Regarding the Law of identity "a is a" is it wrong to argue that a is not a because one a is on the left side of the copula and the other a is on the right side, and having different properties they are clearly not identical — jlrinc
What about the second part of the question. Has there been an equivalent of noneuclidean geometry in Aristotelian logic? Is the law of identity kin to euclidean axioms? Thanks for the help though, this is a complex field and its easy to think you've got a handle on things when you dont. — jlrinc
Has Aristotelian logic been subjected to the same critiques as Euclid's geometry. In other words is there a non Aristotelian logic to be derived by a critical examination of it's axioms? — jlrinc
Regarding the Law of identity "a is a" is it wrong to argue that a is not a because one a is on the left side of the copula and the other a is on the right side, and having different properties they are clearly not identical. — jlrinc
The law of identity (A=A) is a logical necessity. — TheMadFool
The law of identity (A=A) is a logical necessity.
Imagine A is not A. We would then have the logical contradiction A & ~A, violating the law of non-contradiction. — TheMadFool
:up:The best way to think of it is as a definition of "=". — Banno
You seem to be using "A" as the name of a proposition rather than the name of a particular. The "=" usually names the numerical identity relation, which obtains between a particular and itself. If you use "A" as the name of a proposition, and thereby "A=A" to express the claim that the proposition A is identical with itself, then, the negation of this claim isn't expressed as "A & ~A" but rather as "~(A=A)". — Pierre-Normand
A :: A is true IFF A<->A is a tautology. So if ~(A :: A) then ~(A<->A) which is a contradiction. But I wonder how one would express the contradiction so obtained? A &~A? You seem to disagree. — TheMadFool
Two pennies are alike, When one contrasts and compares two pennies, they might say that they are the same, opposed to being different.
But they are not the same penny. — 3rdClassCitizen
The best way to think of it is as a definition of "=". — Banno
The law of identity (A=A) is a logical necessity.
Imagine A is not A. We would then have the logical contradiction A & ~A, violating the law of non-contradiction.
Obviously the left a is not the right. This only means that, when talking about the law of identity, we do not mean the self-identity of the letters, when saying "a is a". If the law of identity was not given the left a would not be the left a. By exclusion this would mean the left a would be the right a and vice versa: a=a would be true. Which would be a contradiction to the law of identity not being given.Regarding the Law of identity "a is a" is it wrong to argue that a is not a because one a is on the left side of the copula and the other a is on the right side, and having different properties they are clearly not identical. — jlrinc
What I ''feel'' is that the Law of Identity is required in logic. May be it's only classical logic that requires it because if you deny that water=water then how are we to have a meaningful discussion on water? Isn't consistency in the meaning of words and terms a requirement for sound argumentation? — TheMadFool
We begin by recalling the infamous Problem of the Identical Particles. According to a widely held interpretation of non-relativistic quantum mechanics, there are many situations in which one cannot distinguish particles of the same kind; they seem to be absolutely indiscernible and that is not simply a reflection of epistemological deficiencies. That is, the problem, according to this interpretation, is seen as an ontological one, and the mentioned indiscernibility prompted some physicists and philosophers alike to claim that quantum particles had "lost their identity", in the precise sense that quantum entities would not be individuals: they would have no identity. Entities without identity such as quantum particles (under this hypothesis) were claimed to be non-individuals."
If, in a discussion, the meaning of ''sex'' changes from gender to intercourse we would have a problem:
Name: John Smith
Age: 24 years old
Sex: Daily with my partner OR Male??!!
So the law of identity is really used independently of the logic, enforcing the idea that if this is the identified thing, which the logic is being applied to, then it and only it, is the thing which the logic is being applied to. — Metaphysician Undercover
The point where I can not follow this is:Identity appears in basically every logic (even non-classical systems), but that's not because it's impossible to modify it or do without it (it just seems like such an obvious thing to assume). Nor does it follow if you limit identity that "water=water" is false. — MindForged
That's an interesting way of looking at it, and it would explain why Aristotle actually didn't formulate the Law of Identity as such, didn't seem to think it that important, and didn't connect it through to the Law of Non-Contradiction (which really was Aristotle's thing). All that - the way we think of the Law of Identity today - seems to be a later development with some of the Schoolmen, Leibniz and Locke. — gurugeorge
There is a natural progression from the law of identity to the law of non-contradiction. The goal is to know, or understand the object. First we identify the object, you might say we point to it, or assign a name to it. If we can do this, then we can say that it has an identify according to the law of identity. Having an identity validates the claim that it exists, as an object. Next, we describe the identified object, and we must do this according to the law of non-contradiction. We cannot assign contradictory properties to the identified object because this is repugnant to the intellect, making the object unintelligible. These principles are designed so as to make the object intelligible, they are what appeals to the intellect in its goal of knowing, or understanding the object. — Metaphysician Undercover
In a nutshell, non-reflexive systems of
logic are systems that violate the so-called ‘Reflexive Law of Identity’ in the form
∀x(x = x). In its ‘metaphysical reading’, the Reflexive Law of Identity is known as a
version of the ‘Principle of Identity’, roughly stating that everything is self-identical.
Versions of this law are restricted in systems of non-reflexive logic, and those systems
are said to incorporate in a rigorous fashion the idea of entities somehow losing their
identity.
Yeah, I agree with that, so far as navigating everyday life goes; but zooming out a bit more, I see identification as secondary (or subsequent to) to discovery, or the knowledge-gathering process. One identifies what is already known, but to bring things into knowledge is a different process, a process of generate-and-test. That's a process of punting, guessing at, possible identities the thing could have (possible coherent bundles of features that are logically interlinked, etc.), and then testing the implications of that possible identity as the object bumps into the rest of the world (including one's experiments and interventions with it). If it doesn't behave as expected, then either we try on another possible identity, or adjust the one we had. — gurugeorge
you imply that identity is what we give to the object. But this is exactly what the law of identity seeks to avoid. — Metaphysician Undercover
I was talking about whether it's possible have a logic to represent the idea that some objects might not be such that Identity is applicable to them. Quoting Arendhart:
"In a nutshell, non-reflexive systems of
logic are systems that violate the so-called ‘Reflexive Law of Identity’ in the form
∀x(x = x). In its ‘metaphysical reading’, the Reflexive Law of Identity is known as a
version of the ‘Principle of Identity’, roughly stating that everything is self-identical.
Versions of this law are restricted in systems of non-reflexive logic, and those systems
are said to incorporate in a rigorous fashion the idea of entities somehow losing their
identity."
As they and others go on to point out, this is a restriction on identity by means of separating the terms of language into those to which identity applies and those of which it does not. — MindForged
"This is also connected with a second point. What exactly is meant when we say that
we deny a tautology (or a logical law, or a logical necessity)? In denying that an axiom
of classical logic is valid in general, don’t we have to accept that this ‘axiom’ is false
in at least one interpretation of an alternative system in which the same formula may
be expressed? Consider, for instance, intuitionistic logic. In denying the validity of
some instances of the law of excluded middle, it is not the case that intuitionists accept
its negation in its place. However, they do accept that the law may be false sometimes
(mostly when we deal with infinite collections)."
[...]
"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. Recall from our discussion in
the previous section that this is the formal counterpart of the idea of something not
having TI. So, if this is correct, the link between metaphysics and logic that underlies
the non-reflexive formulation of the RV is reasonable, in fact, but it does not go in the
same lines as we think it is reasonable to reject some classical principles of logic in
any non-classical logic with the same vocabulary."
I know :)I somewhat confused. In the part you were quoting, I was talking about whether it's possible have a logic to represent the idea that some objects might not be such that Identity is applicable to them. — MindForged
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