Propositional Logic Formal Syntax and Semantics. Computability and Logic

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1 Propositional Logic Formal Syntax and Semantics Computability and Logic

2 Syntax and Semantics Syntax: The study of how expressions are structured (think: grammar) Semantics: The study of the relationship between expressions and what they represent (think: meaning)

3 Propositional Logic Syntax and Grammar Propositional logic (sometimes called sentential logic or truth-functional logic) is the logic concerning propositions (statements, claims, sentences) Syntax: Atomic (individual) sentences (P, Q, R, etc) and combinations thereof (~P, P and Q, etc) Semantics: Assigning truth-values to atomic and complex sentences

4 Formal Syntax and Formal Semantics So far we have kept syntax and semantics rather informal But, in metalogic we want to prove things about logic This requires us to get really precise about syntax and semantics We are going to give syntax and semantics of propositional logic a mathematical treatment This is called formal syntax and formal semantics

5 Formal Syntax Sentences in propositional logic are linear expressions of symbols: Every atomic sentence (A, B, C,.. P, Q, R, ) is a sentence and are sentences With ϕ and ψ sentences: ϕ is a sentence (ϕ ψ) is a sentence (ϕ ψ) is a sentence (ϕ ψ) is a sentence (ϕ ψ) is a sentence Nothing else is a sentence

6 Proving Syntactical Properties of Our Language: A Simple Example Theorem (Parentheses Law): Every sentence has an equal amount of left and right parenthesis Proof: (by Mathematical (Structural) Induction) Base: Every atomic sentence, has 0 left and right parentheses. Same for and. Step: Suppose (inductive hypothesis) that ϕ and ψ both have equal amounts of left and right parentheses (say m and n respectively). Then: ϕ has equal amount (m) of left and right parentheses (ϕ ψ) has equal amount (m + n + 1) of left and right parentheses (ϕ ψ) has equal amount of left and right parentheses Etc. HW question: Prove by mathematical induction that every sentence is of finite length.

7 Formal Semantics Where L is the set of all syntactical sentences, a truth-assignment h is a function h:l {True, False} that satisfies the following conditions: h( ) = True h( ) = False h( ϕ) = True iff h(ϕ) = False h(ϕ ψ) = True iff h(ϕ) = True and h(ψ) = True h(ϕ ψ) = False iff h(ϕ) = False and h(ψ) = False h(ϕ ψ) = False iff h(ϕ) = True and h(ψ) = False h(ϕ ψ) = True iff h(ϕ) = True and h(ψ) = True or h(ϕ) = False and h(ψ) = False

8 Connection Between Truth-Tables and Formal Semantics The rows in a truth-table correspond to possible (classes of) truth-assignments The basic truth-tables for not, and, or, etc express their formal semantics Complex truth-tables reveal how truth-conditions of complex statements are a function of truth-conditions of component statements in accordance to the formal semantics of operators involved Truth-tables are more informal, and easier to read and use, certainly for specific sentences or arguments However, to prove general (metalogical) theorems about propositional logic, formal semantics often works better, since it can be hard to make general statements about truth-tables.

9 Defining Semantical Properties Using Formal Semantics: Some examples A sentence ϕ is a tautology iff there is no truthassignment h such that h(ϕ) = False Two sentences ϕ and ψ are equivalent iff for all truthassignments h: h(ϕ) = True iff h(ψ) = True. We write this as ϕ ψ A sentence ϕ implies sentence ψ iff there exists no truth-assignment h such that h(ϕ) = True and h(ψ) = False. We write this as ϕ ψ A set of sentences Γ = {ϕ 1,, ϕ n } implies a sentence ψ iff there exists no h such that h(ϕ i ) = True for all sentences ϕ i in Γ and h(ψ) = False. We write: Γ ψ

10 Some Metalogical Theorems Regarding Semantical Properties Theorem: ϕ is a tautology iff ϕ is a contradiction Proof: ϕ is a tautology iff (definition tautology) there is no h such that h(ϕ) = False iff (semantics ) there is no h such that h( ϕ) = True iff (definition contradiction) ϕ is a contradiction

11 Another Example Theorem: For any statement ϕ: ϕ (i.e. a contradiction implies anything) Proof: Take any statement ϕ and any truthassignment h. By semantics of : h( ) = False. So, by definition of implication, implies ϕ, i.e. ϕ.

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