Chapter 9 Overview
Despite the power and rigor of categorical logic, it lacks flexibility. Its methods really only
apply to syllogisms in which each of the two premises can be translated into a standard
form categorical claim. For this reason, even an introductory treatment of logic calls for
some discussion of modern symbolic logic. Truthfunctional or propositional logic is the
simplest part of symbolic logic, though you will find it both rigorous enough to let you carry
out systematic proofs and broad enough to handle a wide range of ordinary arguments.
This chapter shows how to work with complex arrangements of individual sentences.
You will use letters to represent sentences, and a few special symbols to represent the
standard relations among sentences: roughly speaking, the relations that correspond to the
English words "not," "and," "or," and "ifthen." Truth tables and rules of proof show how the
truth values of the individual claims determine the truth values of their compounds, and
whether or not a given conclusion follows from a given set of premises.
1.
Truthfunctional logic
is a precise and useful method for testing the
validity of arguments.
a.
Also called propositional or sentential logic, truthfunctional logic is the logic of
sentences.
b.
It has applications as wideranging as set theory and the fundamental principles
of computer science, as well as being useful for the examination of ordinary
arguments.
c.
Finally, the precision of truthfunctional logic makes it a good introduction to
nonmathematical symbolic systems.
2.
The vocabulary of truthfunctional logic consists of
claim variables
and
truthfunctional symbols
.
a.
Claim variables are capital letters that stand for claims.
i.
In categorical logic, we sometimes used capital letters to represent terms
(nouns and noun phrases). Keep those distinct from the same capital letters
that now represent whole sentences.
ii.
Each claim variable stands for a complete sentence.
b.
Every claim variable has a truth value.
i.
We use T and F to represent the two possible truth values.
ii.
When the truth value of a claim is not known, we use a truth table to indicate
all possibilities.
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iii.
Thus, for a single variable P, we write:
P
T
F
c.
Whatever truth value a claim has, its
negation
(contradictory claim) has the
opposite value.
i.
Using ~P to mean the negation of P, we produce the following truth table:
P
~P
T F
F T
ii.
This truth table is a definition of negation.
iii.
~P is read "notP." This is our first truthfunctional symbol.
d.
The remaining truthfunctional symbols cover relations between two claims.
i.
Each symbol corresponds, more or less, to an ordinary English word; but you
will find the symbols clearer and more rigid than their ordinarylanguage
counterparts.
ii.
Accordingly, each symbol receives a precise definition with a truth table, and
never deviates from that definition.
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 Spring '11
 Miller
 Logic, ruth table, condit ional

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