PL (propositional logic) syntax
Next chapter will be about semantics.
Syntax – a closed off system, regardless of the outside world - validity, proofs, derivations, etc.
Semantics – truth values of the premisses – how does language relate to the world? – focus on meaning.
Four letters: P, Q, R, S.
The prime: ‘. (P’, Q’, P’’, etc.)
Inference marker: (so, therefore)
Comma: ,.
Schematic variables are filled in with Greek letters.
- Every PL language has a finite number of atomic propositions.
Well-formed formulas (wff’s).
Parse tree.
Every wff has a unique parse tree.
P v Q ^ R – does not have a unique parse tree, so it’s not a wff. ‘(P v Q) ^ R’ – this if a wff tho.
Main connective – the connective at the top of the tree.
Subformula B of A is only a subformula occurs somewhere in the parse tree of A.
(P v Q) ^ R – the scope of the connective is the lowermost subformula in the parse tree.
Organized
Propositional Logic (PL): Syntax
Syntax vs. Semantics
Syntax: Concerns the structure and rules of the logical system, focusing on validity, proofs, and derivations. It operates as a closed system, independent of external references or the outside world.
Semantics: Focuses on truth values of premises, relating language to the world by emphasizing meaning.
Fundamental Elements
Atomic Propositions: Represented by letters (e.g., P, Q, R, S).
Prime Notation: Denoted by a prime symbol (e.g., P’, Q’, P’’) to represent modified versions of propositions.
Key Symbols
Inference Marker: Indicates logical progression, such as “so” or “therefore.”
Comma (,): Functions as a separator within statements.
Schematic Variables: Often represented with Greek letters, they act as placeholders within formulas.
Well-Formed Formulas (wff)
Definition: A valid arrangement