How Identity Works | Symbolic Logic Tutorial | Attic Philosophy
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- เผยแพร่เมื่อ 25 ก.ค. 2024
- In this tutorial video, we look at how identity works in logic. We're talking about numerical identity, where a=b says that a and b are the very same thing.
00:00 - Intro
01:00 - Different kinds of identity
01:26 - Numerical identity
03:34 - Puzzle of identity
05:10 - Identity in logic
06:43 - Semantics for Identity
08:15 - An Equivalence Relation
09:38 - Leibniz’s Law
10:51 - Identity of Indiscernibles
11:26 - The Substitution Principle
11:54 - Wrap-Up
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Your videos are great, thanks for doing them and so well. I'm re reading my Logic with Trees book which haven't touched since my undergraduate and these have been super helpful. Thanks again, keep 'em coming.
You're very welcome! I'm planning to do some more on proof trees soon.
Please never stop making videos on Logic! I was wondering if you plan to do a series on inductive inference or logical positivism in the future? They are more advanced, but I hope that others will enjoy them as much as i know i will do.
Thanks, I’ll try to keep going! No plans for those topics yet but they’re good suggestions, I’ll add them to the list.
First of all great videos keep them coming. My problem is that I always get confused here: So when we are giving the truth conditions of an identity statement in first order logic (our object language), we are using identity in a metalanguage. Isn't it still possible to hold that that definition is circular, because I explain identity with identity just expressed in different languages? And second regarding a system of deduction: once I fix the semantics for identity, while I can see that the rules for identity in the deductive system are plausible, how can I be sure that the rules are all I need (I know that one gets there with a soundness and completeness proof of the deductive system in question), but is there any intuitive way to see that while constructing a system ?
Thanks! We're using English (metalanguage) to interpret the logical language (object language). Same deal for 'or', 'and' & so on.
On proof rules: in general, completeness is what tells you the rules are strong enough. So what you often do is start a completeness proof and see what rules you need to drive it through. For identity, the rules are usually pretty standard: substitution plus whatever you need for a=a to be valid.
I may be mistaken
but is there not at least two forms
of identity (in the "numerical identity" field)
an "absolute identity" where "for all a, a = a"
and a "functional identity" where
"looking at these characteristics only, a=b"
It seems thst the latter is the more useful.
I assume (and that is always a dangerous thing)
that the way you were defining it with a model.
Is that correct?
In logic, there's just one form of identity: being one and the same thing, expressed as a=b. There's also 'being the same with respect to a property F' - perhaps that's what you mean> This just means: not differing with respect to F: either Fx & Fy, or ~Fx & ~Fx. This isn't identity, since x and y might be distinct.
Philosophically, there's an issue of identity relative to a sortal, like, being the same cat. But in logic, for x and y to be the same cat, just means: x = y and x is a cat. (So identity simpliciter is the primary notion.)
Hi, congratulation for the video. I have a doubt about identity in second-order logic: why a=b is defined as For all X (Xa iff Xb)? I mean, intuitively I understand that among a's properties there is the property (rather special) of being identical to b. So, since b has every property that a has, also b has the property of being identical to a. But how can we formalize this in second order logic?
It’s not defined like that. Or rather, some people attempt to definite it like that, but most take first-order identity to be primitive. Most agree that the equivalence is true: a=b iff AX(Xa Xb), and some hold it even if those properties don’t include properties like being identical to b.
@@AtticPhilosophy Thank you for your comment. Do you think you could make a video on second-order and higher-order logic? It would be extremely useful as you have the rare gift of clarity
@@donatolisio3186 Great shout, I'll add it to the to-do list.
just tell one thing if i have 3 A"s on my table are they identical with each other?
If there's 3 of them, they're not the same thing! They can be the same TYPE of thing (same type of letter), but not the same entity (not the same token of the letter-type 'A')
Consider a ball formed of atoms. We know that atoms aren't identical to one another, but since atoms made a ball, according to Leibniz's law, everything is identical to itself, a ball is identical to atoms because atoms make up a ball.
This is a complex topic, but in short, the atoms can’t be identical to the ball in your example, because there’s many atoms but just one ball. One thing can’t be identical to different things, or to many things. The closest we might get is that the *collection* of atoms, as a whole, is identical to the ball. But even that probably isn’t true, since the atoms that make up the ball change over time, yet the ball retains its identity. So it can’t at any time be identical to the collection of atoms that makes it up at that time.
I can't believe you brought "identity politics" in the definition of Logic Identity.
Important to distinguish different senses of 'identity', no?
thank you for these videos!!! - logic is making me want to drop out of my Philosophy degree 🥲💔
No problem! You might start to enjoy logic as you get more used to it, but even if you don't, you can probably drop it and enjoy the rest of philosophy - I hope you do!