**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 **T**s 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 `p ` 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 **F**s, on the other hand, is said to be self-contradictory. For example:

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 **T**s and **F**s. 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:**

- (pÉq)≡p
- (p≡q) É q
- (pÉq)≡ (pÉq)
- (pÉq)
**·**(qÉp) É q - (pÉq)≡ (qÉp)
**·**q

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

** **

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