Lecture_15_09 - Lecture 15 Lecture 15 Syntax Syntax...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Lecture 15 Lecture 15 Syntax Syntax Signature: constant symbols, relation symbols, e.g. H (unary), R (binary), c (constant). Terms: C (constants) and V (variables), and more if we allow function symbols. Language is defined inductively Base case: atoms built from terms and relation symbols, e.g. R(c,x) and H(x). How about R(H(x),y)? Syntax Syntax Inductive clauses (one for each connective): If ϕ ∈L, then ¬ ϕ ∈L If ϕ ,ψ∈L, then ϕ ∧ L ψ∈ If ϕ ∈L and x ∈V, then ∀x ϕ ∈L If ϕ ∈L and x ∈V, then ∃ x ϕ ∈L Bound and Free Variables Bound and Free Variables Scope Bound occurrences Free occurrences Sentences Definability Definability If M is an interpretation and ϕ is a formula with x as its only free variable, then ϕ defines the following subset of D: {a ∈ D : M ╞ ϕ (a)} Alert: a ∈ D need not be a symbol in our first­order language. So, if ‘ϕ (a)’ is not a wff, then what is meant by M ╞ ϕ (a)? This idea extends to subsets of Dn. Axiomatizability Axiomatizability If ϕ is a sentence in L, then ϕ partitions the class of all L­interpretations into those interpretations that satisfy ϕ and those that do not. If S is a collection of L­interpretations, then S is axiomatizable if there is a collection of sentences Γ ⊆ L such that Mod(Γ) = S. ...
View Full Document

Ask a homework question - tutors are online