I was thinking of this thread drawing fractions for my son. You can draw a half many different ways. A circle with a vertical line bisecting it and one side shaded, a circle with a horizontal line bisecting it and one half shaded, a circle with two lines and two 1/4th sections shaded that sit diagonal from each other, a circle chopped into 8ths making what looks like the nuclear waste symbol, etc.

None of these looks the same. Arguably, for almost all (maybe all) the systems we experience in everyday life, they actually are not the same. No pie is the same all the way through. No system is set up so that the distribution of molecules and the energy of individual molecules is evenly distributed throughout. Getting one slice of pie that is the size of half a pie is different from getting four slices that add up to half the area of the pie.

Got me thinking about how much of what we can do in the sciences is the results of being able to abstract away differences. This thread didn't go anywhere but it brings me back to the idea of synonymity. If things only exist as they exist for other things (e.g., information theoretic approaches) than you have a shifting amount of synonymity between different identities. Water = H20 = identity for many reactions. However, push enough water in one direction and tiny differences in energy levels result in (as of today's physics) totally unpredictable and chaotic turbulence. You move from differences that don't make a difference, to differences that make a large, macroscopic difference.

When people say the three laws of thought are

A=A ... identity
A v ~A .... excluded middle
~(A & ~A) ... non-contradiction

they are using 'A' for two different things.

For identity, 'A' ranges over objects.

For excluded middle and non-contradiction, 'A' ranges over propositions.

Indeed, that is not a good presentation. 'A' should not be mixed up that way.

A better statement is:

For all individuals x, we have x=x.

For all propositions P, we have P v ~P and we have ~(P & ~P).

But the "three laws of thought" paradigm does not express the full scope of reasoning about identity or reasoning about propositions. There are other principles that are also needed for reasoning about identity and for propositional logic. The paradigm has been surpassed by those of symbolic logic that are more comprehensive.

Excellent post. So many of the issues with the "laws of thought" were dissipated by Frege. His logic clarifies the distinctions muddled with here.

His invention of the predicate calculus is great intellectual wisdom.

I do recall reading the identity laws in an introductory text on logic. As you said, identity is about individuals rather than propositions.

If it's not too much trouble, can you please refresh my memory on the 3 identity laws in logic? Danke!

There's one that I haven't forgotten:

1. x = y
2. Px
Ergo,
3. Py

I can't remember the name of the rule though. :sad:

There are 2 more, one's called symmetry. That's all I have on the identity laws. :smile:

I can add a pear and a twig and get to two.

x=x ... reflexivity
with
(x=y & Px) -> Py ... indiscernibility of identicals (aka substitutivity)

is a complete axiomatization of identity theory and they imply:

x=y -> y=x ... symmetry
and
(x=y & y=z) -> x=z ... transitivity

The converse of the indiscernibility of identicals is the identity of indiscernibles. Interestingly, if the language has infinitely many predicates, then the identity of indiscernibles is not expressible.

Another complete axiomatization (from Wang) is:

Ex(x=y & Px) <-> Py

That proves

x=x
and
(x=y & Px) -> Py

:ok:

I made a bad typo. I just now corrected it.

This doesn't remove the problem I showed in the paradox.

I made a bad typo. I just now corrected it. — TonesInDeepFreeze

No problemo, you corrected it before anyone saw it! :smile:

Did you read this? I'm sure you're in the know about what relativist is talking about.

I have a minor use-mention quibble with the penultimate sentence, but otherwise it seems to me that he or she gave a reasonable explanation of the identity of indiscernibles. Why do you ask?

Just found relativist's post interesting! It clears up my misconception.

That's because there is no paradox to solve. You're just equivocating "apple" with "1". Just stop doing that.

What's your take on two cars of the same model? Would you still say identity of indiscernibles or would you switch to equivalence of indiscernibles? — Agent Smith

Neither, strictly speaking, because there will be differences (e.g. the VIN number).

These identities lead to consideration of essentialism and natural kinds. "Electron" is a natural kind: all electrons share the same set of properties (except for spatiotemporal location). That set of properties is the essence of electron-ness. Any object possessing that exact set of properties, is necessarily an electron.

:ok:

What's your take on two cars of the same model? Would you still say identity of indiscernibles or would you switch to equivalence of indiscernibles? — Agent Smith

... there will be differences (e.g. the VIN number).
These identities lead to consideration of essentialism and natural kinds. "Electron" is a natural kind: all electrons share the same set of properties (except for spatiotemporal location). That set of properties is the essence of electron-ness. Any object possessing that exact set of properties, is necessarily an electron. — Relativist

Sure, but what if there are not differences, since you bring up electrons? Two electrons Bill and Ted enter from opposite directions a shared space and interact, and leave via different trajectories than their incoming one. Which exiting electron is Ted? Do particles have identity? They seem very much not to. A molecule perhaps does, but a molecule is nearly a classical thing. There's no evidence that they have spatiotemporal location until measured, so that doesn't distinguish them. The topic is about identity of particulars, not shared properties of a universal.

I'm not talking about an epistemological distinction. I'm not asking if it's possible to measure which one is Ted. I'm just asking if one of them is in fact Ted, however much Bill has the same properties.

<sigh> Time for a math lesson

Given relation R and elements a, b, and c, we may define many properties, but there are 3 of interest,

• Reflexive Property : aRa
• Symmetric Property : if aRb, then bRa
• Transitive Property : if aRb and bRc, then aRc

When a relation has these 3 properties it is called an equivalence relation. Examples are congruence, similarity, "has the same birthday as", and (of course) =.

But there are many common relations which violate one or more of these properties and, thus, are not equivalence relations. Consider

• "is the son of" : violates all 3 properties
• "is the ancestor of" : violates Reflexive and Symmetric (but Transitive holds)
• < : violates Reflexive and Symmetric
• <= : violates Symmetric only
• "is the sibling of" : violates Reflexive only (assuming sibling means sharing the same mother and father)

and so on

A good question by all accounts. If the question had been about billiard balls on a billiard table, it would be an easy answer because we can see the balls bouncing off of each other and then moving in the direction opposite to their initial trajectory.

Electrons however can't be observed like billiard balls above. Nobody knows whether two electrons simply pass through each other without any interaction or behave exactly like billiard balls as described above. In other words, after two electrons, call 'em A and B, have been put on a collision course and they occupy the same spatial region (collision occurs in case of billiard balls), we can't identify which is A and which is B. However, this isn't exactly a counterexample to Leibniz's identity of indiscernibles rule as the electrons can be identified if we get our hands on all the information we need.

"is the son of" : violates all 3 properties — Real Gone Cat

The Son/Father is the father/son of The Son/Father!

"is the sibling of" : violates Reflexive only (assuming sibling means sharing the same mother and father) — Real Gone Cat

The Father/Son is the sibling of The Son/Father!

Sancta Trinitas Unus Deus :pray: :pray: :pray:

A = A is simply the most basic form of saying that ~A = A is false. It is the axiom that tells us that contradictions are always false. — ArmChairPhilosopher

Then why the roundabout way of stating ~A = A is false? Is it hat we don't want to introduce the "not equal" connective?

For example, there are no triangles outside actual triangular objects in trope theory. — Count Timothy von Icarus

And mathematically there are no actual triangular objects in the physical world, merely approximations.

Saying they are two different balls because they are in two different locations is not that helpful either. Relative location is a derived trait, one that changes with context. If such derived traits are part of identity then you would be a different person when you're north of your house than you are when you're south of it. — Count Timothy von Icarus

Good point

The next thing is that the law of identity allows that a thing might be continuously changing, yet maintain its status as the same thing. This is very difficult to conceive of . . — Metaphysician Undercover

unless change is part of the thing's identity, as a whirlpool for instance, or the human body's continuous process of food intake and subsequent evacuation.

He wants to move past propositions such as, "the apple is red," that take the apple and its redness as existing outside of the perceiving mind. Identity has to be different because identity changes and grows more complete over time as our knowledge grows (as the dialectical progresses). And he doesn't want to look just at the apple as being a part of an individual subject's mind, since he is not a solipsist or subjective idealist, but how it is for all minds. — Count Timothy von Icarus

You might find E-Prime relevant to the above.
https://en.wikipedia.org/wiki/E-Prime#:~:text=E%2DPrime%20(short%20for%20English,conjugations%2C%20contractions%20and%20archaic%20forms

Perhaps it is not directly A. What is meant by A? His form, his composure?

I think we are identified by our pulse, thus A is the man is wrong by your standard of A(being his presence). Having the presence of a man is equal to taking his head- you have his head, metaphorically.

Identity seems to be stringently the pulse as it identifies all of a bio.

unless change is part of the thing's identity, as a whirlpool for instance, or the human body's continuous process of food intake and subsequent evacuation. — Art48

These are activities, not things. Activities are attributed to things, as what a thing is doing, so the law of identity doesn't apply. This is partly why it is very difficult for us to gather a complete understanding of activities.

Then why the roundabout way of stating ~A = A is false? Is it hat we don't want to introduce the "not equal" connective? — Art48

It is that we want to keep the axioms in the most basic form possible. ~A != A (or ~A = A is false) are derived statements that can be simplified to A = A.

:up:

This just in. A mysterious caller tips philosophers off:
Metaphysics (identity & change) of/in Logic.