PHIL222 - In Class Test 1

Formation Rules

Explicit rules for constructing well-formed formulas (wffs) in formal languages.

Basic Propositions

Single capital letters: A, B, C, …, Z. Extended notation can be A₁, B₂, C₃ if more are needed.

Atomic Propositions

Propositions with no internal logical structure (e.g., 'A: It is raining').

Logical Connectives

Symbols that connect propositions, resulting in a new proposition.

Negation (¬)

Logical connective meaning 'not'; e.g., ¬A indicates 'It is not raining'.

Conjunction (∧)

Logical connective meaning 'and'; e.g., A ∧ B indicates 'It is raining and cold'.

Disjunction (∨)

Logical connective meaning 'or' (inclusive); e.g., A ∨ B indicates 'It is raining or cold'.

Conditional (→)

Logical connective meaning 'if…then'; e.g., A → B indicates 'If it rains, then it is wet'.

Biconditional (↔)

Logical connective meaning 'if and only if'; e.g., A ↔ B indicates 'It rains iff it is wet'.

Well-Formed Formulas (WFFs)

Constructed formulas that adhere to specific syntactical rules.

Examples of WFFs

P, ¬Q, (P ∧ Q), (¬P ∨ (Q → R)).

Non-WFFs

Invalid constructions; e.g., P Q R (missing connectives), ((P ∧ Q)) (extra parentheses), (P → ↔ Q) (ill-formed).

Importance of Parentheses

Determines scope and logical structure of propositions.

Main Connective

The last connective applied in the construction of a formula.

Example of Main Connective

In (¬P ∧ (Q ∨ R)), the main connective is ∧.

Truth Tables

A tabular method for representing the truth values of logical connectives.

Truth Table for Negation (¬)

Shows output for negation.

Truth Table for Conjunction (∧)

Shows output for conjunction.

Truth Table for Disjunction (∨)

Shows output for disjunction.

Truth Table for Conditional (→)

Shows output for conditional.

Truth Table for Biconditional (↔)

Shows output for biconditional.

Evaluating Complex Propositions

Examples for evaluating propositions using truth values.

Logical Properties

Characteristics of logical propositions: tautology, contradiction, contingent, satisfiable, valid.

Tautology

A proposition that is always true, e.g., P ∨ ¬P.

Contradiction

A proposition that is always false, e.g., P ∧ ¬P.

Validity Testing

Process of determining if an argument is valid using truth tables.

Sound Argument

An argument that is valid and has true premises.

Truth Trees

A method for testing validity and satisfiability using a diagrammatic representation.

Why Use Trees over Truth Tables

Trees are more efficient for complex arguments; visual branching helps track logical dependencies.

Tree Rules for Negation (¬α)

Branch to ¬α if true; branch to α if false.

Tree Rules for Conjunction (α ∧ β)

Write α and β if true; branch to ¬α or ¬β if false.

Tree Rules for Disjunction (α ∨ β)

Branch to α and β if true; write ¬α and ¬β if false.

Tree Rules for Conditional (α → β)

Branch to ¬α or β if true; write α and ¬β if false.

Tree Construction Steps

1. Write premises + negation of conclusion at the top. 2. Apply non-branching rules. 3. Check for closure.

Testing Validity with Trees

Write premises and negation of conclusion for validity testing.

Applications of Trees

Validity testing, satisfiability, tautology check, equivalence testing.

Summary of Key Concepts

Overview of the main concepts related to propositional logic.

Logical Relationships

Different types of relationships that can exist between propositions.

Shortcuts for Efficient Evaluation

Tips for quickly assessing truth values of propositions.

Conclusion

Overall summary of propositional logic concepts.