Statement Forms
In exactly the same sense that individual arguments may be substitutioninstances of general argument forms, individual compound statements can be substitutioninstances of general
statement forms.
In addition, just as we employ truthtables to test the validity of those arguments, we can use truthtables to exhibit interesting logical features of some statement forms.
A statement form whose column in a truthtable contains nothing but Ts is said to be tautologous.
Consider, for example, the statement form:
p ∨ ~p
Notice that whether the component statement
p is true or false makes no difference to the truthvalue of the statement form; it yields a true statement in either case.
But it follows that any compound statement which is a substitutioninstance of this form—no matter what its content—can be used only to make true assertions.
A statement form whose column contains nothing but Fs, on the other hand, is said to be selfcontradictory.
For example:
p • ~p
Again, the truthvalue of the component statement doesn't matter; the result is always false.
Compound statements that are substitutioninstances of this statement form can never be used to make true assertions.
Of course, most statement forms are neither tautologous nor selfcontradictory; their truthtables contain both Ts and Fs.
Thus:
p  q  p ⊃ ~ q


T  T  F


T  F  T


F  T  T


F  F  T


p ⊃ ~q
Since the column underneath it in the truthtable has at least one
T and at least one
F, this statement form is
contingent.
Statements that are substitutioninstances of this statement form may be either true or false, depending upon the truthvalue of their component statements.
Because all five of our statement connectives are truthfunctional, the status of every statementform is determined by its internal structure. In order to determine whether a statement form is tautologous, selfcontradictory, or contingent, we simply construct a truthtable and inspect the appropriate column.
Consider, for example, the statement form:
p  q  (p ∨ ~q)  ⊃  ~(p • q)


T  T  T  F  F


T  F  T  T  T


F  T  F  T  T


F  F  T  T  T


(p ∨ ~q) ⊃ ~(p • q)
Since the truthtable shows that statements of this form can be either true or false, depending upon the truthvalues of their components, the statement form is contingent.
A particularly interesting and useful group of cases comprises those tautologous statement forms whose main connective happens to be a ≡ .
In order for the ≡ statement to be true on every line, the statement forms on either side of it must always have exactly the same truthvalue.
Statements that are substitutioninstances of these two component statementforms are then said to be logically equivalent: no matter what their content may happen to be, the conditions for their truth or falsity are exactly the same.
Consider a few examples that recur frequently enough to warrant special names:
Double Negation (abbreviated as D.N.) has the form:
p ≡ ~ ~ p
As the truthtable to the right clearly shows, this is a tautologous biconditional.
No matter what simple or compoud statement we substitute for
p ,
the same statement with two
~ s in front of it will have exactly the same truthvalue as the original statement.
A pair of more complex tautologous biconditionals are called De Morgan's Theorems (DeM., for short).
p  q  ~(p • q)  ≡  (~p ∨ ~q)


T  T  F  T  F


T  F  T  T  T


F  T  T  T  T


F  F  T  T  T


One form in which DeM.
occurs is:
~(p • q) ≡ (~p ∨ ~q)
As the truthtable at right shows, the statement forms on either side of the
<⊃ always have the same truthvalue.
p  q  ~(p ∨ q)  ≡  (~p • ~q)


T  T  F  T  F


T  F  F  T  F


F  T  F  T  F


F  F  T  T  T


The other form of DeM.
is:
~(p ∨ q) ≡ (~p • ~q)
The truthtable at right demonstrates the logical equivalence of these two statement forms.
Taken together, De Morgan's Theorems establish a systematic relationship between • statements and ∨ statements
by providing a significant insight into the truthconditions for the negations of both conjunctions and disjunctions.
Material Implication (Impl.) has the form:
p  q  (p ⊃ q)  ≡  (~ p ∨ q)


T  T  T  T  T


T  F  F  T  F


F  T  T  T  T


F  F  T  T  T


(p ⊃ q) ≡ (~p ∨ q)
This tautologous biconditional amounts to a logical definition of the
⊃ connective in terms of
∨ and the
~ .
Since expressions of these two forms are logically equivalent, we could make conditional assertions without using the
⊃ symbol at all, though our compound statements would be a bit more complicated.
In similar fashion, Material Equivalence (Equiv.) provides alternative definitions of the ≡ connective.
p  q  p≡q  ≡  (p⊃q)•(q⊃p)


T  T  T  T  T


T  F  F  T  F


F  T  F  T  F


F  F  T  T  T


Its first form defines
≡ in terms of
⊃ , justifying the use of the term "biconditional:"
[p≡q]≡[(p⊃q)•(q⊃p)]
p  q  p≡q  ≡  (p•q)∨(~p•~q)


T  T  T  T  T


T  F  F  T  F


F  T  F  T  F


F  F  T  T  T


Its second form defines
≡ by pointing out its basic truthconditions:
[p≡q]≡[(p•q)∨(~p•~q)]
Again, the logical equivalence of these three expressions provides us with a convenient way to comprehend and employ what is asserted in any statement of material equivalence.
Finally, consider Transposition (Trans.):
p  q  (p⊃q)  ≡  (~q⊃~p)


T  T  T  T  T


T  F  F  T  F


F  T  T  T  T


F  F  T  T  T


(p ⊃ q) ≡ (~q ⊃ ~p)
The truthtable at the right demonstrates the legitimacy of this tautology, which shows the logical equivalence of any
⊃ statement
with another statement that results from switching its antecedent and consequent and negating both.
But beware of what happens if we confuse Trans.
with either of two superficially similar statement forms:
p  q  (p⊃q)  ≡  (q⊃p)


T  T  T  T  T


T  F  F  F  T


F  T  T  F  F


F  F  T  T  T


The fallacy of
converting the conditional switches the antecedent and consequent without negating them:
(p ⊃ q) ≡ (q ⊃ p)
The truthtable at the right shows that these statement forms are not logically equivalent, since the biconditional connecting them is contingent.
p  q  (p⊃q)  ≡  (~p⊃~q)


T  T  T  T  T


T  F  F  F  T


F  T  T  F  F


F  F  T  T  T


Similarly, the fallacy of
negating the antecedent and the consequent negates both elements without switching them:
(p ⊃ q) ≡ (~p ⊃ ~q)
Again, the truthtable shows that these statementforms are not logically equivalent.
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Last modified 12 November 2011.
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