PHIL222 Revision

Atomic Proposition

The simplest form of a statement in logic that cannot be broken down further.

Logical Connectives

Operators that combine atomic propositions to form complex logical expressions.

Negation (¬)

An operator that reverses the truth value of a proposition.

Conjunction (∧)

An operator that yields true only if both connected propositions are true.

Disjunction (∨)

An operator that yields true if at least one of the propositions is true.

Implication (→)

Indicates that if one proposition is true, then another must also be true.

Biconditional (↔)

Holds true if both propositions are either true or false together.

Well-Formed Formula (WFF)

A valid expression in propositional logic based on specific construction rules.

Rules for WFFs

Guidelines for constructing valid logical expressions.

Valid WFF Example

(A → ¬B) suggests that if A holds, then B must not hold.

Invalid WFF Example

((A ∧ B)∧)A is malformed due to improper placement of brackets.

Translation Patterns

Key structures used for translating natural language into logic.

"Only if" Translation

Translates to Q → P.

"If and only if" Translation

Translates to P ↔ Q.

Truth Table

A method for systematically organizing truth values of logical propositions.

Argument Validity

An argument is valid if there is no case where premises are true and conclusion is false.

Semantic Trees

A method for visualizing logical relationships and testing validity.

Universal Quantifier (∀)

Quantifies over all objects in a domain, indicating a universal condition.

Existential Quantifier (∃)

Denotes that at least one object in the domain satisfies the predicate.

Monadic Predicate Logic

Focuses on properties of a single object rather than relationships between multiple objects.

Polyadic Predicates

Predicates involving two or more arguments.

Countermodel

A model that demonstrates the falsity of an argument or statement.

Translation Example: "Bill heard Alice"

Translates to Hba, indicating a relation of hearing.

Translation Example: "Clare is taller than Dave"

Translates to Tcd, indicating a comparison.

Logical Structure

The arrangement of terms and predicates within logical statements.

Negation in Semantic Trees

Used to indicate that a proposition's truth value is reversed.

Closure in Trees

Occurs when all branches of a semantic tree are satisfied, confirming validity.

Semantic Analysis

The evaluation of the meaning and validity of logical expressions.

Tautology

A statement that is always true regardless of the truth values of its components.

Model Theory

Study of the relationships between formal languages and their interpretations.

Translation Nuance

Consideration for word order and quantifier placement in translations.

Predicate Extension

The specific subset in a domain where a predicate holds true.

Predicate Logic Scope

Focus on quantifier effectiveness within logical statements.

Counterexample Identification

The process of finding examples that prove a statement false