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.
It is precisely simple systems like syllogistic and propositional logics that Gödel's Incompleteness Theorems do NOT apply to! In order for a logic system to be subject to Gödel's Incompleteness Theorems it must be sufficiently expressive and consistent (e.g. peano arithmetic). There's even some arithmetics which are complete.
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))?
It’s a strange formulation, but in fact, it’s just the bit of distribution that’s otherwise missing from the system. Using this axiom, you can prove (A&B)v(A&C) from A&(BvC). Can you work out how? You need distribution twice, plus &-elim, &-intro, and commutativity. 7 lines, I think.
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
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!
Think of it as saying: if ~B leads to a contradiction (A and ~A), then B is true. Written as a rule, it’s easier to understand: ~B -> ~A ~B -> A ________ B That’s one way to write the Reductio ad absurdum natural deduction rule, btw. So that’s basically what this axiom is doing.
@@AtticPhilosophy well ... Both change. If you drop the axiom, you lose information about → AND you lose characterization of A as the axiom implies. I know it doesn't look like that's the case, but the modern understanding about Logic and Math says that. To be more precise, A may be the same, as a proposition, but it is not the same considering its relations with the other propositions. So, in some sense, A changes too. Hehe.
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))
That’s a useful trick, especially for abbreviating things more complex than p->p. Just got to be careful: I is neither a sentence letter nor a meta variable!
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.
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..
@@AtticPhilosophy point is some are "obviously " valid (in the sense of being tautologies) while others are not. Im not quite clear on what you mean by "classically valid". If we are going to devise a test for "axiomatic validity" what should it be based on? Seems to me the axioms should be as few as possible and as simple as possible. I think most , if not all, axioms can be "validated" by truth tree construction but that adds a complicated and different layer to the axiomatic system.
@@pfroncole1 A list of axioms defines a logical system. They don't need to be obvious, but should be classically (truth-table) valid, and together define what counts as valid. In sub-classical systems, such as relevant logic, you drop some of the classically valid axioms to avoid things you don't want, eg the "irrelevant" p->(q->p).
I suppose p>(q>p) is "irrelevant " in the sense that it adds nothing to a system with p>p as an axiom. I've often wondered how the axioms are being selected apriori other than with the multi-valued truth table test for independence?
I suppose p>(q>p) is "irrelevant " in the sense that it adds nothing to a system with p>p as an axiom. I've often wondered how the axioms are being selected apriori other than with the multi-valued truth table test for independence?
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?
I suppose it depends on whether one deems it useful to make such axioms explicit . For example, the principle of non-contradiction was implicit all the way up through Plato. It wasn't useful to axiomatize it. Aristotle's formalization of the logical principle of non-contradiction was useful for him however when he decided to refute sophistical arguments that took advantage of ambiguities in grammar. Much in the same way Wittgenstein thought it was unnecessary to show how we use words until they are misused, in which case we require addressing their use explicitly in order to clear up the errors that come up from their misuse.
Most of the big 20th C analytic philosophers were also logicians - Frege, Russell, (early) Wittgenstein, Carnap, Quine, Kripke, Lewis. Much of their work relies on developments in logic. Later Wittgenstein is probably the exception here. You can definitely understand a lot of philosophy without much logic, but it's hard to do work in metaphysics, mind, language, especially without some basic logic.
Well, it definitely depends what you take under consideration. If you are doubtful about the entire formal logic project, I am very confused, as its use is quite obvious, showing that certain invalid arguments very similar to valid arguments are indeed invalid. Abstraction through formalization helps to capture each argument with the same form. On the other hand, if you are skeptical towards axiomatic proofs only, Mark said nicely at the begining of the video that natural deduction or truth trees are much easier for general use. The main use of axiomatic proofs was in philosophy of mathematics other than logic (a part of philosophy when I last checked). It is ok not to be interested in something, but it is not ok to say that something has no value just because you are not interested in it.
@@AtticPhilosophy Some basic logic, you're absolutely right: fundamentals of propositional, first order and modal logic. Nothing very fancy. But I find the textbooks of philosophical logic overly technical and little helpful to understand or elaborate real philosophical arguments. And I still don't get the point of proving that p --> p ! ^^
@@matepenava5888You're answering to a straw man. I don't say I'm "doubtful about the entire formal logic project". I just say I'm skeptical about the interest of those developments (at least most of them) for actual philosophy. It's not a pure intuition, but confirmed by what I know of current philosophical researches in various fields.
@@AtticPhilosophy I wasn't referring to the quantity of comments from philosophers. I meant that many of the comments from philosophers / people interested in Philosophy don't seem to have a great foundation in logic.
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.
It is precisely simple systems like syllogistic and propositional logics that Gödel's Incompleteness Theorems do NOT apply to! In order for a logic system to be subject to Gödel's Incompleteness Theorems it must be sufficiently expressive and consistent (e.g. peano arithmetic). There's even some arithmetics which are complete.
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))?
It’s a strange formulation, but in fact, it’s just the bit of distribution that’s otherwise missing from the system. Using this axiom, you can prove (A&B)v(A&C) from A&(BvC). Can you work out how? You need distribution twice, plus &-elim, &-intro, and commutativity. 7 lines, I think.
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
Modus Ponens isn't an axiom, it's a rule! And you need to use it to reason with category theory (or any other theory) - just try without.
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!
Think of it as saying: if ~B leads to a contradiction (A and ~A), then B is true. Written as a rule, it’s easier to understand:
~B -> ~A
~B -> A
________
B
That’s one way to write the Reductio ad absurdum natural deduction rule, btw. So that’s basically what this axiom is doing.
Just check all possible combinations for true values of "A" and "B" (there are 4) and you'll see that it's tautology. 🤷
I really appreciate your work especially your logic series
Thanks!
4:52 weakening is invalid 5:10 that longer one is valid, and it's not an axiom, you can compute it quite trivially
Depends on the logic but weakening is valid in most logics, invalid in relevant logics.
What do you mean by "weakening is invalid"?
4:40 I don't read A→(B→A) like that. I interpret it like characterizing A as everything that implies A.
Ah, I was thinking of it as saying something about ->, not about A. So if you drop this axiom, what changes: A or -> ?
@@AtticPhilosophy well ... Both change. If you drop the axiom, you lose information about → AND you lose characterization of A as the axiom implies. I know it doesn't look like that's the case, but the modern understanding about Logic and Math says that. To be more precise, A may be the same, as a proposition, but it is not the same considering its relations with the other propositions. So, in some sense, A changes too. Hehe.
Please make some videos on Speech act theory 🙏🏻
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))
That’s a useful trick, especially for abbreviating things more complex than p->p. Just got to be careful: I is neither a sentence letter nor a meta variable!
Great video
Thanks!
What do you think about Schrödinger logics?
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.
I easily understood everything
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..
Of course - you don’t want axioms that aren’t classically valid!
@@AtticPhilosophy point is some are "obviously " valid (in the sense of being tautologies) while others are not. Im not quite clear on what you mean by "classically valid". If we are going to devise a test for "axiomatic validity" what should it be based on? Seems to me the axioms should be as few as possible and as simple as possible. I think most , if not all, axioms can be "validated" by truth tree construction but that adds a complicated and different layer to the axiomatic system.
@@pfroncole1 A list of axioms defines a logical system. They don't need to be obvious, but should be classically (truth-table) valid, and together define what counts as valid. In sub-classical systems, such as relevant logic, you drop some of the classically valid axioms to avoid things you don't want, eg the "irrelevant" p->(q->p).
I suppose p>(q>p) is "irrelevant " in the sense that it adds nothing to a system with p>p as an axiom. I've often wondered how the axioms are being selected apriori other than with the multi-valued truth table test for independence?
I suppose p>(q>p) is "irrelevant " in the sense that it adds nothing to a system with p>p as an axiom. I've often wondered how the axioms are being selected apriori other than with the multi-valued truth table test for independence?
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?
I suppose it depends on whether one deems it useful to make such axioms explicit . For example, the principle of non-contradiction was implicit all the way up through Plato. It wasn't useful to axiomatize it. Aristotle's formalization of the logical principle of non-contradiction was useful for him however when he decided to refute sophistical arguments that took advantage of ambiguities in grammar. Much in the same way Wittgenstein thought it was unnecessary to show how we use words until they are misused, in which case we require addressing their use explicitly in order to clear up the errors that come up from their misuse.
Most of the big 20th C analytic philosophers were also logicians - Frege, Russell, (early) Wittgenstein, Carnap, Quine, Kripke, Lewis. Much of their work relies on developments in logic. Later Wittgenstein is probably the exception here. You can definitely understand a lot of philosophy without much logic, but it's hard to do work in metaphysics, mind, language, especially without some basic logic.
Well, it definitely depends what you take under consideration. If you are doubtful about the entire formal logic project, I am very confused, as its use is quite obvious, showing that certain invalid arguments very similar to valid arguments are indeed invalid. Abstraction through formalization helps to capture each argument with the same form. On the other hand, if you are skeptical towards axiomatic proofs only, Mark said nicely at the begining of the video that natural deduction or truth trees are much easier for general use. The main use of axiomatic proofs was in philosophy of mathematics other than logic (a part of philosophy when I last checked). It is ok not to be interested in something, but it is not ok to say that something has no value just because you are not interested in it.
@@AtticPhilosophy Some basic logic, you're absolutely right: fundamentals of propositional, first order and modal logic. Nothing very fancy. But I find the textbooks of philosophical logic overly technical and little helpful to understand or elaborate real philosophical arguments. And I still don't get the point of proving that p --> p ! ^^
@@matepenava5888You're answering to a straw man. I don't say I'm "doubtful about the entire formal logic project". I just say I'm skeptical about the interest of those developments (at least most of them) for actual philosophy. It's not a pure intuition, but confirmed by what I know of current philosophical researches in various fields.
#RM3 conjunction is left adjoint to implication --- but "truth" is replaced with "validity" --- The Liar is valid, it is both true and false.
🎉
I'm a Math Degree student, and reading the comments I realise Philosophers actually have a very poor foundation in logic... 😅
Who’s to say the comments are from philosophers?
@@AtticPhilosophy I wasn't referring to the quantity of comments from philosophers.
I meant that many of the comments from philosophers / people interested in Philosophy don't seem to have a great foundation in logic.