that system L underlying Q is described in a non-reflexive metalanguage; just like classical logic has a semantics developed in classical set theory, this particular system of logic (i.e. L) has a semantics developed in a non-reflexive set theory (more on that topic soon). So, the system in question is not exactly classical logic, but it formally coincides with classical logic, although it is semantically different (since its semantics is provided in a non-reflexive metalanguage)." — Classical Logic or Non-Reflexive Logic? A case of Semantic Underdetermination
Terms to which identity holds and terms to which it does not. — MindForged
How so if "∀x(x = x)" does not mean "∀x(x = x)" but something else? You cannot make a statement which does not assert itself without... an extra-ordinary amount of freedom what can be written without any possiblilty of someone marking it as an error.so that there is a failure of application when you try to assert something like "∀x(x = x)". — MindForged
Do you mean that sentence to be taken as truth?Truth-predication isn't even directly involved, I think. — MindForged
We allow the usual connectives, quantifiers, an identity relation symbol, and punctuation symbols. For one of the kinds of terms (variables and constants), let us say T p T2, ..., terms of the first kind, we allow that identity holds as usual. In the intended model they represent the individuals. For terms of the second kind, tp t2, ..., identity is not an allowed relation. In the intended interpretation those terms denote non-individuals, items with no identity conditions
Do you mean that sentence to be taken as truth? — Heiko
Expressing such a formalism in itself is:That I believe what I'm saying is true does not entail that it's impossible to give a coherent formalism where objects are not self-identical. — MindForged
It'd be unclear what "those" refers to if the "terms" would not be the "terms", don't you think?For terms of the second kind, tp t2, ..., identity is not an allowed relation. In the intended interpretation *those terms* denote non-individuals, items with no identity conditions
"As we have said in the previous section, in non-reflexive logics we do not accept the negation of the reflexive law of identity. Also, we don’t have to accept that it must fail in at least some interpretations. Rather, we adopt its restriction in the form of its inapplicability. Here, ‘inapplicability’ is couched in terms of identity not making sense, not being a formula, for some kinds of terms." - Arenhart
Sure it is. It models that a mental object that was defined stays the same. A quantum particle, in contrast to it's definition, does not.The law of identity is not a law about reference — MindForged
Why do you think the Law of Identity is required in classical logic? I'm guessing here that, as one of the three laws of thought, it is a necessity for logic. — TheMadFool
Without this basic agreement conversation would be impossible right? — TheMadFool
How would you recognize a change in meaning if the term wasn't identical to itself?or by changing the meaning of terms mid discussion — MindForged
Anyway, I think the Law of Identity has to do with symbols and semantics both:
1. Symbolic: ''Box'' here at one time = ''Box'' there at another time
2. Semantic: ''Box'' means a container and this ''box'' = that ''box'' at another place in the conversation
Without this basic agreement conversation would be impossible right? — TheMadFool
Correct. But the symbol "a" just establishes an abstract identity. This is why you can know that I am talking about the same symbol when I now write, "a" was introduced at an earlier time.it is concerned with the particular thing which is identified through the use of the symbol. — Metaphysician Undercover
If I define "x" as "a sentence that does not exist.", what do we have then?
"x" - as a letter which refers to
"x" - as the idea of it being a variable which refers to
nothing (a sentence which does not exist)
We still can talk about x and sentences that do not exist: Such x'es do not require much typing. — Heiko
But that there may be some class of objects where applying identity doesn't make sense (like a category mistake). — MindForged
Such an idea, however, is subjective rather than objective. Continuing with the electron example, let's take electron A and electron B. No scientific analysis can distinguish A from B. So, we conclude A = B or, in your case, we give up the notion of identity altogether.
Not to be nitpicky but there is a difference between A and B electrons. They're at different loci in space. Don't you think, therefore, that we can still retain the concept of identity for such situations? — TheMadFool
There is no ''discernible'' difference between two electrons, for example. So, identity, as in uniquness, is a problem for electrons. Am I getting your point? — TheMadFool
So, if at all there's a problem with the concept of identity it lies with our inability to see the difference between electron A and electron B. It's subjective. But we know there IS a difference in location between A and B. That's objective.
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