How to Prove Completeness | Logic tutorial | Attic Philosophy

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: https://youtu.be/pR7nQM6iSQQ Soundness and Completeness for proof trees: https://youtu.be/MBEhF3WKWVg Proof by induction: https://youtu.be/9w20J4j5-0Y 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 Get in touch on Social media! Instagram: https://www.instagram.com/AtticPhilosophy Twitter: https://twitter.com/PhilosophyAttic #logic #philosophy #proof