Chapter 8 – Symbolic Logic
Professor D’Ascoli
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Because the appraisal of arguments is made
difficult by the peculiarities of natural
language, logicians have formulated an
artificial, symbolic language system that is
not encumbered by linguistic defects.
In some respects, it helps logicians to
accomplish intellectual tasks without having
to think as much.
Thus, symbolic logic is not tied to syllogisms,
but can go directly to assessing the internal
structure of propositions and arguments.
Symbolic Logic
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Simple and compound statements are logically connected
in a few simple ways.
In English, we use “and,”“not,”“or,” and “if……then.”
Modern symbolic logic uses symbols to represent these
relationships more precisely.
Conjunction (“and”) is symbolized with a dot (•).
Negation (“not”) is symbolized with a tilde ~.
Disjunction (“or”) is symbolized with a wedge (∨).
Material implication (“if……then,” or more precisely,
“implies”) becomes a horseshoe ().
The language of symbolic logic, like all other languages,
uses punctuation marks to disambiguate complex
statements.
Punctuation includes the use of parentheses, brackets, and
braces.
Symbolic Logic
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Using symbols, the definition of “valid” can be made
more precisely.
One method, similar to the method of logical analogy,
says simply that if any two arguments have the same
logical form, they are both either valid or invalid—no
matter what their content.
Simple arguments can also be tested in truth tables,
which are arrays of T and F values.
Common arguments forms such as, modus ponens,
modus tollens, and the disjunctive syllogism are
easily shown to be valid using truth tables.
On the other hand, constructing the appropriate truth
table clearly shows common fallacies (such as
affirming the consequent) to be invalid.
Symbolic Logic
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p q
p•q
____________________
T T
T
T F
F
F T
F
F F
F
Truth Table p and q (conjunction)
p
~p
 _______________
T
F
F
T
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Truth table p
not p (negation)
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p
q
pˇq
___________________
T
T
T
T
F
T
F
T
T
F
F
F
Truth table p or q (disjunction)
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p q
~q
p•~q
~p(p•~q) p ⊃q
--------------------------------------------T T
F
F
T
T
T F
T
T
F
F
F T
F
F
T
T
F F
T
F
T
T
Truth table if p then q (conditional
statements)
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p
q
p ⊃q
_________________________
T
T
T
T
F
F
F
T
T
F
F
T
Simplified if p then q table
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Statement forms are sequences of symbols containing statement
variables, but not statements; when statements are substituted
for the symbols, they result in statements.
Statements that are substituted for a symbolic representation are
substitution instances of that form.
Every statement form with only true substitution instances is a
tautology.
One with only false substitution instances is self-contradictory.
One with at least one true and one false substitution instances is
contingent.
When two statements have the same truth value (either both true
or both false), they are said to be materially equivalent; the three
bar sign, ≡, symbolizes this (“if and only if,” in English).
De Morgan’s theorems are examples of important equivalences,
as are the principle of double negation and the definition of
material implication.
Symbolic logic
The three “laws of thought”—the
principles of identity, non-contradiction,
and excluded middle—are indeed
important principles.
 They do not occupy any primary or central
place in logic, however.
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Symbolic Logic
Principle of Identity – if a statement is
true then it is true – p ⊃ p.
 Principle of non-contradiction – no
statement can be both true and false
p•~p.
 Principle of excluded middle – every
statement is either true or false – pˇ ~p
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Symbolic Logic