Chapter 2: Syntax and Symbolization
Sentential logic uses a formal language to represent logical relationships. Its syntax defines the rules for forming well-structured formulae.
Constructing and Identifying Formulae
A formula in sentential logic might be: (P ∧ Q) → R. You can construct such formulae using atomic propositions (P, Q, R) and logical connectives (¬, ∧, ∨, →, ↔).
Parse Trees
Parse trees break down a formula into its components, showing the hierarchical structure of logical operations. They help clarify how complex statements are built from simpler parts.
Logical Structure of English Sentences
We can discern logical structure by identifying connectives in natural language. For example:
"If it rains, then the ground will be wet" translates to R → W.
Grammar of Sentential Logic
Grammar rules specify that atomic sentences are well-formed formulae, and more complex formulae can be built using connectives, parentheses, and existing formulae.
Symbolizing English Sentences
Example: "Either Alice will study or she will fail the test" → S ∨ F.
Chapter 3: Semantics
Semantics is the study of meaning in sentential logic. It focuses on truth-values and how they are assigned to formulae.
Truth-Tables
Truth-tables show the truth-value of a formula for all possible truth-value assignments of its atomic components.
Truth-Value Assignments
A truth-value assignment gives each atomic proposition a value: True (T) or False (F). This determines the truth of the entire formula.
Tautological, Contingent, and Contradictory Formulae
- Tautology: Always true (e.g.,
P ∨ ¬P). - Contradiction: Always false (e.g.,
P ∧ ¬P). - Contingent: True in some cases, false in others.
Counterexamples
To show an argument is invalid, find a truth-value assignment where all premises are true but the conclusion is false.
Truth-Conditions for Connectives
- Negation (¬P): True if P is false.
- Conjunction (P ∧ Q): True if both P and Q are true.
- Disjunction (P ∨ Q): True if at least one is true.
- Conditional (P → Q): False only if P is true and Q is false.
- Biconditional (P ↔ Q): True if P and Q have the same truth-value.
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