PHIL 121: Methods of Reasoning
April 8, 2013
Instructor:Karin Howe
Binghamton University

• statement or proposition
• arguments, specifically deductive arguments
• validity/invalidity
(no, these things never go away)
 consistency/inconsistency
 logically equivalent statements
 contradictory statements

• 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 .

• 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!

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
8. U
& E, 2
 E, 6,7
8 lines vs. 32 lines … which would YOU rather do??

• 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 )

• 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

• 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

• 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)
• 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