A Class-Room Introduction to Logic

May 4, 2009

Unit-XVIII: Statement Forms

Filed under: Statement Forms — Dr. Desh Raj Sirswal @ 5:45 am

Statement: A proposition; what is typically asserted by a declarative sentence, but not the sentence itself. Every statement must be either true or false, although the truth or falsity of a given statement may be unknown.

Statement Forms: Any sequence of symbols containing statement variables but no statements, such that when statements are consistently substituted for the statement variables, the result is a statement.

In exactly the same sense that individual arguments may be substitution-instances of general argument forms, individual compound statements can be substitution-instances of general statement forms. In addition, just as we employ truth-tables to test the validity of those arguments, we can use truth-tables to exhibit interesting logical features of some statement forms.

Tautology

A statement form whose column in a truth-table contains nothing but Ts is said to be tautologous. Consider, for example, the statement form:

pÉp

p

p

pÉ p

T

T

T

F

F

T

Notice that whether the component statement is true or false makes no difference to the truth-value of the statement form; it yields a true statement in either case. But it follows that any compound statement which is a substitution-instance of this form—no matter what its content—can be used only to make true assertions.

Contradiction

A statement form whose column contains nothing but Fs, on the other hand, is said to be self-contradictory. For example:

p ~p

p

~p

p ~p

T

F

F

F

T

F

Again, the truth-value of the component statement doesn’t matter; the result is always false. Compound statements that are substitution-instances of this statement form can never be used to make true assertions.

Contingency

Of course, most statement forms are neither tautologous nor self-contradictory; their truth-tables contain both Ts and Fs. Thus:

p É ~p

p

~p

p É ~p

T

F

F

F

T

T

Since the column underneath it in the truth-table has at least one T and at least one F, this statement form is contingent. Statements that are substitution-instances of this statement form may be either true or false, depending upon the truth-value of their component statements.

Assessing Statement Forms

Because all five of our statement connectives are truth-functional, the status of every statement-form is determined by its internal structure. In order to determine whether a statement form is tautologous, self-contradictory, or contingent, we simply construct a truth-table and inspect the appropriate column.  

For Exercise:

  1. (pÉq)≡p
  2. (p≡q) É q
  3. (pÉq)≡ (pÉq)
  4. (pÉq) · (qÉp) É q
  5. (pÉq)≡ (qÉp) · q

NOTE : É  represents implication and /\ represents disjunction and Ú represents conjunction.

 

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