Logic 3: Quantifiers (univ. & exist.), Proofs part 1 - Tutorial 3/4

[@Saturos02]

The unique existence quantifier ∃! may also be mentioned, useful for example in defining function properties f: X -> Y is surjective iff ∀ y ∈ Y ∃ x ∈ X s.t. f(x) = y f: X -> Y is bijective iff ∀ y ∈ Y ∃! x ∈ X s.t. f(x) = y Keep up the good work!

[@Bisqwit]

Very good! Thank you for that addition.

[@AT-zr9tv]

You are 20 years late for me 😂 but thank you for creating this invaluable ressource. I particularly appreciate how you strike a perfect balance between being super rigorous and silly. Your content is so enjoyable. I hope your studies are going well.

[@MrMastrsushi]

I appreciate the wide array of educational videos you provide

[@lorenzodomenicolaurito911]

10 minutes and more complete than an entire university semester.

[@ross817]

I have no idea why this was recommended to me but I learned a lot! thanks so much for making these videos

[@plantwateringcitizen8787]

Great material once again, Bisqwit. You are a talented teacher.

[@Bisqwit]

Thank you. Parts 3 and 4 were much more difficult to make than parts 1 and 2, but I tried my best within the available time. I just wish this is useful to somebody. I make material that I wish existed when I needed that knowledge.

[@jxq2855]

"For ∀ll" and "There ∃xists" are fantastic mnemonic devices to help newcomers learn these symbols - would've helped me learn them more quickly back in the day! 👍

[@xravenx24fe]

Excellent video, I've watched the first two videos multiple times so far and each time I retain a little bit more information so here's to hoping I can fully understand this jargon eventually.

[@NightpireVideos]

I really appreciate all the effort you put in your videos. It's a skill in itself to explain something difficult and complex in simple terms.

[@bitterlemonboy]

Wow. This is neat. It must be fun to learn all of this in depth at university!

[@cabalistik]

Very nice and educational videos, keep it up!

[@TheWandererTiles]

I like the part where the gap between logic and modelling the real world is bridged.

[@tigeruby]

very based 🌿👽🌿

[@homo_sapiens_sapiens]

😳😳😳

[@NonTwinBrothers]

Based... based on whut

[@mendelovitch]

@@NonTwinBrothers Base 16.

[@michaelwarnecke3474]

7:55 The right table is that of an equivalence, but should be that of an and gate, right?

[@Bisqwit]

I don’t see a truth table of equivalence at that point. It has M₁ ∧ M₂, which is an AND gate, and it is correctly filled: the green column has 1 only when both M₁ and M₂ have 1. Equivalence would also have 1 when both M₁ and M₂ have 0.

[@michaelwarnecke3474]

@@Bisqwit oh, I looked at M1 for some reason

[@TheMimoJimi]

Lovely

[@sabriath]

You can solve the last one as a binary. Animal is a binary, Cat is a binary, Mammal is a binary, Tiger is a binary; therefore (using * for AND): A*C = M (all animals that are cats are mammals) (A*T) -> C (all animals that are tigers is a cat) The truth table will show that these two expressions are equal, therefore (A*T)->C = M as well. Everything can be turned into logic.

[@Bisqwit]

This is but a simple example for the purpose of making an example palatable. It gets much more complex once you get to problems such as this: prove the following implication: ⊢∀x₀(A→B) → (∀x₀¬B → ∄x₀A), or, prove that ∀x₀(P₀(x₀)→∀x₀P₁(x₀)) ⊬ ∀x₀P₁(x₀), or prove that from the following: { ∀x₂P₀(x₂), ∃x₀(P₀(x₀)→P₁(x₁)) } we can conclude that ∃x₀(P₀(x₀)∧P₁(x₀)). (All three of these were actual exercises in the university class. In the next video, I show how to solve them.) Even then, you have made a false equivalence. Now, you assume that only cats can be mammals.

[@sabriath]

@@Bisqwit I can't read that jargon...ALL things can be placed into simple binary logic statements, and THAT I can read. I have not assumed that only cats can be mammals, the statement is always one directional when it comes to binary....otherwise you could reverse the CPU to find previous position states, you and I both know that's not how it works. So the Mammal may consist of more statements, such as Dog that is also an animal would be D*A + C*A = M...the list will forever grow; however, the identity of the statement put forth stands alone with no extra information, so it is in reduced form.

[@Bisqwit]

You seem to have made up your own rules of logic. But if jargon is the problem, let me formulate these examples as follows: ”Prove, that if for everyone it applies that if a car is white, then there is an engine, then also if for everyone it applies that there is not an engine, then there is nobody for whom we can say that a car is white.” ”Suppose that for everyone it applies that if they like kiwis, then every one of those likes pears. Prove, that it is impossible to conclude from these premises, that everyone likes pears.” “Suppose that everyone likes kiwis. Suppose that there exists someone such that if they like kiwis, then Milo likes pears. Prove, that from these premises we can conclude that there exists someone that likes both kiwis and pears.” Extra credit: Prove, that from the last two premises it is invalid to conclude, that everyone likes both kiwis and pears. Let me know when you have a solution for each ready!

[@Knechtspieler]

Interesting

[@MarquisDeSang]

Too much for me after a day of debugging and merging git branches.

[@Bisqwit]

Maybe another day then? It all bears down to what you can do lefthanded while concentrating on something else entirely…

[@alejandroalzatesanchez]

This is what AI does but AI does this with brute force lol

[@GarrettRose]

English is not super helpful for understanding logic it seems.