Hello, I am French and I regret that there are only a few resources concerning logic. So when I came across your video which is well done and which has less than 1000 views, I am disappointed. Your work is fantastic!

Thanks for the video. I've been self learning set theory and recently explained this to myself with the Cartesian Plane. Seeing this video helped me to be validated. I was also not aware of the first two ways of writing an ordered pair.

@AtticPhilosophy

9 ай бұрын

Glad it was helpful - good luck with the self-study, it's not easy but is rewarding.

Almost got the last question lmao, but I got the idea !!

The more I watch your videos on logic, the more striking it is that there are significant similarities between our respective fields. In software development we use exactly the same kind of concepts, albeit with different syntax and naming conventions. I reckon' anyone who would like to learn logic, might also benefit from learning the basics of a simple programming language; no doubt the reverse may also be true. Is there any such thing as a programming language geared up for philosophers? (Something that can run on a turing machine and could perhaps mock up or visualise logical constructs / statements etc?)

@AtticPhilosophy

2 жыл бұрын

There's a history of logic-based programming languages using in AI and knowledge representation: languages like Prolog. Typically, logic-based languages are declarative, so good for logic but often not so good for actual programming. Then there's type-theory based languages like Haskell, which map nicely onto natural deduction proofs. Defining data-types like TREE in Haskell isn't that far from defining concepts like ORDERED PAIR in set theory.

Would you ever consider doing videos on the epistemology of modality or is that outside of your interests?

@AtticPhilosophy

2 жыл бұрын

As it happens, I did a talk on epistemology of modality a few weeks back & recorded it, so it might show up here soon!

@DarrenMcStravick

2 жыл бұрын

@@AtticPhilosophy Holy crap I'm really looking forward to seeing that if you upload it! Keep working your philosophical magic 👍🏻

I'd like to try to attempt Hausdorf's to the triplets: = {{{{a},0},{{b}}},0},{{c}}}

@AtticPhilosophy

Жыл бұрын

I think it's even uglier than that! (I'm assuming you mean the Weiner definition?) We can write out the pair definition twice like this: = {{{a},0},{{b}}} = {{{X},0},{{c}}} And then cut-and-paste in for X to get: = = = {{{{{{a},0},{{b}}}},0},{{c}}}

@philosophyversuslogic

Жыл бұрын

@@AtticPhilosophy Oh, this is true! Do thank you. Yes, I missed some brackets. (Me teacher of logic would kill me for making this mistake =) ...No, I'm joking of course...)

Thanks for you video! I have a few questions about tupels (which might even be worth a video?): 1. Where is the problem that with Hausdorff's definition, ⟨2,1⟩={{2,1}}={{1,2}}? It still is the case that ⟨2,1⟩≠⟨1,2⟩ since ⟨1,2⟩={{1,1},{2,2}}={{1},{2}} and {{1,2}}≠{{1},{2}}. You still can tell from {{1,2}} that 1 is first and 2 is second because the only way how to define this set with tupels is with ⟨2,1⟩. Plus, if you go with von Neumann's set-theoretic definition of the natural numbers, 1 and 2 are very simple sets, which is not the case for the extra a in Kuratowski's definition (unless a and b are urelements or numbers smaller than 2). So why would you adopt Kuratowski's definition over Hausdorff's if the latter (a) is capable of defining tupels and (b) yields sets which are in most cases way less complex? 2. Here's how to talk about the first and second component of a 2-tupel (!): x is the first component of ⟨a,b⟩ iff x is a member of all elements of ⟨a,b⟩: Fx :↔ ∀y (y∈⟨a,b⟩ → x∈y) - alternatively, x∈⋂⟨a,b⟩ x is the second component of ⟨a,b⟩ iff x is a member of some element of ⟨a,b⟩ but not of the other: Sx :↔ ∃y(y∈⟨a,b⟩ ∧ x∈y) ∧ ∀y∀z (y∈⟨a,b⟩ ∧ z∈⟨a,b⟩ ∧ y≠z → (x∉y ∨ x∉z)) Now that's all fine. But what if we want to talk about the first or second component of ⟨a,b,c⟩? By definition, ⟨a,b,c⟩=⟨⟨a,b⟩,c⟩={ {{{a}, {a,b}}}, {{{a}, {a,b}}, c} }. "Sx" now filters out c, the third component, and very generally, for ever n-tuple, "Sx" filters out the nth element. "Fx" now filters out ⟨a,b⟩, but not a. So this way of talking about the components of tupels does not work if you define n-tupels as nested 2-tupels. To me it seems that for any n-tupel, you need a special predicate for every component that is not n (because for n, "Sx" suffices). This is less than desirable. Is there a way out? 3. There are two tupel definitions that you did not talk about in the video - the definitions of the 0-tupel and of 1-tupels: ⟨ ⟩ := ∅ ⟨a⟩ := a The second definition makes sense to me. Let A={a,b,c}. Then ⟨a,b,c⟩∈AxAxA and ⟨a,b⟩∈AxA, so it is desirable that ⟨a⟩∈A, which the definition that ⟨a⟩ := a guarantees. But what about the first one, that ⟨ ⟩ := ∅? Why do you define it that way? Best wishes from Münster, Germany!

@AtticPhilosophy

2 жыл бұрын

Great questions! I'll do my best ... 1) You're right, we can tell the difference between e.g. and on that definition. So I guess there's no real worry, other than that its hard to tell that e.g. {{1,2}} is a tuple. (Good exercise to set a class though!) 2) If we define n-tuples from pairs recursively, we'll need to define predicates on n-tuples recursively too. So, First(x) will be defined primitively for pairs, for triples we'll use First(First(x)), and so on. That's not to say we need a different predicate for each case: rather, we have one predicate, with a recursive definition. 3) These are extra cases of a tuple, which don't follow from the general definition (which reduces tuples to pairs). But they're useful, particularly the 1-tuple. Then, for example, every n-place predicate is interpreted as a set of n-tuples (including the 1-place case, which is then just a subset of the domain). So, as you say, thinking of n-tuples over X as members of X^n is convenient. So X^1 = X, X^2 = X x X etc. But note: this notation (X^n) is defined in terms of n-tuples, not the other way around! If we want a value for X^0, we'd first need to pick a value for . As far as I know, the only sensible option, if we want this to be a set, it to take it to be {}. This gives the right result (in terms of preserving order - trivially!) and any other choice of set would be problematic - e.g. if we picked {a}, then we'd get = which seems problematic. So I think it's a pragmatic choice: stipulate this result because it works nicely.

Thks & a couple questions; ?Is there a difference between the logic of philosophy & math? ?If-so, what is it?

@AtticPhilosophy

2 жыл бұрын

Same underlying logic, maybe different emphasis. Philosophers tend to be more interested in a variety of different logics, e.g. modal logic & non-classical logics, whereas mathematicians focus more on classical 1st order logic.

@tombouie

2 жыл бұрын

@@AtticPhilosophy Thks

Can you have an ordered pair (or n-tuple) just consisting of one object, e.g., ? If so, how is this treated?

@AtticPhilosophy

Жыл бұрын

Yes, the definition is completely general. So you can have {{a},a}

Which uni do you teach at? also is it too late to switch unis?! ;)

@AtticPhilosophy

2 жыл бұрын

Nottingham. Most unis accept transfers of credits (so, modules you've already studied could count towards getting your final degree, even if you transfer). That said, there are great teachers at just about every uni. Part of the trick is picking modules taught by lecturers who get you & the way you learn.

@BillboMC

2 жыл бұрын

@@AtticPhilosophy I was kidding for the most part XD some profs however don’t seem to realise how new the jargon is and they then proceed to use said jargon multiple times in a short sentence. It takes more than a sentence worth of time to understand and process what “the codomain of the set which semantically entails the sum of the functions of Z” means.

@AtticPhilosophy

2 жыл бұрын

Are you on a maths course? I find maths to be 50% jargon of easy concepts and 50% actually quite hard concepts. The advantage of videos is, you can go back & watch again!

@BillboMC

2 жыл бұрын

@@AtticPhilosophy computer science the module specifically is called foundations of computer science. Propositional logic is fine but I struggle to understand the questions. Linear algebra is very close to a level further maths so I was ok with that. But set theory kills me and seemingly everyone in the lectures XD

this is my first time seeing a definition of ordered pairs in set theory. it's interesting, but it seems cumbersome

@AtticPhilosophy

2 жыл бұрын

That'a kinda how it goes in set theory. Because there's just one basic concept, membership, the definitions don't always look simple. The good news is, once you've defined ordered pair (or whatever), you can use them as they are & don't have to worry about the definition!

As these 3 sets are equal: {{a}, {a,b}} = {{a,b}, {a}} = {{b,a}, {a}}, how the order of a ordered pair is defined?

@AtticPhilosophy

Жыл бұрын

They’re all the pair . The singleton {a} means that a comes first in the pair.

@andreisljusar5452

Жыл бұрын

"The singleton {a} means that a comes first in the pair." Why?