True, False, Other | Non-Classical Logic | Attic Philosophy
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- เผยแพร่เมื่อ 9 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 look at logics with 3 truth-values: True, False, and some other value, which might be Neither true nor false, for even Both true and false. We'll look at how to understand the logical connectives. In a follow-up video, we'll look at the some different 3-valued logics.
00:00 - Intro
01:28 - The Basic Idea
02:16 - Relational semantics
03:19 - Three-Valued Semantics
04:24 - Truth-Functional Semantics
04:53 - Negation
05:33 - Truth Matrices
06:19 - Conjunction
08:10 - Disjunction
09:04 - Semantic clauses
12:11 - Fuzzy Logic
13:46 - Implication
16:19 - Lukasiewitz matrix
More videos on non-classical logic coming next! 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
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I‘m so glad you made these videos, they‘re really fun to watch 😊
Thanks for the video explaining this! I have a group meeting tomorrow to discuss non-classical logic and I had never even heard of such a thing until a week ago. Now I'm (a little) more prepared!
No problem, glad it helped!
Hey there, I'm studying Philosophy in Germany (still undergraduate tho) and I just wanted to tell you how much I love your logic videos. I've had a seminar on non classical logics this semester (we're reading "from if to is") and am using your videos to help me study for my oral exam next week.
So, once again, love your content and thanks for making these very informative vids.
Thanks! That’s a great book to learn from.
thanks, mark
your videos are always helpful,
greetings from Colombia
Thanks!
Is there a typo in the formula for the truth value of the conditional in L3? It says (VA - VB) if B < A. But if B = 0 and A = 1 then we get 1-0 = 1 which makes A --> B = 1?
Good spot! It should be 1-(VA-VB). So the smaller the gap between A and B, the more true A->B will be.
this is awesome
is it just me, or is that basically just classic logic and Rule of inference with variables and algebra ?
its really still classical logic.
we are just dealing with undetermined values, but none of those assumptions, will imply that the "REAL" or "ultimate" value is outside of being either true of false.
because it is clearly what we are trying to achieve in all of those procedures, we just want to find the actual value of all those none determined values or for more accurate name variables.
Hi Mark,
I came across your TH-cam channel. I’m a student in my final year of university studying Philosophy (love it!).
Do you have any books you’d recommend regarding Theory of Truth and aesthetics?
Thank you, sir!!
Thanks! Here's some book suggestions:
Truth:
Doubt Truth be a Liar by Graham Priest
Truth in virtue of meaning, by Gillian Russell
Truth by Paul Horwich
What Truth is by me
Aesthetics:
Languages of Art, by Nelson Goodman
After the End of Art or by Arthur Danto
Is there an intuitive way of choosing infinitely many truth values in fuzzy logic? I want to learn something more about the fuzzy logic what do you suggest to me?
You are mentioning something called infinite hierarchy of truth values on the "true and false?Dialethism" video.Maybe this can be a candidate for creating a fuzzy logic around.
Option_2=Both true and (both true and false) sentence
Option_3=Both true and [both (both true and false) and false] sentence?
//////////////////////////////////
Or maybe i would assign Options like this
V(T->O1)=O2
V(T->O2)=O3... Something similar to lambda calculus or Haskell's kind
(I don't know just thinking loudly)
The truth-values of fuzzy logic are all the real numbers, 0 to 1, with 0 being fully false and 1 fully true. The nearer a statement gets to 1, the ‘more true’ it is.
Hi Mark, is possible to use truth matrices in place of truth tables in all cases or are they limited to only showing values for one connective at a time? I can't seem to find any info on them anywhere...
Also, will you ever be doing a video specifically on truth matrices? Thanks!
That’s a great question. Matrices are a good way of displaying the semantics of a connective, which would otherwise need 9 truth table lines. I also find a matrix much easier to remember than a 9-line table, because of the patterns they form. But truth tables can be extended with new (sub)sentences, one per column, whereas matrices can’t. You need a new matrix for each subsentence. So if you want to calculate the value of a complex sentence under all possible valuations, a truth table is usually the best way. You’re right - I should probably add a new video on tables & matrices!
You might look up karnaugh maps. It's been a bit since I studied digital logic (which is where I learned about them) so I don't remember the details, but the truth matrices look really similar to them.
@@Lucky10279 Funnily enough my comp sci course covered karnaugh maps and that parallel struck me as well, very useful stuff
It would be nice if you gave some concrete examples in reality as you were teaching because it would help with understanding. One of my philosophy instructors told me that if I didn't use examples then, "No one will think you know what you are talking about."
Yes you should always use examples in your philosophy essays! This video was about explaining the technical side of 3-valued logics. There's a video with examples and applications here: th-cam.com/video/fZLg1r1zapg/w-d-xo.html
fuzzy logic sounds alot like probability theory where the value of p(~A) = 1-p(A).
Let me rephrase the question 😅
At 5:00, if Other stood for Both, then ~Other should stand for Neither, and vice versa. So we cannot say that Other = ~Other.
And at 11:05 Both and Neither would have the same Truth Value, 1/2.
In the above cases the tables show correctly that Both/Neither belong to Other, but they cannot specife which is which! So Łukasiewicz's theory is wrong imo.
Negation takes Neither to Neither and Both to Both. The reason is, if A is T and F, then ~A is both F and T. So, however the 3rd value O is interpreted, ~O = O.
Ok, the O functions well as an "unknown truth value" in the tables. Except in ¬ case, there things lose sense.
The way I look at propositions with a value that isn't true yet isn't false is to look at what it means to be true or false.
If it *is* true then it's *equivalent* to true, i.e. A is true is just shorthand for AT. Similarly, A is false is just shorthand for AF.
Since some of these implications are redundant, you can just use one-way implications: T->A and A->F.
If you accept the rules of classical logic (but not the idea that all propositions have a truth value), you'll notice that even if neither implication is valid, if you or them together you'll find that T -> (T->A) v (A->F), or in other words, (T->A) v (A->F) is true.
I don't like 3-valued logic. Let's say you have some proposition A with value O. ~A should also have a value O just for the sake of symmetry. Now, it can be seen that A&~A has value O&O, which is O, and not F as A&~A should be. If you want to preserve that particular rule, you have to sacrifice A&A = A. Surely you should be able to &intro and &elim from one side to the other there.
4-valued logic is fine, I think, (and you don't have to think of the extra two states as "neither" and "both". You can instead stick with arbitrary truth values X and Y) but I don't see why one should assert that "every proposition is equivalent to one of these four model propositions" when one doesn't have to.
~T = F
~X = Y
~Y = X
~F = T
(regular & and v rules apply:)
X&Y = F
X&X = X
Y&Y = Y
XvY = T
XvX = X
YvY = Y
implication rules:
A->T = T
F->A = T
A->F = ~A
T->A = A
A->A = T
~A->A = A
One problem here is you have F -> F equivalent to T, so ~F equivalent to T, ie any A is true iff not false. But that’s exactly what many-valued approaches deny, eg if you have a happy system, sentences can be neither T nor F.
@@AtticPhilosophy It is _true_ that (A is true) iff (A is not false) with classical rules. All I'm saying is that from there, it doesn't then follow that either (true entails A) or (A entails false) are valid entailments.
Let's take the four valued logic I gave:
All classical rules apply so it can be seen that
(T->A) ~(A->F)
for all truth values of A. (If you wish, you could check both ways' evaluations on the values X and Y yourself.)
Despite being equivalent to each others' negation, T->A and A->F are both invalid whenever A has value X or Y even though we designed our logic so the rules of classical logic apply.
edit: I'll leave proving that the rules hold in this 4 value system as an exercise to the reader. However, I assure you that I've confirmed it myself.
@@AtticPhilosophy I think a more instructive statement would be (AT) v (AF). Within this 4-valued logic, this can be seen to hold in spite of both (AT) and (AF) being invalid for some values of A. This seems to defy the law of excluded middle since "the disjunction of two invalid statements is valid", however that's not what the law of the excluded middle says. The law of the excluded middle only says "you can always derive that Av~A is valid". The only sacrifice from "normal" classical logic here is that among A and ~A, they may be both invalid.
I also rub my chin like that whenever in deep thought 15:53
Haha! It’s a bad habit, hard to break!