Download Introduction - Villanova Department of Computing Sciences

Language, Proof and
Logic
Introduction
Introduction
0.a
The special role of logic in rational inquiry
What do the fields of astronomy, economics, finance, law, mathematics, medicine,
physics and sociology have in common? Not much in the way of subject matter, that’s
for sure. And not all that much in the way of methodology.
What they do have in common, with each other and with all other areas of
intellectual activity, is their dependence on a certain standard of rationality. In other
words, they presuppose an underlying acceptance of basic principles of logic.
In this sense logic --- the study of correct ways of reasoning --- is most basic of all
sciences. Its laws are even more fundamental and less subject to repeal than the laws of
physics. We can imagine a world where water flows up hill. But we can’t imagine a
world where both are and are not nine planets.
Logic is considered a full-fledged subfield of three disciplines: philosophy,
mathematics, and computer science.
It is mainly philosophy by subject matter, mainly mathematics by methodology, and
mainly computer science by applications. This is why the course is cross-listed under
CSC, MAT and PHI.
0.b
What is logic?
Logic = Science about correct reasoning.
As such, it is only interested in the form rather than content.
Every human is mortal
Socrates is a human
---------------------------Socrates is mortal
Every H is M
S is an H
---------------------------S is M
0.b
What is logic?
Logic = Science about correct reasoning.
As such, it is only interested in the form rather than content.
Every hemin is melin
Solik is a hemin
---------------------------Solik is melin
Every H is M
S is an H
---------------------------S is M
0.b
What is logic?
Logic = Science about correct reasoning.
As such, it is only interested in the form rather than content.
Every human is mortal
Socrates is a human
---------------------------Socrates is mortal
Every H is M
S is an H
---------------------------S is M
Some humans are mean
Socrates is a human
---------------------------Socrates is mean
Some H are M
S is an H
---------------------------S is M
0.c
True or false?
There is a student in this class who likes all those
(and only those) who do not like themselves.
0.d
Liar’s paradox
True or false?
The sentence written in this frame is false.
0.e
Prisoner’s paradox
JUDGE
“You will be executed at noon on one of the days of this week;
yet so that in that morning you will not know it is your last day.”
PRISONER
Can I be executed on Saturday?
Friday?
Thursday?
Wednesday?
EXECUTED!
Tuesday?
Monday?
Today (Sunday)?
No!
No!
No!
No!
No!
No!
No!
0.f
The main aims of this course
The main goal of this course is to introduce you to some of the most important
concepts and tools of logic.
More particularly, there are two main aims:
1. To help you learn a new language, the language of first-order logic (FOL).
It serves as a prototypal example of an artificial language, and is used every day by
mathematicians, philosophers, computer scientists, linguists, and practitioners of
artificial intelligence. Like English, it is a truly universal language: there is nothing on
earth you cannot say in it. But, unlike English, it is mathematically precise and hence
totally free of any ambiguities, abundant in natural languages.
2. To help you learn about the notion of logical consequence,
and about how one goes about establishing some claim does or does not logically
follow from other accepted claims. Modern logic teaches us that one claim is a logical
consequence of another if there is no way the latter could be true without the former
also being true.
0.g
Essential instructions about homework exercises
Read the same-title subsection of the textbook.
You should familiarize yourself with Tarski’s World, Fitch, Boole,
Submit, and Grade Grinder. We will be heavily relying on these software
tools throughout the course.
Heaven forbid you copy somebody’s file, or even just a tiny fragment
of such. The Grade Grinder is smart, and will immediately alarm the
instructor about suspicious activity. Such a report will be considered a
proof of cheating, no matter what actual story stands behind it.