I don't get it, isn't the mistake made in the very first part? 1:07 The argument is that C = C -> P This implies C = ( C -> P ) -> P And recursively C = ( ( C -> P ) -> P ) -> P In general: C = ( ( ... { ( ( C -> P ) -> P ) -> P } n times But isn't it the case that because it is a self containing sentence that answer is a derivative? Therefor: C" = C -> P Because in order for C" to be true, both C and P are required to be true. My apologies if my terminology is faulty and/or imprecise, I haven't studied philosophy but I'm really interested in it and I'm trying to understand this paradox.
It’s not a mistake, but a weird feature. What the reasoning needs is a sentence C equivalent to true(C)->P. It’s guaranteed to exist in strong enough systems. Then as you say, C is also equivalent to (C->P) -> P etc, assuming equivalence of any P with true(P). But it’s not right that C->P requires both C,P to be true, only that P is true if C is.
I wasn't familiar with the paradox (or had forgotten reading about it if so), but as soon as you said "can be used to prove whatever you like" I immediately knew it was going to involve self-reference. My own approach (no idea if its proper) would be observing that the only conclusion you can come away with is what you started with. Since it is self-contained, as it were, it guarantees that it will never be possible for you to establish an argument where C is NOT assumed to be true which can prove C. Something like "This sentence has five words" can be proven without assuming its truth, as can every other valid statement. As a consequence, I think you can never reduce the statement beyond "If C, then P". Reducing it to P is misformed, because the argument presented only implies P when C is true, and on and on.
If the only conclusion is what you start with, does that mean that the only valid inference is: A entails A? But then logic is completely useless! On the other hand, if you can get conclusions which aren’t premises, then you might get paradoxes like this. Most logicians try to avoid the paradoxes by thinking about which logical rules they can keep (and which to reject) without the dodgy consequences.
Hi Mark, How about Yablo Paradox in which the self-referentiality is construed as a (possibly) infinite hierarchy of liar sentences? And also there is the Yablurry Paradox, arguably the hardest of them all, constructed by Roy Cook as the union of Yablo and Curry Paradox. It would be very interesting if you explore these challenges in your next videos. I would like to know whether intensional logics can give a satisfactory explanation of these paradoxes (in which context of statement and context of evaluation is not necessarily identical like in classical logic or LP). Cheers
Good call! The Yablo-style paradoxes are basically pointing out that you can get the problem without self-reference. A good solution to the liar should also work for Yablo’s. (I don’t think banning self-reference is a good solution!) And similarly, a good solution for the Curry should also work for the Yablo-Curry version.
I thought we cannot use C -> P in a deductive proof, unless C -> P is true. I guess I am missing something here, but why can't we say that we shouldn't include C -> P in a deductive proof because it is false?
We can make assumptions to see what follows. If we assume A and show that B follows, we’ve proved A->B. In the Curry, we assume C, show P follows, so we’ve proved C->P. But that’s identical to C, so P follows: bad news!
Thank you very much for explaining!! I'm currently working with your articles concerning truthmaker maximalism and negative facts. What an amazing approach in a interesting debate! I would love to see a video summarizing the challenges of truthmaker theory. (Just in case you consider video suggestions) Thank you very much for your channel. It explains a lot to me and I enjoy your videos.
Good question! I’d suggest starting with ‘Social Philosophy’ and maybe some videos from ‘Truth’ playlists, and when you’re on the course, have a look at ‘How to Write an Essay’. And if you’re taking a logic class, have a look at the first few videos of the “what is logic?” playlist. Good luck on your course!
How about semantic theory of truth? Or even simpler - I dont see any reason why shouldnt we ban self reference in the object language as a rule. Then we could allow speaking about language only in its metalanguage. I'm not sure if this solves anything, but i think this is the first condition any considerable solution Has to meet
In artificial logical languages, you can ban self-reference, but at a cost: you can't then do basic arithmetic or have a proper object-language concept of truth. In natural language, self-reference is there whether we like it or not!
@@AtticPhilosophy Can't we really do arithmetic this way? I'm reading a book on the topic and I'm not sure so I have to double check it, but some axiomatizations of arithmetic are fine. I don't know why object language truth would be considered proper when it raises paradoxes. And yes in natural language we have self-reference, but it depends on the definition of natural language. If we consider anything that isn't fromal language a natural language then some of natural languages wont have self-reference. In othe words some natural languages like scientific theories could be semantically open. edit; I opened the book again, and Jan Woleński clearly says that we can make a semantic definition of truth (Tarski's theory) for arithmetic. Based on the fact that such a definition requires the language to be semantically open, because we dont want it to have paradoxical sentences. Now it turns out self-reference is allowed to a certain extent, but we can only have semantical terms in language L that refer to language L. I'm new to the topic so thanks for the occasion to learn
I'm stupid. Can someone help me? Where is the problem? The consequence of this sentence does only follow if it assume to be true. But if we just assume to be false?
No, not just if you assume it’s true. The problem is you can prove the sentence, ie you can prove that it’s true, so the bad consequences follow whatever you assume (or even if you assume nothing).
@@AtticPhilosophy Okay, I think I slowly get it. We prove what we first assume. But this self-referential move from: "we have C" to "C is nothing else than: "C-->P" and then we use the first C to infer via modus ponens that P feels weird (no crtic; just weird for me; I'm not very good at logic). But thank you for your response.
@@hegelsmonster5521 @hegelsmonster5521 yes, and it's not an official move in standard natural deduction systems. A more formal way to set this up is with a sentence C which is provably equivalent to: if C then [whatever]. For then, having proved the former, you can infer C and hence [whatever] just using standard proof rules.
My consistent thought so far throughout this series is that there seems to be a conflation of the concept of truth and logical truth. We map one onto the other, but that does not mean they are the same.
Logical truth = truth in all models (of the relevant kind). Paradoxes of truth like the liar/Curry paradoxes turn on truth (simplicitier), not logical truth.
@@AtticPhilosophy I’m going to have to disagree there. In the concept creation phase, any sort of “meaning” that truth is going to have will need to be built upon a preexisting logical structure, else there is no means by which to build the concept at all. The moment we attempt to create a three-valued logical model that maintains this mapping, we call into question the means by which the concepts were built, and a failure to address this more empirical aspect detracts from the meaningfulness of the system. Instead of actually addressing the problems with these paradoxes, it seems as though it only “cleans up”, so to speak, so we can feel more comfortable with ourselves. I don’t mean to suggest that there aren’t valid concepts of truth that are compatible with a three-value system, but seems to me this is a necessary part of the discussion. Another point to be made is that by disjoining concepts of truth with logic is that it becomes possible to create new systems of logic that concepts of truth might not cleanly map to. To raise a three-valued example, in which the values become “red, “green”, and “blue” (arbitrary labels), one can organize the values such that logical transformations such as “not”, conjunctives, and disjunctives simply don’t exist in the same way they do in classical systems, as the matrices we create are not merely three-valued, but 3-dimensional. Instead of inversion, you could have translation. Instead of “or” and “and”, which are functionally identical to “truth” and “false” presence checkers, you have checks for a value being 2/3 present, and so on. Such a system is distinctly different than the “true or false or other” systems, as it frees us from the constraints of thinking about logic in a particular epistemological context. Of course, the utility of such a system can still be questioned - how could it possibly be applied to a proposition? But in my mind, that would be a worthwhile investigation.
@@ZMattStudio In formal logic, a sentence is logically true (=valid) iff it's true on all interpretations (=true in all models). So "logical truth" is a derived concept. That's just a fact about how modern mathematical logic works.
Okay, it seems as if I need to use some different terms then, though I’m not sure what. By “logical truth”, contrasting to a concept of truth, I mean the role we apply the label of “truth” to in a logical system. In Boolean logic, true and false are identical except that they are not each other, and can be swapped (so long as operators are appropriately inverted as well) while preserving the system. We could use any other label - on and off, left and right, pink and orange, it makes no difference, because they are just signs for abstractions of possible informational states. In this sense, marking one or the other for preservation (validity) is an arbitrary choice. We care about truth preservation as opposed to falseness preservation only because of our concept of truth, that is, we want to apply the system so we can make arguments about the natural world, about ethics, etc, which we perceive to have a quality of being “true”. And that applies also to any proposition we have, eg, the mentioned paradoxes. So any attempt to address these paradoxes by altering the underlying system without also addressing how we think about the concepts in the first place strikes me as incomplete. As for modern mathematical logic, could you clarify? Are you referring to a particular foundational theory such as ZF set theory or group theory? As I understand them, both rely on axioms which would require some prior logical basis to even define.
IF there exists a proposition C such that C C->P then you can work by cases: If C is false, C->P must be false as it is equivalent to false. However, looking at C->P, it's clear that it's true as falsehood implies anything. This leaves us with a contradiction so we can disregard this case. If C is true, C->P must be true as it is equivalent to true. Looking at C->P, the fact that C is true indicates that P is true. Therefore P is true. This is actually perfectly valid logic. The only issue was the very first word. Is there really a proposition C that is equivalent to its implication of P? Well we've just done all the work to show that, by contradiction, no, whenever P is false. If P is true then any true statement will do. If P isn't true or false (i.e. there is no implication T->P or P->F) then maybe the existence of such a statement C is similarly in a middle state.
Some may take issue with the usage of working by cases. If that's you, I want to talk to you. I want to talk to someone who uses intuitionistic logic about why they reject excluded middle but accept ex falso quodlibet (or, more specifically, that for any proposition P, P∧¬P -> F, where F is the statement that all propositions are true)
That's right, from C (C -> P), P follows. Part of the problem is that, in FOL with a truth predicate + basic arithmetic, you can show (using diagonalization) that there is guaranteed to be such a sentence! Clearly something is wrong, so the question is, which bit of FOL, truth, or basic arithmetic should we chuck out?
@@AtticPhilosophy Let G(X) be the godel number of X. Diagonalization only states that "For all sentences A(x), there is a sentence C such that C A(G(C))". To turn this into the form we want, we need there to be a predicate on numbers A that can be written down using base symbols such that A(G(X)) X->P. Now we're back to the question: how would we even dream of writing down such an A? Does one exist? Edit: Such a predicate clearly exists. If you have a number n, either it's the godel number of a statement or it isn't. If n = G(X) then A(n) is equivalent to X->P. But I make a clear distinction here. In order to use diagonalization, you need to be able to write down A using base symbols. Sometimes you can't do this kind of thing. A halting problem type argument shows that sometimes this kind of thing is impossible: Assume there's a sentence predicate H(x,y) such that H(G(X), n) X(n) for any sentence predicate X. Let R(n) = ¬H(n,n). What is R(G(R))? This contradiction implies there is no sentence that represents H. If there was, you should be able to get a godel number for R. Edit2: Actually, the "halting predicate" can be created from A if you choose P = F. ¬A(n) is a universal truthseer i.e. it tells you whether or not n is the godel number of a true statement. From here H(x,y) is just "disassemble x, replace the free variable with y, reassemble, then plug that into A". Therefore there is no sentence for A if you choose a P that is equivalent to false.
@@AtticPhilosophy Sorry for the contentless re-reply. I just wanted to tell you I edited my last reply to say that I chuck out (or at least challenge) that there really is guaranteed to be such a "sentence".
If what I'm saying is true, then anything. But what I'm saying doesn't have to be true. It doesn't have to be false, either. We need to teach our children that "I don't know" is a valid truth value.
“I don’t know” isn’t in contrast to true or false: something might be true (or false) even though we don’t know which. But sure, it’s good to say “I don’t know” when you don’t know!
@@AtticPhilosophy Sorry! Meant to say 1:40 i.e. shortly before you mentioned our good old mate modus ponens for the first time. Gosh, how did I manage to tag the very end of the video? Doh! P.S. the original point was intended to be that when you first mentioned modus ponens I knew this was my kind of channel.
2:11 this is inifnite rekrusion C->(C->(C->......(C->P))) you never proof it insdie is inifnitie end by the way C and C->P i is in loop definiable its some nonsens and question its calculable in finite time by turing machine ?
I think it might be interesting to investigate the iterated version of curry's paradox: C_{n} : C_{n+1} -> P; Then we would have C_0: C_1 -> P ((C_2 -> P) -> P) ((((C_4 -> P) -> P) -> P) -> P) ((((((((C_8 -> P) -> P) -> P) -> P)-> P) -> P) -> P) -> P) It seems like the thing about repeating a lie until it becomes true
Interesting idea, but not convinced there are sentences like that. Each sentence has to be finitely long, but specifying that Cn includes Cn+1 as its antecedent makes sentences infinitely long.
@@AtticPhilosophy It'd have to be an omega order logic (omega being the first infinity ordinal). But you can stop at some arbitrary place in the sequence and explore other ways of representing it. Then by induction generalize those equivalent forms. Perhaps it can be represented in a product of sums (and of ors) way and then it'd be possible to study the behavior of this as a sequence of propositions. My conjecture is that this sequence alternates between true and false forever so that's why it doesn't have a definite value at infinity. Similar to how the sequence 0,1,0,1,... doesn't have a last value
OMG! Given a true sentence as the consequent, the Curry statement is true. Given a false sentence as the consequent, the Curry statement is the Liar statement! Also, suck it dialetheism.
Yeah, dialethism doesn't do so well with the Curry sentence - I think that's a strong argument against dialethism, since intuitively, Liar and Curry should have the same solution.
You always need proof rules. You can either use axioms + rules or just (more) rules. Here, I:m using natural deduction proof rules. Axioms (+ modus ponens) are equivalent. What's useful using just rules is that we can see which ones are responsible for the problem. Here, just the rules for 'if then' and 'true'.
@@GODemon13 That's a common initial reply to the liar - "it's meaningless", but it's very hard to substantiate the claim. It's meaningful words put together perfectly grammatically. But the paradox remains irrespective of the meaningfulness of the English sentence, since in strong enough systems there provably exists a sentence provably equivalent to its own negation.
Just don't allow sets to contain themselves, and say no to infinite recursion. Stop with all this tail chasing and begging the question. The only set that should contain itself is the null or empty set, that's how we get countable numbers, every other instance is just a shadow of that concept.
I don't get it, isn't the mistake made in the very first part? 1:07
The argument is that
C = C -> P
This implies
C = ( C -> P ) -> P
And recursively
C = ( ( C -> P ) -> P ) -> P
In general:
C = ( ( ... { ( ( C -> P ) -> P ) -> P } n times
But isn't it the case that because it is a self containing sentence that answer is a derivative? Therefor:
C" = C -> P
Because in order for C" to be true, both C and P are required to be true.
My apologies if my terminology is faulty and/or imprecise, I haven't studied philosophy but I'm really interested in it and I'm trying to understand this paradox.
It’s not a mistake, but a weird feature. What the reasoning needs is a sentence C equivalent to true(C)->P. It’s guaranteed to exist in strong enough systems. Then as you say, C is also equivalent to (C->P) -> P etc, assuming equivalence of any P with true(P). But it’s not right that C->P requires both C,P to be true, only that P is true if C is.
I wasn't familiar with the paradox (or had forgotten reading about it if so), but as soon as you said "can be used to prove whatever you like" I immediately knew it was going to involve self-reference. My own approach (no idea if its proper) would be observing that the only conclusion you can come away with is what you started with. Since it is self-contained, as it were, it guarantees that it will never be possible for you to establish an argument where C is NOT assumed to be true which can prove C. Something like "This sentence has five words" can be proven without assuming its truth, as can every other valid statement. As a consequence, I think you can never reduce the statement beyond "If C, then P". Reducing it to P is misformed, because the argument presented only implies P when C is true, and on and on.
If the only conclusion is what you start with, does that mean that the only valid inference is: A entails A? But then logic is completely useless! On the other hand, if you can get conclusions which aren’t premises, then you might get paradoxes like this. Most logicians try to avoid the paradoxes by thinking about which logical rules they can keep (and which to reject) without the dodgy consequences.
Hi Mark,
How about Yablo Paradox in which the self-referentiality is construed as a (possibly) infinite hierarchy of liar sentences? And also there is the Yablurry Paradox, arguably the hardest of them all, constructed by Roy Cook as the union of Yablo and Curry Paradox. It would be very interesting if you explore these challenges in your next videos. I would like to know whether intensional logics can give a satisfactory explanation of these paradoxes (in which context of statement and context of evaluation is not necessarily identical like in classical logic or LP). Cheers
Good call! The Yablo-style paradoxes are basically pointing out that you can get the problem without self-reference. A good solution to the liar should also work for Yablo’s. (I don’t think banning self-reference is a good solution!) And similarly, a good solution for the Curry should also work for the Yablo-Curry version.
Great content! Procurei por todo o youtube, por várias linguas, e este vídeo com certeza é o mais bem produzido e didático! Congrats mate :)
Thanks! Much appreciated.
I thought we cannot use C -> P in a deductive proof, unless C -> P is true. I guess I am missing something here, but why can't we say that we shouldn't include C -> P in a deductive proof because it is false?
We can make assumptions to see what follows. If we assume A and show that B follows, we’ve proved A->B. In the Curry, we assume C, show P follows, so we’ve proved C->P. But that’s identical to C, so P follows: bad news!
Thank you very much for explaining!!
I'm currently working with your articles concerning truthmaker maximalism and negative facts. What an amazing approach in a interesting debate! I would love to see a video summarizing the challenges of truthmaker theory. (Just in case you consider video suggestions)
Thank you very much for your channel. It explains a lot to me and I enjoy your videos.
Thanks very much! And thanks for the suggestion - always open to ideas for video topics!
Hello sir,
I am about to take philosophy as my UG course. I just found your channel. Can you suggest from which video or playlist shall I start?
Good question! I’d suggest starting with ‘Social Philosophy’ and maybe some videos from ‘Truth’ playlists, and when you’re on the course, have a look at ‘How to Write an Essay’. And if you’re taking a logic class, have a look at the first few videos of the “what is logic?” playlist. Good luck on your course!
@@AtticPhilosophy Thank you sir. Looking forward to your content.:-)
How about semantic theory of truth? Or even simpler - I dont see any reason why shouldnt we ban self reference in the object language as a rule. Then we could allow speaking about language only in its metalanguage. I'm not sure if this solves anything, but i think this is the first condition any considerable solution Has to meet
In artificial logical languages, you can ban self-reference, but at a cost: you can't then do basic arithmetic or have a proper object-language concept of truth. In natural language, self-reference is there whether we like it or not!
@@AtticPhilosophy Can't we really do arithmetic this way? I'm reading a book on the topic and I'm not sure so I have to double check it, but some axiomatizations of arithmetic are fine. I don't know why object language truth would be considered proper when it raises paradoxes. And yes in natural language we have self-reference, but it depends on the definition of natural language. If we consider anything that isn't fromal language a natural language then some of natural languages wont have self-reference. In othe words some natural languages like scientific theories could be semantically open.
edit; I opened the book again, and Jan Woleński clearly says that we can make a semantic definition of truth (Tarski's theory) for arithmetic. Based on the fact that such a definition requires the language to be semantically open, because we dont want it to have paradoxical sentences. Now it turns out self-reference is allowed to a certain extent, but we can only have semantical terms in language L that refer to language L.
I'm new to the topic so thanks for the occasion to learn
I'm stupid. Can someone help me? Where is the problem? The consequence of this sentence does only follow if it assume to be true. But if we just assume to be false?
No, not just if you assume it’s true. The problem is you can prove the sentence, ie you can prove that it’s true, so the bad consequences follow whatever you assume (or even if you assume nothing).
@@AtticPhilosophy Okay, I think I slowly get it. We prove what we first assume. But this self-referential move from: "we have C" to "C is nothing else than: "C-->P" and then we use the first C to infer via modus ponens that P feels weird (no crtic; just weird for me; I'm not very good at logic). But thank you for your response.
@@hegelsmonster5521 @hegelsmonster5521 yes, and it's not an official move in standard natural deduction systems. A more formal way to set this up is with a sentence C which is provably equivalent to: if C then [whatever]. For then, having proved the former, you can infer C and hence [whatever] just using standard proof rules.
My consistent thought so far throughout this series is that there seems to be a conflation of the concept of truth and logical truth. We map one onto the other, but that does not mean they are the same.
Logical truth = truth in all models (of the relevant kind). Paradoxes of truth like the liar/Curry paradoxes turn on truth (simplicitier), not logical truth.
@@AtticPhilosophy I’m going to have to disagree there. In the concept creation phase, any sort of “meaning” that truth is going to have will need to be built upon a preexisting logical structure, else there is no means by which to build the concept at all. The moment we attempt to create a three-valued logical model that maintains this mapping, we call into question the means by which the concepts were built, and a failure to address this more empirical aspect detracts from the meaningfulness of the system. Instead of actually addressing the problems with these paradoxes, it seems as though it only “cleans up”, so to speak, so we can feel more comfortable with ourselves. I don’t mean to suggest that there aren’t valid concepts of truth that are compatible with a three-value system, but seems to me this is a necessary part of the discussion.
Another point to be made is that by disjoining concepts of truth with logic is that it becomes possible to create new systems of logic that concepts of truth might not cleanly map to. To raise a three-valued example, in which the values become “red, “green”, and “blue” (arbitrary labels), one can organize the values such that logical transformations such as “not”, conjunctives, and disjunctives simply don’t exist in the same way they do in classical systems, as the matrices we create are not merely three-valued, but 3-dimensional. Instead of inversion, you could have translation. Instead of “or” and “and”, which are functionally identical to “truth” and “false” presence checkers, you have checks for a value being 2/3 present, and so on. Such a system is distinctly different than the “true or false or other” systems, as it frees us from the constraints of thinking about logic in a particular epistemological context. Of course, the utility of such a system can still be questioned - how could it possibly be applied to a proposition? But in my mind, that would be a worthwhile investigation.
@@ZMattStudio In formal logic, a sentence is logically true (=valid) iff it's true on all interpretations (=true in all models). So "logical truth" is a derived concept. That's just a fact about how modern mathematical logic works.
Okay, it seems as if I need to use some different terms then, though I’m not sure what. By “logical truth”, contrasting to a concept of truth, I mean the role we apply the label of “truth” to in a logical system. In Boolean logic, true and false are identical except that they are not each other, and can be swapped (so long as operators are appropriately inverted as well) while preserving the system. We could use any other label - on and off, left and right, pink and orange, it makes no difference, because they are just signs for abstractions of possible informational states. In this sense, marking one or the other for preservation (validity) is an arbitrary choice. We care about truth preservation as opposed to falseness preservation only because of our concept of truth, that is, we want to apply the system so we can make arguments about the natural world, about ethics, etc, which we perceive to have a quality of being “true”. And that applies also to any proposition we have, eg, the mentioned paradoxes. So any attempt to address these paradoxes by altering the underlying system without also addressing how we think about the concepts in the first place strikes me as incomplete.
As for modern mathematical logic, could you clarify? Are you referring to a particular foundational theory such as ZF set theory or group theory? As I understand them, both rely on axioms which would require some prior logical basis to even define.
IF there exists a proposition C such that C C->P then you can work by cases:
If C is false, C->P must be false as it is equivalent to false. However, looking at C->P, it's clear that it's true as falsehood implies anything. This leaves us with a contradiction so we can disregard this case.
If C is true, C->P must be true as it is equivalent to true. Looking at C->P, the fact that C is true indicates that P is true.
Therefore P is true.
This is actually perfectly valid logic. The only issue was the very first word. Is there really a proposition C that is equivalent to its implication of P? Well we've just done all the work to show that, by contradiction, no, whenever P is false. If P is true then any true statement will do. If P isn't true or false (i.e. there is no implication T->P or P->F) then maybe the existence of such a statement C is similarly in a middle state.
Some may take issue with the usage of working by cases. If that's you, I want to talk to you. I want to talk to someone who uses intuitionistic logic about why they reject excluded middle but accept ex falso quodlibet (or, more specifically, that for any proposition P, P∧¬P -> F, where F is the statement that all propositions are true)
That's right, from C (C -> P), P follows. Part of the problem is that, in FOL with a truth predicate + basic arithmetic, you can show (using diagonalization) that there is guaranteed to be such a sentence! Clearly something is wrong, so the question is, which bit of FOL, truth, or basic arithmetic should we chuck out?
@@AtticPhilosophy Let G(X) be the godel number of X.
Diagonalization only states that "For all sentences A(x), there is a sentence C such that C A(G(C))". To turn this into the form we want, we need there to be a predicate on numbers A that can be written down using base symbols such that A(G(X)) X->P. Now we're back to the question: how would we even dream of writing down such an A? Does one exist?
Edit: Such a predicate clearly exists. If you have a number n, either it's the godel number of a statement or it isn't. If n = G(X) then A(n) is equivalent to X->P. But I make a clear distinction here. In order to use diagonalization, you need to be able to write down A using base symbols. Sometimes you can't do this kind of thing. A halting problem type argument shows that sometimes this kind of thing is impossible: Assume there's a sentence predicate H(x,y) such that H(G(X), n) X(n) for any sentence predicate X. Let R(n) = ¬H(n,n). What is R(G(R))? This contradiction implies there is no sentence that represents H. If there was, you should be able to get a godel number for R.
Edit2: Actually, the "halting predicate" can be created from A if you choose P = F. ¬A(n) is a universal truthseer i.e. it tells you whether or not n is the godel number of a true statement. From here H(x,y) is just "disassemble x, replace the free variable with y, reassemble, then plug that into A". Therefore there is no sentence for A if you choose a P that is equivalent to false.
@@AtticPhilosophy Sorry for the contentless re-reply. I just wanted to tell you I edited my last reply to say that I chuck out (or at least challenge) that there really is guaranteed to be such a "sentence".
If what I'm saying is true, then anything. But what I'm saying doesn't have to be true. It doesn't have to be false, either. We need to teach our children that "I don't know" is a valid truth value.
“I don’t know” isn’t in contrast to true or false: something might be true (or false) even though we don’t know which. But sure, it’s good to say “I don’t know” when you don’t know!
6:20 You say Modus Ponens, I say Subscribed!
(and belled.)
Haha, Thanks!
@@AtticPhilosophy Sorry! Meant to say 1:40 i.e. shortly before you mentioned our good old mate modus ponens for the first time.
Gosh, how did I manage to tag the very end of the video? Doh!
P.S. the original point was intended to be that when you first mentioned modus ponens I knew this was my kind of channel.
@@mattbox87 👍
2:11 this is inifnite rekrusion C->(C->(C->......(C->P))) you never proof it insdie is inifnitie end by the way C and C->P i is in loop definiable its some nonsens and question its calculable in finite time by turing machine ?
No, you can prove it, the proof is in the video.
I think it might be interesting to investigate the iterated version of curry's paradox:
C_{n} : C_{n+1} -> P; Then we would have C_0: C_1 -> P ((C_2 -> P) -> P) ((((C_4 -> P) -> P) -> P) -> P) ((((((((C_8 -> P) -> P) -> P) -> P)-> P) -> P) -> P) -> P)
It seems like the thing about repeating a lie until it becomes true
Interesting idea, but not convinced there are sentences like that. Each sentence has to be finitely long, but specifying that Cn includes Cn+1 as its antecedent makes sentences infinitely long.
@@AtticPhilosophy It'd have to be an omega order logic (omega being the first infinity ordinal). But you can stop at some arbitrary place in the sequence and explore other ways of representing it. Then by induction generalize those equivalent forms. Perhaps it can be represented in a product of sums (and of ors) way and then it'd be possible to study the behavior of this as a sequence of propositions. My conjecture is that this sequence alternates between true and false forever so that's why it doesn't have a definite value at infinity. Similar to how the sequence 0,1,0,1,... doesn't have a last value
OMG! Given a true sentence as the consequent, the Curry statement is true. Given a false sentence as the consequent, the Curry statement is the Liar statement!
Also, suck it dialetheism.
Yeah, dialethism doesn't do so well with the Curry sentence - I think that's a strong argument against dialethism, since intuitively, Liar and Curry should have the same solution.
Dont you need some kind of axiom to make a proof? I seriously dont understand the attempt to prove anything without axioms
You always need proof rules. You can either use axioms + rules or just (more) rules. Here, I:m using natural deduction proof rules. Axioms (+ modus ponens) are equivalent. What's useful using just rules is that we can see which ones are responsible for the problem. Here, just the rules for 'if then' and 'true'.
Like the Liar's Paradox, it is not a logical statement and therefore should not be treated as such.
What's not logical about it? You can use logic to prove the sentence exists, and to "prove" it!
Just like the liars paradox, it is a nonsensical statement. It's not hard to see that.@@AtticPhilosophy
@@GODemon13 That's a common initial reply to the liar - "it's meaningless", but it's very hard to substantiate the claim. It's meaningful words put together perfectly grammatically. But the paradox remains irrespective of the meaningfulness of the English sentence, since in strong enough systems there provably exists a sentence provably equivalent to its own negation.
It's certainly paradoxical, which is exactly why it is to be disregarded.@@AtticPhilosophy
Just don't allow sets to contain themselves, and say no to infinite recursion. Stop with all this tail chasing and begging the question.
The only set that should contain itself is the null or empty set, that's how we get countable numbers, every other instance is just a shadow of that concept.
Nothing here about sets or recursion, just plain logic!
Subscribed then unsubscribed!
....that is the truth
If this sentence is true, then this sentence is false.
That’s equivalent to the original liar sentence!