University of New Hampshire  Main Campus  DUPLICATE
theory of computation
COMPUTER S CS 659

Winter 2016
Propositions are not Enough
Our proposition based system of proof is not powerful enough
to do everyday reasoning.
Example:
Predicate Logic
Consider a "typical" mathematical proposition:
All even numbers are divisible by 2 and some even numbers are
divisi
University of New Hampshire  Main Campus  DUPLICATE
theory of computation
COMPUTER S CS 659

Winter 2016
Proofs
Translate the sentence.
If this course is interesting or I study hard then I will
pass the course or the instructor is bad. If the course
is uninteresting then the instructor is bad. I dont
study hard and the instructor is good. Therefore I
pass th
University of New Hampshire  Main Campus  DUPLICATE
theory of computation
COMPUTER S CS 659

Winter 2016
Formal Proofs
Next we want to write formal proofs in predicate calculus.
Formal Proofs in Predicate Logic
The rules of inference of propositional calculus will be the
basis.
We will add new rules to deal with quantiers.
CS659
First a denition
Fall 2015
Te
University of New Hampshire  Main Campus  DUPLICATE
theory of computation
COMPUTER S CS 659

Winter 2016
The Problem
We often reason and make arguments in everyday language.
Elementary Logic 1
An arguments consists of a list of propositions (premises)
followed by a proposition (called the conclusion).
How do we know if our reasoning is sound?
Logic models th