Truth-Functional Definitions

Now that we've learned how to construct truth-tables, we can begin using them to test whether sentences and arguments have certain properties.
Remember the definitions we learned early in the semester. (Those definitions were somewhat imprecise, and we will be gradually refining them with more precise analogues. Truth-tables allow us to make our first such refinements.)

Truth-functional truth, falsity and indeterminacy:

A sentence p of SL is truth-functionally true if and only if p is true on every truth-value assignment.
A sentence p of SL is truth-functionally false if and only if p is false on every truth-value assignment
A sentence p of SL is truth-functionally indeterminate if and only if p is neither truth-functionally true nor truth-functionally false. (That is, if p is true on at least one truth-value assignment and false on at least one truth-value assignment.)

To test whether a sentence is truth-functionally true, false or indeterminate we simply construct a truth-table for the sentence and then look at the column under the main connective. If that column contains only T's, the sentence is truth-functionally true. If that column contains only F's, the sentence is truth-functionally false. If that column contains both T's and F's, the sentence is truth-functionally indeterminate.

Truth-functional equivalence:

Sentences p and q of SL are truth-functionally equivalent if and only if there is no truth value assignment on which p and q have different truth-values.

To test whether two sentences are truth-functionally equivalent we simply construct a truth-table for the sentences and then look at the columns under the main connectives. If those columns are exactly the same (if they agree on every row) then the sentences are equivalent. If not (if there is at least one row where the one sentence has a T and the other has an F, or vice versa) then the sentences are not equivalent.

Truth-functional consistency:

A set of sentences of SL is truth-functionally consistent if and only if there is at least one truth-value assignment on which all the members of the set are true.
A set of sentences of SL is truth-functionally inconsistent if and only if it is not consistent.(That is, if there is no truth-value assignment on which all the members of the set are true.)

To test whether a set of sentences is truth-functionally consistent or inconsistent we simply construct a truth-table for the sentences and then look at the columns under the main connectives. If there is at least one row where every sentence is true (has a T under the main connective) then the set is consistent. If not (if on every row at least one of the sentences is false) then the set is inconsistent.

Truth-functional entailment:

A set G of sentences of SL truth-functionally entails a sentence p if and only if there is no truth-value assignment on which every member of G is true and p is false.

Truth-functional validity:

An argument of SL is truth-functionally valid if and only if there is no truth-value assignment on which all of the premises are true and the conclusion is false.
(Put another way, an argument is truth-functionally valid if and only if the premises truth-functionally entail the conclusion.)

Truth-functional invalidity:

An argument of SL is truth-functionally invalid if and only if it is not truth-functionally valid.
(That is, an argument is truth-functionally invalid if and only if there is some truth-value assignment on which all of the premises are true and the conclusion is false.)

To test whether an argument is truth-functionally valid or invalid we simply construct a truth-table for the argument and then look at the columns under the main connectives of the premises and conclusion. If there is no row where the premises are all true (have T's under the main connective) and the conclusion false (has an F under the main connective) then the argument is valid. If there is one (or more) rows where the premises are all true and the conclusion is false, then the argument is invalid.