Language, Proof and

1
Introduction
Language, Proof and Logic
Introduction
2
The special role of logic in rational inquiry
0.a
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, thats 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 cant 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.


3
What is logic?
0.b
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


4
What is logic?
0.b
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


5
What is logic?
0.b
Logic Science about correct reasoning. As
such, it is only interested in the form rather
than content.
Some humans are mean Socrates is a
human ---------------------------- Socrates is
mean
Every human is mortal Socrates is a
human ---------------------------- Socrates is
mortal
Some H are M S is an H ---------------------------
- S is M
Every H is M S is an H ---------------------------
- S is M


6
True or false?
0.c
There is a student in this class who likes all
those (and only those) who do not like
themselves.


7
Liars paradox
0.d
True or false?
The sentence written in this frame is false.


8
Prisoners paradox
0.e
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.
No!
No!
No!
No!
EXECUTED!
No!
No!
No!
9
The main aims of this course
0.f
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.
10
Essential instructions about homework exercises
0.g
Read the same-title subsection of the
textbook. You should familiarize yourself
with Tarskis World, Fitch, Boole, Submit, and
Grade Grinder. We will be heavily relying on
these software tools throughout the course.
Heaven forbid you copy somebodys 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.