CHAPTER 2
Logic
1. Logic Deﬁnitions
1.1. Propositions.
Definition
1.1.1
.
A
proposition
is a declarative sentence that is either
true (denoted either T or 1) or
false (denoted either F or 0).
Notation: Variables are used to represent propositions. The most common variables
used are
p, q,
and
r
.
Discussion
Logic has been studied since the classical Greek period ( 600300BC). The Greeks,
most notably Thales, were the ﬁrst to formally analyze the reasoning process. Aristo
tle (384322BC), the “father of logic”, and many other Greeks searched for universal
truths that were irrefutable. A second great period for logic came with the use of sym
bols to simplify complicated logical arguments. Gottfried Leibniz (16461716) began
this work at age 14, but failed to provide a workable foundation for symbolic logic.
George Boole (18151864) is considered the “father of symbolic logic”. He developed
logic as an abstract mathematical system consisting of deﬁned terms (propositions),
operations (conjunction, disjunction, and negation), and rules for using the opera
tions. It is this system that we will study in the ﬁrst section.
Boole’s basic idea was that if simple propositions could be represented by pre
cise symbols, the relation between the propositions could be read as precisely as an
algebraic equation. Boole developed an “algebra of logic” in which certain types of
reasoning were reduced to manipulations of symbols.
1.2. Examples.
Example
1.2.1
.
“Drilling for oil caused dinosaurs to become extinct.” is a propo
sition.
21
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Example
1.2.2
.
“Look out!” is not a proposition.
Example
1.2.3
.
“How far is it to the next town?” is not a proposition.
Example
1.2.4
.
“
x
+ 2 = 2
x
” is not a proposition.
Example
1.2.5
.
“
x
+ 2 = 2
x
when
x
=

2
” is a proposition.
Recall a
proposition
is a declarative sentence that is either true or false. Here are
some further examples of propositions:
Example
1.2.6
.
All cows are brown.
Example
1.2.7
.
The Earth is further from the sun than Venus.
Example
1.2.8
.
There is life on Mars.
Example
1.2.9
.
2
×
2 = 5
.
Here are some sentences that are not propositions.
Example
1.2.10
.
“Do you want to go to the movies?” Since a question is not a
declarative sentence, it fails to be a proposition.
Example
1.2.11
.
“Clean up your room.” Likewise, an imperative is not a declar
ative sentence; hence, fails to be a proposition.
Example
1.2.12
.
“
2
x
= 2 +
x
.” This is a declarative sentence, but unless
x
is
assigned a value or is otherwise prescribed, the sentence neither true nor false, hence,
not a proposition.
Example
1.2.13
.
“This sentence is false.” What happens if you assume this state
ment is true? false? This example is called a paradox and is not a proposition, because
it is neither true nor false.
Each proposition can be assigned one of two
truth values
. We use T or 1 for true
and use F or 0 for false.
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 Fall '08
 PenelopeKirby
 Logic, Proposition, exclusive or, LOGIC DEFINITIONS

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