Excellent course on intuitionist logic 💯 It’s like having a personal tutor who communicates the vital aspects with clarity and vitality. I couldn’t grasp the formalised language of logic without this channel 👏
Intuitionistic logic is similar to classical logic in this respect: propositional and predicate versions are sound and complete (e.g. for the Kripke semantics) but intuitionistic number theory is incomplete. In some ways, however, intuitionists are less bothered by this: if some truths aren’t provable, so what?
@@AtticPhilosophy I think the Intuitionist would, at least implicitly, assert that Godel's Theorem (number 6 in the original paper) demonstrates that some statements about Natural numbers have a truth value which is neither True nor False (perhaps analogous to that of the Epimenides sentence), and that we can either exclude these rendering Number Theory incomplete, or include them and have a non (i.e., para) consistent Number Theory.
As far as I understand it, that's a technique with a very specific application in modelling human expert decision making. It isn't (as far as I know) a general approach to logic. Fuzzy sets or fuzzy logic is often good for representing our not-quite-certain attitudes towards things. But logic is often concerned with the facts themselves, and what they entail, rather than just what we believe about them.
@@AtticPhilosophy epistemologically, how else do you know you know anything if not through your belief that we know things? It takes away absolutism from the equation, which drives polarization nowadays.
@@marcomongalo3328 Most of the time, when we're reasoning or evaluating arguments, we're interested in what's true, not just in what people believe. Some people think truth comes in degrees, and if it does, fuzzy logic looks a good way to reason. For reasoning about people's degrees of belief, people often use Bayesian reasoning (using the probability calculus).
Love the content I’m finding it very exciting. What are your thought on what might the consequence of Intuitionistic logic be for Friston’s free energy principle?
Thanks! Consequences for free energy principle: probably none. Intuitionistic logic behaves just like classical logic for decidable cases and, I'm guessing, facts of biology are (at least in principle) decidable.
Tbh they don’t feature prominently in much philosophical logic. Type theory sometimes in passing as Russell’s solution to set-theoretic paradoxes, and sometimes in more technical material on truth. Category theory doesn’t show up much in philosophy.
@@AtticPhilosophy ok but when do you teach higher order logics . Is it too hard and maybe obscure. I can’t find many/any books that aren’t expensive and look like they are for phd students
It’s big is CS departments, where constructive reasoning is the boss, but less so in mathematics departments, where classical logic is usually the default. In philosophy departments, it’s usually there as non-technical background, when thinking about the past, or verificationist theories, etc.
Structuralism might be what you're after: the idea that numbers (& other mathematical entities) really exist, but have no intrinsic nature. They're just positions in a structure: 1 after 0, 2 after 1, etc. Beyond that, there's nothing to the ontology of the numbers.
I was reading a book and the author denies the validity of "the law of excluded middle". So that makes him an anti realist? I don't understand the jargon can you explain in layman's terms?
Anti-realism is one reason to deny Excluded Middle, but not the only reason. So the author may or may not be an anti-realist. That means, very roughly, that thinking reality is to a large extent created or determined by our thought or language.
I don't get the connection between natural numbers being grounded in 'the way we think' and the rejection of the excluded middle. Just because some numerical formula is beyond our current capacities to satisfy constructively, that doesn't mean that it is not either true or false, or that there is not some form in which could ultimately be asserted and decided on. Ability is not necessarily determined by a single, limited state of affairs, as anyone who has taken a test will know. Seems to me to be somewhat flimsy grounds for chucking away one of the strongest pillars of rationality that we have.
It’s something like this: to prove a numerical statement, for intuitionists, you need to construct the number, and to disprove it, you need to show there’s no possible construction. Validity (or truth) is then equated with proof. So, for cases where no construction or proof-of-no-construction is possible, you don’t have pv~p. Not everyone agrees of course!
@@AtticPhilosophy I think maybe there is a place for intuitionism as a logic of empiricism - which might be why it maps onto system states quite nicely. Good for CompSci, perhaps, but I can’t really see it as a foundation for metaphysics.
You definitely deserve more viewers! I think Philosophy is so interesting.
Thanks! It really is interesting.
Excellent course on intuitionist logic 💯 It’s like having a personal tutor who communicates the vital aspects with clarity and vitality. I couldn’t grasp the formalised language of logic without this channel 👏
Very informative!! Thank you so much
Thanks!
What is the intuitionist's attitude regarding Godel's theorems? I assume they reject them, based on the video?
Intuitionistic logic is similar to classical logic in this respect: propositional and predicate versions are sound and complete (e.g. for the Kripke semantics) but intuitionistic number theory is incomplete. In some ways, however, intuitionists are less bothered by this: if some truths aren’t provable, so what?
@@AtticPhilosophy I think the Intuitionist would, at least implicitly, assert that Godel's Theorem (number 6 in the original paper) demonstrates that some statements about Natural numbers have a truth value which is neither True nor False (perhaps analogous to that of the Epimenides sentence), and that we can either exclude these rendering Number Theory incomplete, or include them and have a non (i.e., para) consistent Number Theory.
What about hesitant intuitionistic fuzzy logic? I think this is the ultimate representation of our consciousness
As far as I understand it, that's a technique with a very specific application in modelling human expert decision making. It isn't (as far as I know) a general approach to logic. Fuzzy sets or fuzzy logic is often good for representing our not-quite-certain attitudes towards things. But logic is often concerned with the facts themselves, and what they entail, rather than just what we believe about them.
@@AtticPhilosophy epistemologically, how else do you know you know anything if not through your belief that we know things? It takes away absolutism from the equation, which drives polarization nowadays.
@@marcomongalo3328 Most of the time, when we're reasoning or evaluating arguments, we're interested in what's true, not just in what people believe. Some people think truth comes in degrees, and if it does, fuzzy logic looks a good way to reason. For reasoning about people's degrees of belief, people often use Bayesian reasoning (using the probability calculus).
@@AtticPhilosophy Where do doxastic or epistemic modal logics come in?
Love the content I’m finding it very exciting. What are your thought on what might the consequence of Intuitionistic logic be for Friston’s free energy principle?
Thanks! Consequences for free energy principle: probably none. Intuitionistic logic behaves just like classical logic for decidable cases and, I'm guessing, facts of biology are (at least in principle) decidable.
Where does type theory and category theory fit in for the modern philosophical but not mathematical logician?
Tbh they don’t feature prominently in much philosophical logic. Type theory sometimes in passing as Russell’s solution to set-theoretic paradoxes, and sometimes in more technical material on truth. Category theory doesn’t show up much in philosophy.
@@AtticPhilosophy ok but when do you teach higher order logics . Is it too hard and maybe obscure. I can’t find many/any books that aren’t expensive and look like they are for phd students
is intuitionist logic for math and computer more than philosophy unless it philosophy of math and cs?
It’s big is CS departments, where constructive reasoning is the boss, but less so in mathematics departments, where classical logic is usually the default. In philosophy departments, it’s usually there as non-technical background, when thinking about the past, or verificationist theories, etc.
There needs to be something between Intuitionistic Anti-Realism and Platonism.
Structuralism might be what you're after: the idea that numbers (& other mathematical entities) really exist, but have no intrinsic nature. They're just positions in a structure: 1 after 0, 2 after 1, etc. Beyond that, there's nothing to the ontology of the numbers.
I was reading a book and the author denies the validity of "the law of excluded middle". So that makes him an anti realist? I don't understand the jargon can you explain in layman's terms?
Anti-realism is one reason to deny Excluded Middle, but not the only reason. So the author may or may not be an anti-realist. That means, very roughly, that thinking reality is to a large extent created or determined by our thought or language.
So are you saying that tables exist? bit weird
My favourite thing about the people who say "there's nothing but mereological atoms": they don't exist!
I don't get the connection between natural numbers being grounded in 'the way we think' and the rejection of the excluded middle. Just because some numerical formula is beyond our current capacities to satisfy constructively, that doesn't mean that it is not either true or false, or that there is not some form in which could ultimately be asserted and decided on. Ability is not necessarily determined by a single, limited state of affairs, as anyone who has taken a test will know. Seems to me to be somewhat flimsy grounds for chucking away one of the strongest pillars of rationality that we have.
It’s something like this: to prove a numerical statement, for intuitionists, you need to construct the number, and to disprove it, you need to show there’s no possible construction. Validity (or truth) is then equated with proof. So, for cases where no construction or proof-of-no-construction is possible, you don’t have pv~p. Not everyone agrees of course!
@@AtticPhilosophy I think maybe there is a place for intuitionism as a logic of empiricism - which might be why it maps onto system states quite nicely. Good for CompSci, perhaps, but I can’t really see it as a foundation for metaphysics.
Talk about god and religion
Good call.