Propositional Logic

Tags: #◇Conceptual #◇Propositional Logic #◇CCDSTRU #◇M1

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◇Key Definitions

Propositional Logic (or Propositional Calculus): The area of logic dealing with propositions, developed by Aristotle. Proposition: A declarative sentence that is either true or false, but not both. (e.g., "the university is located in Manila.") Compound Proposition: Formed by combining one or more existing propositions using logical operators. Propositional Variables: Letters used to represent propositions, conventionally lowercase (e.g., $p$, $q$, $r$). Logical Connectives: Operators used to form compound propositions.

Logical Connectives Negation (Not, $\neg p$): The opposite truth value of $p$. "It is not the case of $p$." Conjunction (And, $p \land q$): True if and only if both $p$ and $q$ are true. Disjunction (Inclusive-Or, $p \lor q$): True when either $p$ or $q$ is true. Exclusive-Or (Xor, $p \oplus q$): True when either $p$ or $q$ is true, but not both. Implication (If-Then, $p \rightarrow q$): Read as "if $p$ then $q$". False only when $p$ is true and $q$ is false. Here, $p$ is the hypothesis/premise, and $q$ is the conclusion/consequence. Biconditional (If and only if, $p \leftrightarrow q$): True when both $p$ and $q$ have the same truth value.

◇Theoretical Framework

Implication Variations

Given an implication statement $p \rightarrow q$: Converse: $q \rightarrow p$ Inverse: $\neg p \rightarrow \neg q$ Contrapositive: $\neg q \rightarrow \neg p$

[!tip] Translation Tips for Implication ($p \rightarrow q$)

• $p$ is sufficient for $q$
• $q$ is necessary for $p$
• $q$ when $p$
• $q$ whenever $p$
• $p$ only if $q$
• $q$ unless $\neg p$
• $q$ follows from $p$

Truth Tables Summary

◇References & Navigation

• Related Notes: ◇Propositional Logic