Chapter Three
Truth Tables
1. Computing Truth-Values
• We can use truth tables to determine the truth-value of any
compound sentence containing one of the five truthfunctional sentence connectives.
• This method can also be used to determine the truth-value
of more complicated sentences.
• This procedure is called truth table analysis.
2. Logical Form
• The assignment of a truth-value to a compound sentence
from the truth-values of its atomic constituents is called a
valuation.
• Expressions that contain only sentence variables and
sentence connectives are called sentence forms.
• If we replace sentence variables with sentence constants
we end up with a substitution instance of the original
sentence form.
• Logical form is not the same as logical equivalence.
3. Tautologies, Contradictions,
and Contingent Sentences
• A sentence that is true in virtue of its logical form is a
tautology.
• Contradictions are sentences that cannot possibly be true.
• The form of a contradiction is a contradictory sentence
form.
Tautologies, Contradictions, and
Contingent Sentences, continued
• A contingent statement is one whose form has at least one
T and one F in its truth table.
4. Logical Equivalences
When two statements are logically equivalent, the truth-value
of one determines the truth-value of the other. That is, each
has the same truth-value under the same truth conditions.
5. Truth Table Test of Validity
• We can use truth tables to determine if any argument in
sentential logic is valid.
• Recall: An argument is valid if and only if it is not possible
for its premises to all be true while its conclusion is false.
• So, if an argument is valid there will be no line in a truth
table in which all the premises are true and the conclusion
false.
Truth Table Test of Validity, continued
We can test an argument for validity by conjoining the
premises into the antecedent of a conditional, putting the
conclusion as the consequent, and testing its form to see if
it is a tautological form. If it is, the argument is valid.
Truth Table Test of Validity, continued
If the corresponding conditional, or test statement form, of an
argument is a tautology, then premises are said to logically
imply or entail the conclusion of the argument.
Truth Table Test of Validity, continued
A logical implication is a tautology who main connective is a
horseshoe.
A counterexample is an assignment of truth-values that will
yield true premises and a false conclusion.
6. Truth Table Test of
Consistency
We can use a truth table to check for consistency by
constructing the truth table for the conjunction of the forms
and looking for a line on which all the substitutions are
true. The set is consistent if and only if there is such a line.
7. Validity and Consistency
• The counterexample set of an argument consists of the
premises of the argument together with the denial of the
conclusion.
• If the counterexample set is consistent then the argument is
invalid.
• All arguments with inconsistent premises are valid.
8. The Short Truth Table Test for
Invalidity
• All it takes to show that an argument is invalid is a single
counterexample—a single line of a truth table on which the
premises are all true and the conclusion false.
• It is often possible to produce such a counterexample by
assigning a truth-value to the entire sentence and then
working to find the appropriate assignment of truth-values
to the atomic constituents.
The Short Truth Table Test for
Invalidity, continued
• We find the logical form of the argument, then assign an F
to the conclusion and the try to assign a T to each premise.
• If we can do this the argument is invalid.
9. The Short Truth Table Test for
Consistency
All it takes to show that a set of sentences is
consistent is to produce a single line in a truth table
that makes them all true.
10. A Method of Justification for
the Truth Tables
We can build the truth table for any connective given the
following information:
1) A set of intuitively valid and invalid arguments
2) In a valid argument, if the premises are true the
conclusion must be true
3) In an invalid argument, there must be the possibility that
the premises are true and the conclusion false
A Method of Justification for the Truth
Tables, continued
To justify a truth table, find valid and invalid argument forms,
using the connective in question, that force the lines of the
truth table.
Key Terms
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Contingent statement
Contingent sentence form
Contradiction
Contradictory sentence form
Corresponding conditional of an argument
Counterexample of an argument
Counterexample set
Logical equivalence
Logically equivalent
Logical implication
Key Terms, continued
• Logically implies
• Sentence form
• Substitution instance
• Tautologous sentence form
• Tautology
• Test statement form
• Truth table analysis
• Valuation
Key Terms, continued
• Truth-function
• Truth-functional operator
• Truth table
• Truth-value
• Wedge
• Well-formed