How to Prove Completeness | Logic tutorial | Attic Philosophy

Описание: The completeness theorem says that we can prove all the genuine entailments. Saying that a proof system is complete means that, if premises X logically entail some conclusion C, then you can prove that conclusion C from those premises X.

In this video, I'll show you how to prove the completeness theorem, focusing on the proof tree system. The technique focuses on how to construct a counter-model from a failed proof. This is pretty straightforward in the case of proof trees, where a failed proof will have a finished open branch.

If you're new to these ideas, you might want to watch these background videos first:

An overview of Soundness and Completeness:    • Soundness and Completeness Tutorial | Atti...  
Soundness and Completeness for proof trees:    • Soundness and Completeness for Proof Trees...  
Proof by induction:    • How to use Mathematical Induction | Logic ...  

00:00 - Intro
01:08 - Completeness is holistic
02:02 - Completeness and satisfiability
04:06 - Finished open branches are satisfiable
04:52 - The overall strategy
05:50 - Building the model
07:09 - An important clarification
08:16 - The model satisfies the open branch
09:43 - The base case: atomic (and negated atomic) sentences
11:45 - The complex sentences
12:10 - Inductive hypothesis
13:22 - Case 1: conjunction
15:32 - Case 2: disjunction
18:07 - Finishing the proof
19:08 - Proof by induction?
19:36 - Wrap up

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

Links:

My academic philosophy page: http://markjago.net
My book What Truth Is: http://bit.ly/JagoTruth
Most of my publications are available freely here: https://philpapers.org/s/Mark%20Jago

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