Rules of Logic and Inference Flashcards

Vocabulary flashcards covering the formal rules of inference, logical notations, general symbolic forms, and descriptions from the logic lecture notes.

Conditional Elimination ($\rightarrow\text{E}$)

If the IF part is true, the THEN part must be true.

Follow the arrow.

If it rains → ground is wet. It rained → ground is wet.

A rule of inference stating that if a conditional ($p \rightarrow q$) and its antecedent ($p$) are true, you can conclude the consequent ($q$). General form: $p \rightarrow q$, $p \therefore q$.

Conditional Elimination ($\rightarrow E$)

A rule of inference stating that if a conditional statement ($p \rightarrow q$) and its antecedent ($p$) are both true, the consequent ($q$) must be true.

Modus Tollens MT

If the THEN part is false, the IF part must be false.

Deny the result → deny the cause.

If it rains → ground is wet. Ground is NOT wet → it did NOT rain.

A rule of inference stating that if a conditional ($p \rightarrow q$) and the negation of its consequent ($\neg q$) are true, you can conclude the negation of the antecedent ($\neg p$). General form: $p \rightarrow q$, $\neg q \therefore \neg p$.

Conditional Introduction ($\rightarrow\text{I}$)

Two conditionals can be chained when the middle statement matches.

Make a chain: p → q → r.

If I study → pass. If I pass → celebrate. Therefore, if I study → celebrate.

A rule of inference stating that if two conditionals connect with the consequent of one matching the antecedent of the other ($p \rightarrow q$ and $q \rightarrow r$), you can conclude a new conditional from the first antecedent to the second consequent ($p \rightarrow r$). General form: $p \rightarrow q$, $q \rightarrow r \therefore p \rightarrow r$.

Conjunction Elimination ($\wedge\text{E}$)

An AND statement lets you take out either part.

AND → take one part out.

I have a dog AND a cat → I have a dog.

A rule of inference stating that if a conjunction ($p \wedge q$) is true, either individual conjunct ($p$ or $q$) can be concluded. General form: $p \wedge q \therefore p$.

Conjunction Introduction ($\wedge\text{I}$)

Two true statements can be joined with AND.

Two things → put AND between them.

I have a dog. I have a cat. → I have a dog AND a cat.

General form: $p$, $q \therefore p \wedge q$.

A rule of inference stating that if two sentences ($p$ and $q$) are true, their conjunction ($p \wedge q$) is also true.

Disjunctive Syllogism. DS

An OR statement plus eliminating one option leaves the other option.

OR + eliminate one → get the other.

Pizza OR tacos. NOT pizza → tacos.

General form: $p \vee q$, $\neg p \therefore q$.

A rule of inference stating that if a disjunction ($p \vee q$) is true and one disjunct is false ($\neg p$), the other disjunct ($q$) must be true.

Disjunction Introduction ($\vee\text{I}$)

If one statement is true, you can make an OR statement by adding another possibility.

One true thing → add OR anything.

I have a dog → I have a dog OR a cat.

A rule of inference stating that if one sentence ($p$) is true, it can be combined with any other sentence using OR to form a disjunction ($p \vee q$). General form: $p \therefore p \vee q$.

DeMorgan’s Laws DeM

When a NOT moves inside parentheses, put NOT on each statement and flip AND/OR.

NOT goes in → flip the connector.

NOT (cake OR ice cream) → NOT cake AND NOT ice cream.

Rules stating that when pushing a negation inward, flip the logical operator: NOT (OR) becomes AND of NOTs ($\neg(p \vee q) \leftrightarrow \neg p \wedge \neg q$), and NOT (AND) becomes OR of NOTs ($\neg(p \wedge q) \leftrightarrow \neg p \vee \neg q$).