1. Introduction to Derivations in Sentential Logic PHIL 121: Methods of Reasoning April 8, 2013 Instructor:Karin Howe Binghamton University

2. Issues from Part I, II and III that are still highly relevant statement or proposition arguments, specifically deductive arguments validity/invalidity (no, these things never go away)  consistency/inconsistency  logically equivalent statements  contradictory statements

3. Most importantly … We will be proving that arguments are valid through a series of (valid) deductive inferences. Given a small number of basic rules, (10 - 2 for each connector), each of which preserves validity, we can derive the conclusion from premises. If a conclusion can derived from its premises through a series of (correct) applications of these 10 basic rules, then the argument is valid.

4. Motivation Consider the following argument: –P & Q, S & T, (P & Q)  [S  (T  U)]  U How many lines would there be in the full truth table for this argument? –2 5 = 32 lines!

5. Compare the following proof: 1.P & Q Pr. 2.S & T Pr. 3.(P & Q)  [S  (T  U)] Pr. /  U 4.S  (T  U)  E, 3,1 5.S & E, 2 6.T  U  E, 4,5 7.T & E, 2 8.U  E, 6,7 8 lines vs. 32 lines … which would YOU rather do??

6. Pros and Cons of Proving Validity via Deductive Inferences Pros: –More fun than truth tables, and usually shorter –Allows you to uncover connections between the premises and conclusion -- lets you see the reasoning behind the argument –More like the way we reason naturally Cons: –It's crap for proving invalidity (doesn't work for this at all)

7. Brief overview of new things we will be learning in Part IV How to derive proofs in sentential logic using the 10 basic rules (okay, 11). Once we have learned how to derive proofs using the 10 basic rules, we will add a number of derived rules (rules that can be derived from the 10 basic rules) [there will be ~20 of these, depending on how you count them] –These rules will help make our proofs faster and easier –They will allow us to attack some more complicated proofs with greater ease than if we just had the 10 basic rules

8. How to prove statements are tautologies (theorems) using the proof method If time allows, we will also learn how to do the following: –prove statements are contradictions using the proof method –prove that two statements are logically equivalent using the proof method –prove that a set of statements are inconsistent using the proof method

9. Logic & Proofs This module will cover the rest of the Logic & Proofs course that you have accessible to you (Ch 4-6, and bits of Ch 7) We will be making extensive use of the Proof Lab in these modules (~half of the homework exercises will ask you to do exercises in the Proof Lab)Proof Lab Therefore, it is essential that you fix any technical issues you may be having with the OLI website immediately, especially in terms of your ability to load the necessary Java applets to run the Proof Lab