MATH 3265 Class Notes 01-29-2025

notelogic

The semantics for a formal language is a way to assign meaning to the expressions in the language.

Valuation

Definition: a valuation is a function $V:\{v_i\mid i\in \mathbb{N}\}\to \{0,1\}$.

• If $V(p)=1$, we say ”$V$ makes $p$ true.”
• If $Vp=0$, we say ”$V$ makes $p$ false.”

Step 2: Once we find a valuation $V$, we want to extend it to a function

$$\hat{V}:\text{ set of all proposition formulae }\to \{0,1\}.$$

We can use recursion to define $\hat{V}$.

$$\hat{V}=\{\begin{array}{ll}V(p)& \text{ if α is a propositional variable p},\\ \dot{\neg }V(\phi )& \text{ if α is ¬φ},\\ \hat{V}(\phi )\dot{\wedge }\hat{V}(\psi )& \text{ if α is (φ∧ψ)},\\ \hat{V}(\phi )\dot{\vee }\hat{V}(\psi )& \text{ if α is (φ∨ψ)},\\ \hat{V}(\phi )\dot{\to }\hat{V}(\psi )& \text{ if α is (φ→ψ)},\\ \hat{V}(\phi )\dot{\leftrightarrow }\hat{V}(\psi )& \text{ if α is (φ↔ψ)}.\end{array}$$

I know this is not particularly helpful but it’s how we did it in class and I don’t want to write the truth tables for the different connectives since I absolutely know them.

Defined Satisfiable