MetaLogic Proofs | Attic Philosophy

How do we know logic works the way it's supposed to work? We prove it! When we do proofs about how logic works, we're doing Metalogic. (As in, the Meta theory of Logic.) Unlike in proofs within logic (like natural deduction or a tree proof), there aren't set rules for metalogic, so knowing what to do can be confusing. In this video, I'll show you how to set out a simple meta-logic proof.

00:00 - Intro
01:07 - Entailment and its properties
01:34 - Proof and entailment
02:14 - Soundness and Completeness
03:23 - Example: commutative conclusions
06:05 - Reason as in natural deduction
10:43 - Recap
11:51 - Summing up the approach
12:16 - General lessons
12:52 - Leaving some steps out?
13:22 - Wrap-up

If there’s a topic you’d like to see covered, leave me a comment below.

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#logic #philosophy

[rickybundy]

hi mate, just wanted to say you single-handedly saved my uni career this last semester and I really appreciate it. thanks so much

[DarrenMcStravick]

Amazing, I've been self-teaching myself formal logic for a little under a year and so far this has been more clear and informative an explanation of metalogic, soundness and completeness than what I've attempted to read in 6 intro textbooks and multiple youtube videos. Thank you.

[irisll]

Whenever I get confused in my uni logic studies, I can get clear explanations from your videos, in almost every topic. I love the clear way you explain things, you are my hero.

[Rampag]

Awesome! Are there any metalogic text books you recommend?!

[karlfriedmann]

Great (and really helpful) explanation as usual. When I first got into contact with metalogic, we just went through the proofs without any backround information and I was really confused about what I was doing. The question I have is can you reason about all different logics the same way ? For example if I have a logic, where proof of contradiction doesn‘t work, can I use an informal version of that proof, when doing metalogic about that logic?

[poklar]

Great video as always! I was wondering, any chance you have or plan to do anything on intensional logic?

[andersedson]

Great video! I’m glad you’re doing meta logic videos since i’ll be starting a meta logic class next semester. But I have a question, how is that proof meta logical if it uses strategies of logic like universal generalization and universal instantiation?

[feigao]

That's really helpful, I get a basic understanding of what metalogic is, and I wonder is it possible to define the rules of metalogic rigorously? For, as far as I know, mathematics is based on ZFC-set theory, ZFC is built upon first-order logic, and rules of first-order logic are governed by metalogic. The rules of Matelogic include natural deduction, some set notion, and something hard to tell. It seems to me there is a loop definition since natural deduction is a part of mathematics.

[pfroncole]

That particular proof seems very particular to binary systems. Supposes we draw from a sack of random sentences. We know what to do with truths, we keep 2 of them, and with any that evaluate as false we simply discard, but those in a 3rd state, we have no way to decide to keep or discard, so if we draw one from the sack, the procedure is stuck.

[philosophyversuslogic]

Even before the start of the video I thought about this. I am stopping the video at the moment, but firstly I want to put my thoughts, and then get back to the video. So, I think that to prove logic we have to use logic (or kinda), so there's no way out except for accept the logic. If we deny it, then we can't say whether or not logic is true: we've just stuck within the bad circle of not knowing what to choose and so on.
Let's say A is a method to prove logic. If A works, then logic works, if it doesn't work, we should be start looking for another one, and another one, but all of any new methods must be provable, so let's say A, B... and any other methods should be P or provable. So, if A (or anyone else) belongs to P, then it is provable, and we can prove the logic, if it doesn't - we never do it. But it is interesting that P itself means that P is provable, so therefore P is logical.
Conclusion: if logic is provable, then logic is logical, and so on - we will get into that circle within logic.
If logic can't be proved, we will never know that: so the other option is just left logic be, without being able to say anything about it.

[ssd]

Is this different to logical pluralism? Thanks for the video ❤️‍🔥

Can this be related to pluralism rather?

[nullvoid]

Great content! I have a question though !!
Can we define interfaces in meta logic ?

[hobojoe]

Attic has a double meaning here because Attica is the historical region containing Athens where Aristotle invented formal logic.