Introduction to Logic Exam 1 Flashcards

Flashcards covering truth table definitions, evaluation of logic statement forms, tautology and contradiction classifications, logical implications, short truth table validity checks, and argument symbolizations from Dr. Mark Vopat's Exam 1.

Logically Equivalent Statements

A pair of statements that have the exact same truth value as each other under every possible truth value assignment of their component variables (they share identical truth table columns).

Consistent Set of Statements

A set of statements for which there exists at least one truth value assignment (one row in a joint truth table) in which all statements in the set are simultaneously true.

Tautology

A statement form that evaluates to true under all possible truth value assignments of its component variables (true in every row of its truth table).

Contradiction

A statement form that evaluates to false under all possible truth value assignments of its component variables (false in every row of its truth table).

Contingency

A statement form that is true under at least one truth value assignment and false under at least one truth value assignment (containing a mix of true and false values in its main column).

Logical Implication

A relationship where statement (1) logically implies statement (2) if and only if there is no truth value assignment under which statement (1) is true and statement (2) is false.

Short Truth Table Method

A technique for testing argument validity by attempting to assign truth values that make all premises true and the conclusion false; if a consistent assignment is found, the argument is invalid, otherwise it is valid.

Truth Value of $(p \lor q) \supset [(p \cdot q) \supset (r \cdot s)]$

Given $p=T$, $q=F$, $r=T$, and $s=T$: $(T \lor F) \supset [(T \cdot F) \supset (T \cdot T)]$ evaluates to $T \supset [F \supset T]$, which yields $T \supset T = T$ (True).

Truth Value of $(p \supset q) \supset [(p \cdot \sim r) \equiv (\sim r \lor q)]$

Given $p=T$, $q=F$, $r=T$, and $s=T$: $(T \supset F) \supset [(T \cdot F) \equiv (F \lor F)]$ evaluates to $F \supset [F \equiv F]$, which yields $F \supset T = T$ (True).

Classification of $[p \supset (q \supset r)] \supset [(p \supset q) \supset (p \supset r)]$

Tautology. This statement form represents the principle of distribution of implication and evaluates to true under all 8 possible truth value assignments for $p$, $q$, and $r$.

Classification of $[p \supset (q \supset p)] \supset [(q \supset q) \supset \sim (r \supset r)]$

Contradiction. Since $p \supset (q \supset p)$ and $q \supset q$ are both tautologies ($T$), and $\sim(r \supset r)$ is a contradiction ($F$), the overall form evaluates to $T \supset (T \supset F) = F$ in all cases.

Logical Implication of (1) $(p \supset q)$ vs (2) $(p \supset q) \lor r$

Statement (1) implies statement (2) (Option a). Disjoining any statement $r$ to $(p \supset q)$ preserves truth whenever $(p \supset q)$ is true.

Logical Implication of (1) $(p \cdot q) \lor \sim r$ vs (2) $p \cdot (q \lor \sim r)$

Statement (2) implies statement (1) (Option b), and statement (1) does not imply statement (2) (Option c). Expanding statement (2) via distribution gives $(p \cdot q) \lor (p \cdot \sim r)$, which is strictly stronger than $(p \cdot q) \lor \sim r$.

Validity of Argument 9: Premises $((p \cdot q) \supset r)$, $(r \supset (s \cdot t))$, Conclusion $\sim s \supset (\sim p \lor \sim q)$

Valid. Setting the conclusion to false requires $\sim s = T$ ($s = F$) and $\sim p \lor \sim q = F$ ($p = T, q = T$). Premise 1 then requires $r = T$, causing premise 2 to evaluate to $T \supset (F \cdot t) = F$, which creates an unavoidable contradiction.

Validity of Argument 10: Premises $(p \cdot q) \supset (r \supset (s \lor t))$, $s \equiv (p \cdot t)$, $\sim t \equiv (q \lor \sim r)$, Conclusion $r \supset (s \lor \sim p)$

Valid. Setting the conclusion to false requires $r = T$, $s = F$, and $p = T$. Propagating these values through the three premises leads to an impossible truth value assignment, proving the argument is valid.

Symbolization and Validity of Problem 11 (Rationality and Determinism)

Symbolization: Let $P$ = people are entirely rational, $A$ = all of a person's actions can be predicted in advance, $D$ = universe is essentially deterministic. Premises: $P \supset (A \lor D)$, $\sim A$. Conclusion: $\sim D \supset \sim P$. Status: Valid.

Symbolization and Validity of Problem 12 (Oil Consumption and Bankruptcy)

Symbolization: Let $C$ = oil consumption continues to grow, $I$ = oil imports increase, $R$ = domestic oil reserves depleted, $B$ = nation will soon be bankrupt. Premises: $C \supset (I \cdot R)$, $(I \lor R) \supset B$. Conclusion: $C \supset B$. Status: Valid.