10-534-52006-5.pdf by ADSD-MAC-005

Author:ADSD-MAC-005 [ADSD-MAC-005]
Language: eng
Format: epub
Published: 2011-01-07T03:04:22.557000+00:00

words, the truth value of the compound proposition is ‘‘contingent’’ upon the truth values of its components. Sometimes it is true, sometimes false, depending on the truth values of the components.

On the other hand, consider the following truth tables:

[ ( G ⊃ H ] • G ] ⊃ H

( G v H ) ≡ (ϳ G • ϳ H )

T T T

T T

T T

T T T

F

F T F F T

T F F

F T

T F

T T F

F

F T F T F

F T T

F F

T T

F T T

F

T F F F T

F T F

F F

T F

F F F

F

T F T T F

The proposition on the left is tautologous (logically true or a tautology) because the column under the main operator is all true. The one on the right is self-contradictory (logically false) because the main operator column is all false. In neither case is the truth value of the compound proposition contingent upon the truth values of the components. The one on the left is true regardless of the truth values of its components—in other words, necessarily true. The one on the right is necessarily false.

If a proposition is either logically true or logically false, its truth value depends merely upon its form and has nothing to do with its content. As a result, such statements do not make any genuine assertions about things in the world. For example, the tautologous statement ‘‘It is either raining or it is not raining’’ provides no information about the weather. Similarly, the self-contradictory statement ‘‘It is raining and it is not raining’’ provides no information about the weather. On the other hand, the contingent statement ‘‘It is raining in the mountains’’ does provide information about the weather.

Comparing Statements

Truth tables may also be used to determine how two propositions are related to each other. Two propositions are said to be logically equivalent if they have the same truth value regardless of the truth values of their simple components, and they are contradictory if they have opposite truth values regardless of the truth values of their simple components. Two (or more) propositions are consistent if there is at least one line on which both (or all) of them turn out to be true, and they are inconsistent if there is no line on which both (or all) of them turn out to be true. By comparing the main operator columns, one can determine which is the case. However, because the first two relations are stronger than (and may overlap) the second two, the first two relations should be considered first.

Columns under

Relation

main operators

logically equivalent

same truth value on each line

contradictory

opposite truth value on each line

consistent

there is at least one line on which

S

the truth values are both true

N

inconsistent

there is no line on which the truth values are both true

L

Truth Tables for Propositions

329

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