Axioms in logic

00:00 - Intro
01:57 - Why use axioms?
03:56 - Typical Axioms
06:42 - Axioms vs schemes
08:03 - Axiomatic systems
08:35 - Proofs
09:18 - Different axiomatic systems
10:39 - Example: the identity axiom
15:04 - Working out axiom instances
15:34 - Example: explosion
19:25 - Difficult case: permutation
20:33 - Proof recipes
21:25 - The Deduction Theorem

If there’s a topic you’d like to see covered, leave me a comment below.

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#logic

[jonc]

Thank you. Also, I liked the brief mention of the Logical Positivists at the start, although given your area is logic, might be good to also explore the limits of axiomatic, indeed the limits of all sentence statements as a result of Gödel & the Incompleteness Theorem in some future presentations.

[luyombojonathan]

I really appreciate your work especially your logic series

[ASMRChess]

I have a question regarding this great video. Can you help me understand this a bit better?

(¬B → ¬ A) → ((¬B → A) → B)

I’ve tried substituting in some sentences for the variables, but it didn’t make sense so I must have messed up.

This is what I tried

B = I have a chessboard
A = I can play chess


From that I seem to get: If it is the case, that if (“if I don’t have a chessboard, then I can’t play chess”) → (“if I don’t have a chessboard then I can play chess”) → (“I can play chess.”)

If it was

(¬B → ¬ A) → ((¬B ∧ A) → B)

Then I think I could get it to make sense.

I know this is a lot to ask, but I would appreciate it very much if you could point out what I have misunderstood or misinterpreted.

Thank you for a great video!

[tomholroyd]

3:20 There's like a dozen "axioms" for RM3, and one of them is "mingle", where the M comes from, which they added because the mathematical structure they accidentally created already had that property. It's really much simpler than that.
Modus Ponens is a theorem you can PROVE using Category Theory

[ayodhyakinkarkabi]

Please make some videos on Speech act theory 🙏🏻

[willemesterhuyse]

At timestamp 3:12: doesn't the "distribution" axiom give an inconsistent system (it is not the same as the usual distribution: A AND (B OR C) ->- (A AND B) OR (A AND C))?

[patrickwithee]

Since this is about axioms, I’m going to leave the axioms for a logic that I call “Weak Negation Logic”. This is for anyone to see and use.

(A→B)→(C→(A→B))

(A→(B→C))→((A→B)→(A→C))

((A→B)→C)→(¬C→¬B)

¬(A→A)→B

(¬(A→B)→B)→(A→B)

(A∧B)→A

(A∧B)→B

(A→B)→((A→C)→(A→(B∧C)))

A→(A∨B)

B→(A∨B)

(A→C)→((B→C)→((A∨B) →C))

(A∧(B∨C))→((A∧B)∨(A∧C))

From A and (A→B), infer B.

P.S., you can distinguish between positive literals and meta-variables, in which case you can change the first axiom to

A→(B→A) where A is a positive literal or A=(C→D).

I call this modified version “Non-Intuitionistic Logic”.

These are both basically propositional S4, but with some modifications.

[ShaunLovesMaths]

What do you think about Schrödinger logics?

[samueldeandrade]

4:40 I don't read A→(B→A) like that. I interpret it like characterizing A as everything that implies A.

[tomholroyd]

4:52 weakening is invalid 5:10 that longer one is valid, and it's not an axiom, you can compute it quite trivially

[pfroncole]

Every axiom can easily be shown to be a tautology by truth tree or RAA. As always, sentences with 100 arrows can be more easily handled by applications of the Deduction Theorem, that is the "meanings" become far clearer when the forms are reduced to their Deduction Theorem equivalent sentences..

[samueldeandrade]

Why people don't use another letter for p→p? For example, P. Or p². Or I, for Identity. This way you avoid making mistakes, I guess.

The proof becomes:

p→(I→p)
(p→(I→p))→((p→I)→I)
p→(I→p)
(p→I)→I
p→I
I

Also, there is a missing parentheses here: 13:27 . Hahaha. It should be

... ((p→(p→p))→(p→p))

[tomholroyd]

#RM3 conjunction is left adjoint to implication --- but "truth" is replaced with "validity" --- The Liar is valid, it is both true and false.

[philociraptor]

I'm a philosophy teacher, did my master thesis on Wittgenstein, and I must say I never got the point of mathematical logic for philosophy (let alone the proof of p ---> p !). None of the great philosophical contributions I've read so far (including in contemporary analytical philosophy) ever needed heavy logical tools. I'm talking about classical philosophical topics: metaphysics, ethics, philosophy of action, philosophy of religion, etc. Should I feel, some day, that I miss all these technical tools, I'll invest some time in it, but until then it looks like a waste of time to me. Could maybe someone give a few examples of significant contributions in some classical fields of philosophical that wouldn't have been possible without axiomatic logic?

[GabriTell]

I'm a Math Degree student, and reading the comments I realise Philosophers actually have a very poor foundation in logic... 😅