Transcript Introduction to Derivations in Sentential Logic

Introduction to Derivations in
Sentential Logic
PHIL 121: Methods of Reasoning
April 8, 2013
Instructor:Karin Howe
Binghamton University
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
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.
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?
– 25 = 32 lines!
Compare the following proof:
1.
2.
3.
4.
5.
6.
7.
8.
P&Q
S&T
(P & Q)  [S  (T  U)]
S  (T  U)
S
TU
T
U
Pr.
Pr.
Pr. / U
 E, 3,1
& E, 2
 E, 4,5
& E, 2
 E, 6,7
8 lines vs. 32 lines … which would YOU rather do??
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)
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
• 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
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)
• 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