Short (brief/indirect) truthtable method
N.Apostol
The Procedure of ShortTable Proof
1. Put argument form in propositional form.
2. Put an F to the left of the conclusion and a T to the left of each premiss.
3. Start with that proposition (whether conclusion or premise) which allows for only one
assignment of truthvalues to make it as required by the values to its left (false, if it is the
conclusion you start with; true, if it is a permise you start with)
4. Fill in the truth values for the rest of the propositions, carrying the "lockedin" values and
assigning new values, as required for the rest of the simple propositions
5. If all the premises can be made true and the conclusion false, the argument is INVALID and
the procedure is complete. If not, then we enter into the "Columbo Syndrome." "Columbo
Syndrome" means that just like the detective of that popular series starring Peter Falk, who knew
who commited the crime but had to prove it, so we know that the argument is valid, but we have
to prove it. Proving it means showing that under every possible truth substitution, one of the
premises must be false.
.
We need show only the instance where the conclusion forces at least
one premise to become false; or all true (relevant) premises force the conclusion to become
true
.
NOTE: never stat by making the conclusion true
never start by making the premises false
Test the following arguments for validity. In other words, determine whether the following
arguments are valid or invalid.
(
Important: all answers require proof; that is, all answers require an adequate
justification, whether arguments are valid or invalid).
CASE ( I ):
A
B
~A
A&B
~(A&B)
AvB
~A>B
A>B
A v B
T
T
F
T
F
T
T
T
:.
~A > B
T
F
F
F
T
T
T
F
F
T
T
F
T
T
T
T
F
F
T
F
T
F
F
T
CASE (I)
P1
P2
C
CASE (II)
C
P2
P1
In order to test for validity, only one
combination of truth values between premises and
conclusion will prove (reveal) the nature of the argument: all
true premises and false conclusion.
1)
If it is possible
to assign certain
truth values to the sentences A and B such that all premises are
true and conclusion false, the argument is invalid.
2)
If it is impossible
to assign any
truth values to the sentences A and B which will make all
premises true and the conclusion false, then the argument is valid.
In other words, every
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View Full Documentsingle combination of truth values between A and B will never produce all true premises and
a false conclusion . This only happens due to the way the sentences are related by the logical
connectives ( &, v, >, <>, ~ ) within the argument. It is that relation which either allows for
all true premises and a false conclusion, proof of invalidity, or it blocks the possibility of all
true premises and a false conclusion, the proof of validity.
Very important: validity and
invalidity are a function (a property) of an argument given the way the atomic
sentences (A, B, etc.) are related. We (or anyone) cannot make a given
argument valid
or invalid, they are valid or invalid as a result of the already existing relation
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 Spring '08
 Apostol
 Logic, Conclusion, premises, Logical connective, Peter Falk

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