Really, really enjoy this channel and your way of expounding knowledge. I’m a graduate student pursuing an MS in Risk Analysis in the US, but have always been extremely fond of mathematics and philosophy…and low and behold Set Theory is the bridge that connects them both! … Keep up the word class videos, lord knows we, the viewers, need and appreciate them!! 🤙🏻🤙🏻
This is a phenomenal RUclips channel! I'm an aspiring mathematician, as well as an undergrad in both pure and applied mathematics, and so many people ask me where they can go to get a fundamental understanding of proofs and logic to improve their mathematical abilities. Namely, in areas like topology(an understanding of set theory is, of course, crucial in topology), real analysis, etc. Now I see why every pretty much mathematician claims that mathematics is just applied philosophy! I was wondering if you had any recommendations for books I could use to study and learn logic from a more philosophical perspective rather than just my typical mathematical perspective. Thanks for the great content!
Thanks so much! There’s plenty of good intro logic textbooks written by philosophers, which cover a bit of the philosophy understanding of logic, eg: - Logic by Greg Restall - Logic by Nick Smith Then, getting more philosophical, - logic for philosophers by Ted Sider And then even more philosophical: - philosophy of logics by Susan Haack Good luck!
Thanks! This one covers relations in terms of sets (ordered pairs): ruclips.net/video/GzUXbyiNIfU/видео.html Plus more coming on set theory & logic in the next few weeks
Some people use the terms interchangeably (or with 'formal logic', as opposed to informal logic). But I would say there's a spectrum from the less to the more mathematical bits of logic. The mathematical side is, e.g., semantics, model theory, proof theory, algebras. Stuff like translations form English to quantified logic, or using truth tables, is further from the mathematical side.
Great video :)....I just got curious thinking about membership: Is membersip (or y is an element of y) introduced as primitive or can it be defined in some way ?
It's primitive. In fact, it's the *only* primitive of set theory: you add membership to first-order logic & that's the concepts you need for set theory. (if you also have things that aren't sets in the domain, you also need a predicate 'is a set'. Or you could take the empty set symbol as a primitive.) An alternative is to take inclusion as primitive and define x's membership in y as inclusion of {x} in y. But that seems more clunky.
Standard set theory is exact, ie everything is either in or not in a set. Fuzzy set theory exists, built on fuzzy logic, but it’s more complex to explain.
@@AtticPhilosophy I am the heir to Lotfi Zadeh. Set theory corresponds to Human Associative Memory (HAM) which is fuzzy. I hope you can join me in my continuing to think through the significance of fuzzy logic.
Hi Professor Jago, I was wondering if we could distinguish two empty sets by distinguishing their complements relative to their respective domains. That is, if the empty set isn't understood simply as a set with _no_ members, but as a set with no members _from a specified domain_, then couldn't each empty set have a distinct property and thus be distinguished from one another? So, suppose one domain is all the constituents of possible world W1, and another is all the constituents of possible world W2, and suppose no constituent of W1 is in W2, and vice versa. Then the empty set relative to W1 will be distinguished from the empty set relative to W2 by its distinct complement. So my first question is, is this sort of move permissible? And my second question is, to prevent this sort of move, must we conceptualize the empty set as empty relative to all possible objects or some totalizing domain (or something like that)?
Nice idea! it's true that, for complement to be defined, we first need to fix the universe. Then sets are defined as subsets of that universe. So there would be no meaningful notion of a set independently of the universe, which is what I think you'd need.
Hi Attic! idk if you take questions in the comments but can you explain the answer to A ^ B |- A -> B I tried assuming A then trying to get to B but ik that A is already true and so is B so assuming something I knew to be true felt weird.
You did it right: assume A, infer B (from A^B), conclude A->B. often with ND, it's best just to apply the rules and not worry too much about what they mean! (But if you think about it: why would it be a problem to assume what you already know? It can't allow you to infer something false since, by assumption, it's true!)
Lmao this channel is carrying me through first year. I love the videos!
Great - hope you're having a really good 1st year.
your best video yet
Thanks! I’m still figuring out what works.
Really, really enjoy this channel and your way of expounding knowledge. I’m a graduate student pursuing an MS in Risk Analysis in the US, but have always been extremely fond of mathematics and philosophy…and low and behold Set Theory is the bridge that connects them both! … Keep up the word class videos, lord knows we, the viewers, need and appreciate them!! 🤙🏻🤙🏻
Thanks very much for the kind words!
All great videos a difficult subject like Philosophy broken down into subdivisions easily explained
Glad you think so!
This is a phenomenal RUclips channel! I'm an aspiring mathematician, as well as an undergrad in both pure and applied mathematics, and so many people ask me where they can go to get a fundamental understanding of proofs and logic to improve their mathematical abilities. Namely, in areas like topology(an understanding of set theory is, of course, crucial in topology), real analysis, etc. Now I see why every pretty much mathematician claims that mathematics is just applied philosophy! I was wondering if you had any recommendations for books I could use to study and learn logic from a more philosophical perspective rather than just my typical mathematical perspective. Thanks for the great content!
Thanks so much! There’s plenty of good intro logic textbooks written by philosophers, which cover a bit of the philosophy understanding of logic, eg:
- Logic by Greg Restall
- Logic by Nick Smith
Then, getting more philosophical,
- logic for philosophers by Ted Sider
And then even more philosophical:
- philosophy of logics by Susan Haack
Good luck!
Thanks!
You’re welcome!
Very helpful video!👍 Could you cover relation and functions in terms of sets?
Thanks! This one covers relations in terms of sets (ordered pairs): ruclips.net/video/GzUXbyiNIfU/видео.html
Plus more coming on set theory & logic in the next few weeks
Can you give lectures on mathematical induction pls?... And explain how it can be used to prove some theorems in logic.
Have a look here: How to use Mathematical Induction ruclips.net/video/9w20J4j5-0Y/видео.html
Dr. Jago, what makes mathematical logic different from plain old logic?
Some people use the terms interchangeably (or with 'formal logic', as opposed to informal logic). But I would say there's a spectrum from the less to the more mathematical bits of logic. The mathematical side is, e.g., semantics, model theory, proof theory, algebras. Stuff like translations form English to quantified logic, or using truth tables, is further from the mathematical side.
Great video :)....I just got curious thinking about membership: Is membersip (or y is an element of y) introduced as primitive or can it be defined in some way ?
It's primitive. In fact, it's the *only* primitive of set theory: you add membership to first-order logic & that's the concepts you need for set theory. (if you also have things that aren't sets in the domain, you also need a predicate 'is a set'. Or you could take the empty set symbol as a primitive.) An alternative is to take inclusion as primitive and define x's membership in y as inclusion of {x} in y. But that seems more clunky.
Is the set theory fuzzy or exact?
Standard set theory is exact, ie everything is either in or not in a set. Fuzzy set theory exists, built on fuzzy logic, but it’s more complex to explain.
@@AtticPhilosophy I am the heir to Lotfi Zadeh. Set theory corresponds to Human Associative Memory (HAM) which is fuzzy. I hope you can join me in my continuing to think through the significance of fuzzy logic.
Hi Professor Jago, I was wondering if we could distinguish two empty sets by distinguishing their complements relative to their respective domains. That is, if the empty set isn't understood simply as a set with _no_ members, but as a set with no members _from a specified domain_, then couldn't each empty set have a distinct property and thus be distinguished from one another? So, suppose one domain is all the constituents of possible world W1, and another is all the constituents of possible world W2, and suppose no constituent of W1 is in W2, and vice versa. Then the empty set relative to W1 will be distinguished from the empty set relative to W2 by its distinct complement. So my first question is, is this sort of move permissible? And my second question is, to prevent this sort of move, must we conceptualize the empty set as empty relative to all possible objects or some totalizing domain (or something like that)?
Nice idea! it's true that, for complement to be defined, we first need to fix the universe. Then sets are defined as subsets of that universe. So there would be no meaningful notion of a set independently of the universe, which is what I think you'd need.
Hi Attic! idk if you take questions in the comments but can you explain the answer to A ^ B |- A -> B
I tried assuming A then trying to get to B but ik that A is already true and so is B so assuming something I knew to be true felt weird.
You did it right: assume A, infer B (from A^B), conclude A->B. often with ND, it's best just to apply the rules and not worry too much about what they mean! (But if you think about it: why would it be a problem to assume what you already know? It can't allow you to infer something false since, by assumption, it's true!)
I haven't studied much mathematics so will I have trouble taking logic.Im ready work hard.Somebody please reply.