@@AtticPhilosophy There's one question , Deductive argument goes from general to specific and inductive argument from specific to general ... But will this always be the case?
Your lessons are so helpful! I'm doing a logic course in university at the moment and you're a huge help. The lessons are always so clearly laid out and very easy to understand. An underrated philosophy channel! Keep it up, thanks so much.
the logic course we've all been missing. one emphasis i added for myself was that entailments rely on the validation of All propositions involved. like "premises entail conclusions" is contingent not only on specific premises, but also specific conclusions, which somehow felt unobvious after discarding the function(argument) notation in favor of "p entails c". i'm sure this is clear for many, but i felt a need to highlight this. thanks for having this exceptional course going! ps. i also wondered how "truism" relates to "tautology, logical truth and valid sentence". i assume it's just a term for their application in rhetoric.
my use of the word "conclusions" remained contentious even above. they were meant to denote separately evaluated premises, not conclusions implied by the other premises as conditions. my point instead was just that they need to share a same axiology. the disambiguation at the end of the video is a sticky one.
I have the same issue with you as I have with Agatha Christy. After I have finished one of her books I'm lost wondering which book to read next. Take your series on logic. I never did the subject at school so now in the autumnal glow of my late eighties I decided to give it a go. You have made it interesting and I'm bound to benefit largely but could you please put the videos in some sort of chronological order, please. I'd appreciate a list of titles that I could methodically work through from the start to the final video. NB Agatha presents the same problem but she isn't with us right now.
19:20 Okay so you’re telling me that this channel is named after the fact that he gives lectures from an attic and not from the region in Greece? Didn’t know!
Instead of using truth table, I can use natural deduction or truth tree when I want to prove that some premises entail a conclusion. But in some books I have seen another construction: instead of using formulas, there are set of formulas, that represent with Greek letters Gamma, Delta and so on, and instead of a concrete formula (like A->B, A->(B/\C) etc.) they use small greek letters, that represent any formula. For instance, I want to prove this: Premise 1: Gamma entails ф ; Premise 2: Delta ф entails omega; Conclusion: Gamma Delta entails omega. This is cutting principle, as it is described in books. Here Gamma Delta are sets of formulas, ф and omega - formula. I think , I can't use here natural deduction or truth tree, because I don't know exactly what concrete formulas here. How to prove, when there is construction like this, when instead of formulas we use set of formulas? Maybe you have a video about this subject?
These are *meta rules*, concerning how certain deductive relationships follow from others. They're very useful in constructing more complex proofs. Cut, for example, effectively allows you to combine two proofs into one (it's a form of transitivity). To prove a meta-rule, there are two routes. One is to show it is sound on the semantics in question (eg truth tables). The other is to establish that it holds within the proof system itself (without going via the semantics). In natural deduction, CUT is easy to establish. Suppose you have a proof P1 of X |- A and a second proof P2 of Y, A |- B. Assume X, Y, write out P1, then (since you now have Y,A) P2, inferring B. You then have X, Y |- B, establishing CUT. Using proof trees, it's a bit less straightforward. Suppose you've proved X |- A and Y, A |- B, ie the X, ~A tree and the Y, A, ~B trees both close. Now start a tree with X, Y, ~B and consider what happens if you add Av~A. It branches, the A-branch closes (since Y, A, ~B branches close) and the ~A-branch closes (since X, ~A branches close). But given soundness, if it closes with Av~A, then it closes without Av~A, hence the X, Y, ~B tree closes. So X,Y |- B, establishing CUT for trees.
@@AtticPhilosophy Thank you. Yes, I understand the main idea. But in books they use turnstile with two horizontal lines, it's so called semantic entailment. Will it be difference in proofs?
i feel like i now have a better idea of what ⊧ is, but what about A⊧B vs A⊢B? (sorry for abusing of your time but may i add the difference between the previous and A⊩B to this question?)
⊧ is semantic entailment: (in every valuation/model) if A is true, then B is true. ⊢ is about proof: B can be proved/derived from A. These are standard symbols with fixed meanings. ⊩ is sometimes used to speak about a sentence being true relative to a possible world, state, situation or whatever: s ⊩ A means that A is true relative to (world/state/situation) s. (Some people also use ⊧ for this.)
This was from 3yrs ago but still saves grades. Thank you!
20:14 my respect for you has increased so much , You have cleared my confusion 😭
Thanks! Glad it helped.
@@AtticPhilosophy There's one question , Deductive argument goes from general to specific and inductive argument from specific to general ... But will this always be the case?
Your lessons are so helpful! I'm doing a logic course in university at the moment and you're a huge help. The lessons are always so clearly laid out and very easy to understand.
An underrated philosophy channel! Keep it up, thanks so much.
That's great to hear! I know how stressful logic courses can be at uni, so glad this has helped.
Hi! Your videos are so helpful. Keep up!
AMAZING LESSONS!!!! CONGRATULATIONS!
Thanks!
Thank you so much for this video, ive been so lost with entailment but now it makes sense!!!
Fantastic, glad it helped!
the logic course we've all been missing. one emphasis i added for myself was that entailments rely on the validation of All propositions involved. like "premises entail conclusions" is contingent not only on specific premises, but also specific conclusions, which somehow felt unobvious after discarding the function(argument) notation in favor of "p entails c". i'm sure this is clear for many, but i felt a need to highlight this. thanks for having this exceptional course going! ps. i also wondered how "truism" relates to "tautology, logical truth and valid sentence". i assume it's just a term for their application in rhetoric.
my use of the word "conclusions" remained contentious even above. they were meant to denote separately evaluated premises, not conclusions implied by the other premises as conditions. my point instead was just that they need to share a same axiology. the disambiguation at the end of the video is a sticky one.
Thanks! I think 'truism' means something commonly understood to be true, like 'politicians lie', rather than anything to do with logic.
finally I found a video that explains it perfectly. thank you for the video :)
Thanks!
Thanks , keep up the great work
Thanks! I’ll try.
Very well explained.
I have the same issue with you as I have with Agatha Christy. After I have finished one of her books I'm lost wondering which book to read next. Take your series on logic. I never did the subject at school so now in the autumnal glow of my late eighties I decided to give it a go. You have made it interesting and I'm bound to benefit largely but could you please put the videos in some sort of chronological order, please. I'd appreciate a list of titles that I could methodically work through from the start to the final video. NB Agatha presents the same problem but she isn't with us right now.
I'm looking to put some 'courses' together - guided playlists, eg, Logic 101, with videos following a (mostly) coherent progression. Bear with me!
So, valuations result in evaluations?
If a valuation in the primitives is T/F, it cascades into the evaluation side of the truth table?
That's the one! Much easy to understand. Why they make everything so confusing when it's barely even explained...
Thanks!
Great lesson. Thanks
Thanks!
thank you so much
19:20 Okay so you’re telling me that this channel is named after the fact that he gives lectures from an attic and not from the region in Greece?
Didn’t know!
It's a terrible pun, I know ...
Would an objection (counterexample) towards “A → B iff A ⊨ B” be something like: “If a large dog is a dog then red is a color”?
Thanks for the suggestion! 'Red is a colour' isn't a logical truth, so in this case, A->B isn't valid and A doesn't entail B.
Instead of using truth table, I can use natural deduction or truth tree when I want to prove that some premises entail a conclusion. But in some books I have seen another construction: instead of using formulas, there are set of formulas, that represent with Greek letters Gamma, Delta and so on, and instead of a concrete formula (like A->B, A->(B/\C) etc.) they use small greek letters, that represent any formula. For instance, I want to prove this: Premise 1: Gamma entails ф ; Premise 2: Delta ф entails omega; Conclusion: Gamma Delta entails omega. This is cutting principle, as it is described in books. Here Gamma Delta are sets of formulas, ф and omega - formula. I think , I can't use here natural deduction or truth tree, because I don't know exactly what concrete formulas here. How to prove, when there is construction like this, when instead of formulas we use set of formulas? Maybe you have a video about this subject?
These are *meta rules*, concerning how certain deductive relationships follow from others. They're very useful in constructing more complex proofs. Cut, for example, effectively allows you to combine two proofs into one (it's a form of transitivity). To prove a meta-rule, there are two routes. One is to show it is sound on the semantics in question (eg truth tables). The other is to establish that it holds within the proof system itself (without going via the semantics).
In natural deduction, CUT is easy to establish. Suppose you have a proof P1 of X |- A and a second proof P2 of Y, A |- B. Assume X, Y, write out P1, then (since you now have Y,A) P2, inferring B. You then have X, Y |- B, establishing CUT.
Using proof trees, it's a bit less straightforward. Suppose you've proved X |- A and Y, A |- B, ie the X, ~A tree and the Y, A, ~B trees both close. Now start a tree with X, Y, ~B and consider what happens if you add Av~A. It branches, the A-branch closes (since Y, A, ~B branches close) and the ~A-branch closes (since X, ~A branches close). But given soundness, if it closes with Av~A, then it closes without Av~A, hence the X, Y, ~B tree closes. So X,Y |- B, establishing CUT for trees.
@@AtticPhilosophy Thank you. Yes, I understand the main idea. But in books they use turnstile with two horizontal lines, it's so called semantic entailment. Will it be difference in proofs?
Thank you
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So "Entailment" to "=>" is what "Triple=" is to "".
Yes: A -> B is valid iff A entails B, and A B is valid iff A is equivalent to B.
you are awesome
Thanks, I try!
What is about A ⇔ B |= A ∨ B.
In this one value of truth of AB is true but it is false for A ∨ B.
So what is the result
That’s not a valid entailment, AB allows both A,B to be false, but then AvB isn’t true.
i feel like i now have a better idea of what ⊧ is, but what about A⊧B vs A⊢B?
(sorry for abusing of your time but may i add the difference between the previous and A⊩B to this question?)
⊧ is semantic entailment: (in every valuation/model) if A is true, then B is true. ⊢ is about proof: B can be proved/derived from A. These are standard symbols with fixed meanings. ⊩ is sometimes used to speak about a sentence being true relative to a possible world, state, situation or whatever: s ⊩ A means that A is true relative to (world/state/situation) s. (Some people also use ⊧ for this.)