PHIL222 - Truth Trees for PL L5

Tree in Propositional Logic

A graphical representation that helps determine if a set of propositions is satisfiable.

Satisfiability

The condition in which at least one truth assignment makes all propositions true simultaneously.

Truth Table Limitations

Truth tables grow exponentially and are inefficient for large sets of propositions, making them unsuitable for predicate logic semantics.

Advantages of Trees

Trees are simpler, faster, and provide a visual representation of logical structures, aiding in comprehending interactions between propositions.

Tree Rules

Rules based on truth tables for logical connectives that help break down logical propositions into simpler forms.

Negation Rule (¬)

If ¬¬α is true, then α must be true, confirming that double negation yields the original proposition.

Conjunction Rule (∧)

If α ∧ β is true, then both α and β must individually be true.

Disjunction Rule (∨)

For α ∨ β to be true, at least one of α or β needs to be true.

Conditional Rule (→)

If α → β is true, then if α is true, β must also be true.

Biconditional Rule (↔)

If α ↔ β holds, then both α and β are either true or false together.

Path Closure

A path through the tree is closed if it contains both a formula and its negation, indicating inconsistency.

Application of Rules on Open Paths

Rules can apply to multiple open paths to ensure consistency in logical deductions.

Efficiency in Trees

Prioritizing non-branching rules first tends to result in fewer overall applications and enhances clarity.