6/9 6. Testing Arguments with Truth Tables

Arguments, validity, and soundness

A deductive argument has one or more premises followed by a conclusion. Its logical form can be represented schematically as:

• Premise 1

• Premise 2

• Therefore, conclusion

For example, let $p$ mean “the alarm is set” and $q$ mean “the light flashes.” The argument “If the alarm is set, the light flashes; the alarm is set; therefore, the light flashes” has the form:

An argument is valid when no possible valuation makes all its premises true and its conclusion false. Validity concerns the connection between premises and conclusion, not whether the statements are factually true. An argument is sound only when it is both valid and composed entirely of true premises.

Takeaway: Truth tables test validity. They do not, by themselves, establish soundness or the factual truth of individual propositions.

Propositional connectives

Truth-table construction begins by translating statements into propositional connectives. The principal connectives are:

• Negation: $\neg p$, meaning “not $p$.” It is true exactly when $p$ is false.

• Conjunction: $p \land q$, meaning “$p$ and $q$.” It is true only when both propositions are true.

• Disjunction: $p \lor q$, meaning “$p$ or $q$.” It is true when at least one proposition is true.

• Conditional: $p \to q$, meaning “if $p$, then $q$.” It is false only when $p$ is true and $q$ is false.

• Biconditional: $p \leftrightarrow q$, meaning “$p$ if and only if $q$.” It is true when $p$ and $q$ have the same truth value.

The conditional requires special attention: when its antecedent is false, the conditional is true under the truth-functional interpretation used in propositional logic.

Takeaway: Translate consistently, and remember that a conditional has only one false combination: a true antecedent with a false consequent.

Constructing a complete truth table

To construct a complete truth table:

1. Identify every basic proposition.

2. Count the basic propositions. If there are $n$, create $2^n$ rows.

3. List every possible assignment of true and false values.

4. Add columns for intermediate expressions.

5. Evaluate the final expression one connective at a time.

With two basic propositions, such as $p$ and $q$, the table has $2^2 = 4$ rows. With three basic propositions, it has $2^3 = 8$ rows. Each row is a valuation: one possible way the basic propositions could be true or false.

For example, the four valuations for $p$ and $q$ are:

• $p$ true, $q$ true

• $p$ true, $q$ false

• $p$ false, $q$ true

• $p$ false, $q$ false

Intermediate columns reduce errors because each compound expression is calculated from values already established in earlier columns.

Takeaway: A complete table is exhaustive: every possible valuation must appear exactly once.

The joint-premise method

The joint-premise method converts an argument into one conditional. For premises $P_1$, $P_2$, and conclusion $C$, construct:

For three premises, construct:

The argument is valid exactly when this final conditional is a tautology. In a truth table, that means the final column contains only $\mathrm{T}$.

Consider:

The corresponding test is:

The final conditional is true on every valuation, so the argument is valid. Rows where the conditional premise $p \to q$ is false do not threaten validity, because those rows do not make all the premises true.

Takeaway: In the joint-premise test, an all-true final column establishes validity.

Countervaluations and invalid arguments

To disprove an argument, look for a countervaluation: a row in which every premise is true and the conclusion is false. One such row is enough to establish invalidity.

Consider:

Set the conclusion $p$ to false. To make the first premise $p \to q$ true while $p$ is false, set $q$ to true. The resulting assignment is:

• $p$ is false.

• $q$ is true.

• $p \to q$ is true.

• The premise $q$ is true.

• The conclusion $p$ is false.

Therefore, this assignment is a countervaluation and the argument is invalid. The form illustrates affirming the consequent: observing $q$ does not prove $p$, because $q$ might have another cause.

Takeaway: A true-premises/false-conclusion row is decisive evidence of invalidity.

Interpreting results and checking your work

A complete table is systematic, but a partial truth-table test can be faster when invalidity is likely:

1. Set the conclusion to $\mathrm{F}$.

2. Try to assign values that make every premise $\mathrm{T}$.

3. If the assignments are consistent, a countervaluation has been found and the argument is invalid.

4. If no consistent assignment works, the argument is valid.

When reading a completed table, focus only on rows where all premises are true:

• If every such row has a true conclusion, the argument is valid.

• If at least one such row has a false conclusion, the argument is invalid.

• Rows with one or more false premises do not refute validity.

Truth tables evaluate propositional structure. If a whole quantified statement, such as “Every student submitted the assignment,” is represented by one letter such as $p$, the table can test how $p$ combines with other propositions. It cannot determine the internal logic of “every” or “some.” Quantifiers such as $\forall x$ and $\exists x$ require predicate-logic methods involving predicates, variables, and domains.

Before finalizing a test, verify that you have identified all basic propositions, used $2^n$ rows when making a complete table, included needed intermediate columns, combined premises with $\land$, and distinguished validity from soundness and factual truth.

Takeaway: Judge validity by the relevant rows, not by rows in which a premise is already false.