Proving Invalidity in Predicate Logic – The Finite Universe
Method
In general, to show that an argument is invalid, we show
that the premises can be true when the conclusion is false,
that is, we show that the truth of the premises does not
guarantee the truth of the conclusion. In predicate logic
we do not have simple truth values for statements, so we do
not have truth tables. So the standard truth table test,
or indirect truth table test, from propositional logic, is
unavailable. However, we can talk about the truth or
falsity of a statement in predicate logic in regard to a
universe. If that universe is THE UNIVERSE (1 in Boolean
systems), then we assume that it contains an infinite
number of things and, of course, we cannot list them. On
the other hand, we can talk about the truth or falsity of
particular statements relative to finite universes and we
can use those to demonstrate invalidity. (Note: we will
assume all universes have at least one member; the empty
universe is of no interest here. Furthermore we will
display universes as sets of names.)
Handy definitions:
Substitution instance: A substitution instance is the
result of eliminating the quantifier from a
quantificational proposition and substituting a name for
every occurrence of the variable of quantification.
Examples:
(let a and b be names)
1. ~ [ Sa  Wa ] is a substitution instance of (x) ~ [ Sx
 Wx ]
2. Wb •
Bx]
Bb is a substitution instance of
3. Note, however, that (x) Wx •
substitution instances because this
quantificational proposition; it is
conjuncts (x) Wx and (x) [Bx •
instances, however.
(x) [ Wx •
(x) [Bx • Tx] has no
is not a
a conjunction. Both
Tx] have substitution
Quantificational Equivalence in a Finite Universe: Given
any predicate logic proposition, we can create its
quantificational equivalent in any finite universe. The
-2procedure is to form all of the substitution instances for
every name in the universe of a quantificational
proposition and link these substitution instances by
conjunction, if the quantifier is universal, or
disjunction, if the quantifier is existential. Other
operators, in this procedure, stay the same. If there are
names in the original, those names must be part of the
finite universe. The result has no quantifiers and
therefore can be treated like a proposition in
propositional logic.
Examples:
Let the universe be {a, b}
1. The quantificational equivalent of (x) ~ [ Sx  Wx ] is
~ [ Sa  Wa ] • ~ [ Sb  Wb ]
2. . The quantificational equivalent of (x) Wx • (x)
[Bx • Tx] is:
(Wa V Wb) • [(Ba • Ta) V (Bb • Tb)]
3. The quantificational equivalent of
~(Wa V Wb)
~(x) Wx is:
4. The quantificational equivalent of
Ha  ~(Wa V Wb).
Ha
 ~(x) Wx is:
Quantificational Validity: An argument is
quantificationally valid if in every universe, when the
quantificational equivalents of the premises are true, the
quantificational equivalent of the conclusion is true as
well. In a given universe, if an argument is such that when
the quantificational equivalents of the premises are true,
the quantificational equivalent of the conclusion is true
as well, we can say that this argument is valid in that
universe. So another way of defining quantificational
validity is to say that an argument that is valid in all
universes is quantificationally valid.
FACT: if an argument is valid in a universe of size n, then
it is valid in all universes smaller than n.
Quantificational Invalidity: An argument is
quantificationally invalid if there is a universe such that
the quantificational equivalents of the premises can be
true when the quantificational equivalent of the conclusion
is false. Or, an argument is quantificationally invalid if
the argument is invalid in some universe.
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-3-
FACT: If an argument is invalid in a universe of size n,
then it is invalid in all universes larger than n.
Proving Invalidity:
The method consists in choosing the smallest universe that
will make the premises true and the conclusion false, form
the quantificational equivalents of the premises and
conclusion, and then apply the indirect truth table test.
The proof of invalidity requires a universe and a set of
values. If an argument turns out to be valid in the
universe you initially choose, then go to the next larger
universe. Finally, it should be noted that if an argument
contains names, these have to be in the universe and if
there are existential quantifiers you usually need a
universe with more than one name. But not always, the
argument
(x) Sx V (x) Hx
(x) Sx
can be shown invalid in a universe of one thing.
Here is an example of the method:
(x) Sx
(x) [Sx  Mx]
Ma
Let the universe = {a,b}
The quantification equivalents of the premises and
conclusion would be:
Sa V Sb
(Sa  Ma) •(Sb  Mb)
Ma
Here are values that make the premises true and the
conclusion false:
Sa
F
Sb
T
Ma
F
Mb
T
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-4Note: Since Ma, the conclusion, must be false, Sa must be
false as well, or the second premise could not be true.
But at least one substitution instance of Sx must be true
in order to make the first premise true. Note that we need
at least two things in the universe to make this work and
one of them -- b -- is such that both Sb and Mb are true.
This set of values makes the premises true and the
conclusion false, and hence is a proof of invalidity.
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