3-Valued Logic | Non-Classical Logic | Attic Philosophy
ฝัง
- เผยแพร่เมื่อ 24 ก.ค. 2024
- In classical logic, sentences are always either true or false (but never both). That rule breaks down in non-classical logic, and things get more interesting! In this video, we take a deeper look into three important 3-valued logics: Strong Kleene logic, Lukasiewitz logic, and the Logic of Paradox. This is a follow-up to my intro to non-classical logic:
True, False, Other • True, False, Other | N...
00:00 - Intro
01:10 - Recap
01:38 - 3 logics
01:55 - Validity
02:17 - Strong Kleene logic
04:11 - Lukasiewitz logic
05:44 - Logic of Paradox
06:40 - Validity in LP
08:38 - Entailment
11:50 - Philosophical Uses
More videos on non-classical logic coming soon! If there’s a topic you’d like to see covered, leave me a comment below.
Links:
My academic philosophy page: markjago.net
My book What Truth Is: bit.ly/JagoTruth
Most of my publications are available freely here: philpapers.org/s/Mark%20Jago
Get in touch on Social media!
Instagram: / atticphilosophy
Twitter: / philosophyattic
#logic #philosophy #semantics
Thanks for this great content.
My pleasure! Hope it was helpful.
i am a philosophy major that just graduated and i love logic. that is my focus and i plan on getting my master and phd in logic. but i will say that i wish i was sponge and could just memorize what i learn. its going to take my a while to wrap my brain around all of this.
Don’t worry, it takes time for everyone! A good idea is to immerse yourself: go to talks, watch lectures online, read different things, & soon the concepts start to fit together much more easily.
Where are you applying? University of Amsterdam has the best logic graduate program in the world. University of Barcelona and Ludwig Maximilian University of Munich also have an extremely good MA in Logic. LMU especially if you didn't take metatheory in undergrad.
@@IndustrialMilitia i have no idea. Ive been looking anywhere and everywhere. Ill take a look at those universities!
@@brandonpropterhoc If you're American, Carnegie Mellon University has an MSc in Logic. It's funded too. I think that's the only North American program. All others are unfunded and in Europe.
Thank you so much man :)
You’re welcome!
nice video, i've been enjoying your channel. what do you mean with entailment and validity?
Thanks! Entailment means: necessarily, if the premises are true, the conclusion is true too. Validity means a sentence that is logically guaranteed to be true.
Strong Kleene's "Other" feels a lot like a null pointer in programming.
Maybe, I'm not sure! Doesn't 'NULL' evaluate to 1 in many languages?
@@AtticPhilosophy I guess I mean it just reminds me of the saying "Garbage in, garbage out" since "Other" seems to bubble-up to the top.
damn he really in the attic
It’s true!
Thanks again for this series. What a strange subject. In what way does any of this actually relate to reality I wonder? 🤔
It’s mostly about how we reason about reality. Some practical applications of many-valued logic include working with databases (which have gaps & may be inconsistent), or modelling answers to questions from different people.
It certainly relates to politics 😂
Thanks for the great video! I am curious about a few things:
1) If LP and FOL have the same set of valid sentences, then ⊨ (p ∧ ¬p) → q in LP because it is the case in FOL as well. But isn't a characteristic that this principle does not hold?
2) Am I right that the deduction theorem does not hold in LP? I think about it this way: Let's assume it does. Since LP and FOL have the same set of valid sentences and by the deduction theorem, you can transform every valid inference into a valid formula (and vice versa), the same inferences must hold. But this is not the case. So the deduction theorem does not hold.
3) Your videos on intuitionistic logic are so damn good that we're going to watch them together in my study group! We're still looking for a intermediate-friendly introduction - can you recommend one?
Keep up the good work!
Yes, (p ∧ ¬p) → q is always either T or O, so valid in LP. But the corresponding entailment, from (p ∧ ¬p) to q, doesn't follow, because Modus Ponens isn't valid in LP. Modus Ponens is equivalent to Disjunctive Syllogism. Where p is both true and false, you have a counter-example to pvq, ~p |- q. And yes, the deduction theorem fails for the same reason. There are *other* conditionals available in paraconsistent logics, which do support modus ponens, but they aren't truth-functional.
@@AtticPhilosophy These are really great videos, Mark! I thought it would be worth clarifying here that there are, indeed, truth-functional implications that support modus ponens _and_ for which the deduction theorem holds in its usual formulation. One could conservatively extend LP, for instance, by adding a 3-valued implication such that v(A→B)=T if v(A)=F, and v(A→B)=v(B) otherwise (see [D. Batens @ Logique & Analyse, 1980]). Note that the negation-free fragment of the corresponding logic shares the validities of the corresponding fragment of Classical Logic. It is true that A→B would no longer be definable as ¬A∨B, but, well, there seems to be no obvious reason why such a pattern should be taken as a golden standard in manufacturing an implication for K3 or for LP; in fact, this pattern would not even work, say, for defining the expected implication in Intuitionistic Logic!
@@quod-libet Thanks Joao! That seems a sensible approach to (truth-functional) implication. It might be a bit strange to allow A->A to be Both, but I suppose we could say that in proving the sentence, we guarantee its truth, but can't thereby rule out its also being false?
@@AtticPhilosophyI guess that what is really crucial is that →(0,F)=F (that's the cell that was modified, right?). This eliminates the countermodel LP produced for modus ponens, when the pattern ¬A∨B was (inadvertently?) considered for implication, and agrees with the intuition for the falsity of classical implication, according to which an implication is false (i.e., it is non-designated) iff the antecedent fails to be false (i.e., it is designated) while the consequent happens to be false (i.e., it is non-designated).
@@quod-libet could you provide a specific paper on this different paraconsistent logic? All I could find is Batens’s book.
Yeah, Set Theory gets a stroke seeing that (3-valued) use of ¬.
Thank you very much for your awesome videos. I notice in this video that (A -> A) is valid in the Tukasiewitz Logic, but (A V not A) is not. However, ( A -> A) is equivalent to (A V not A) according to definition of implication. So, we have Validity is equivalent to no Validity. It seems to me like a contradiction in the Tukasiewitz Logic or am I missing something?
In many non-classical logics like these, A->B isn’t equivalent to ~AvB. Rather, -> is defined by its truth matrix, not by some combination of other connectives. If you define A->B as ~AvB in 3-vale’s logic, you get the K3 matrix for ->, rather than the L3 matrix. And then, as you say, A->A wouldn’t be valid.
@@AtticPhilosophy Thank you😊
I dont understand all of this yet but it seems as if when people say he did this to me so its ok when i do it to him, because he is evil. You cant be just good by oposing evil, you can be evil and opose evil or you cant just be unbiased and get to the truth you have to be biased towards truth to see it. When you are unbiased you arent seeig at all. Im not sure if thrse concepts apply in these cases but i get the vibe it does i feel like 1+1=2. In going to be dissapointed i think
I don’t think that’s a good example of 3-valued logic. ‘Neither good nor evil’ is fine with classical 2-values logic, since evil doesn’t mean the same as not good. Things can be neither.
@@AtticPhilosophy why not?
@@petarpejic1468 Because, like I said, 'neither good nor evil' doesn't require more than a 2-valued logic.
You need to work on pedagogy. What is it like to not know what you know? Work from there.
Wow! This is totally useless!
wow, good luck bro
Doubt I can really convince you otherwise since it's been a year and your comment has a very set-in-stone attitude, but a quick motivation I can give for trivalent logical systems is the evaluation of statements about the future. If I say "tomorrow I will go to the theatre," can you definitively say that the statement is either true or false? If so, that has some weird implications about future events, like they already have a pre-determined truth value, which obviously has some philosophical baggage. So we probably (unless we're like, strict determinists about the future) want to evaluate that as neither true nor false.
@@animore8626 Do trivalent logical systems create anything interesting? Like is there any field in mathematics with interesting theorems or results that is founded in a trivalent logical system?