How do I just determine if a predicate statement is false or true based on the σ?

Question

Is there a simple rule of thumb and if so, what is it for predicate logic?

Answer 1

In the semantic of predicate calculus, we consider structures. An $\mathcal{L}$-structure $\mathcal{A}$ for a language $\mathcal{L}$ is a set $A$ such that:

$\bullet$ for each constant symbol $c$ in $\mathcal{L}$, there is a member $c^{\mathcal{A}}$

$\bullet$ for each $n$-ary predicate $P$ in $\mathcal{L}$, there is a $n$-ary relation $P^{\mathcal{A}} \subseteq A^{n}$

These are the $\textit{interpretations}$ of the symbols of $\mathcal{L}$.

The terms of $\mathcal{L}$ are the variables along with the constant symbols. Given an $\mathcal{L}$-structure $\mathcal{A}$ and an evaluation $f: Var \to A$ of the variables in $\mathcal{L}$, the interpretation $t^{\mathcal{A}}(f)$ of a term $t$ is $c^{\mathcal{A}}$ if $t$ is the constant symbol $c$, and $f(x_i)$ if $t$ is the variable $x_i$.

Given a $\mathcal{L}$-formula $\phi$, a structure $\mathcal{A}$ and an evaluation $f: Var \to A$ of the variables of $\mathcal{L}$, we say that $\phi$ holds in $\mathcal{A}$ under the evaluation $f$ - we write '$\mathcal{A} \vDash \phi(f)$' - iff:

$\bullet$ If $\phi$ is $P(t_1, ..., t_n)$, where $P$ is a $n$-ary predicate symbol and $t_1, ..., t_n$ are terms, $(t_1^{\mathcal{A}}(f), ..., t_n^{\mathcal{A}}(f)) \in P^{\mathcal{A}}$

$\bullet$ If $\phi$ is $\neg \psi$, where $\psi$ is an $\mathcal{L}$-formula, it isn't the case that $\mathcal{A} \vDash \psi(f)$

$\bullet$ If $\phi$ is $(\psi \land \theta)$, where $\psi, \theta$ are $\mathcal{L}$-formulas, $\mathcal{A} \vDash \psi(f)$ and $\mathcal{A} \vDash \theta(f)$

$\bullet$ If $\phi$ is $(\psi \lor \theta)$, where $\psi, \theta$ are $\mathcal{L}$-formulas, $\mathcal{A} \vDash \psi(f)$ or $\mathcal{A} \vDash \theta(f)$

$\bullet$ If $\phi$ is $(\psi \rightarrow \theta)$, where $\psi, \theta$ are $\mathcal{L}$-formulas, it isn't the case that $\mathcal{A} \vDash \psi(f)$ or $\mathcal{A} \vDash \theta(f)$

$\bullet$ If $\phi$ is $(\psi \leftrightarrow \theta)$, where $\psi, \theta$ are $\mathcal{L}$-formulas, $\mathcal{A} \vDash \psi(f)$ iff $\mathcal{A} \vDash \theta(f)$

$\bullet$ If $\phi$ is $\forall x_i \psi$, where $x_i$ is a variable and $\psi$ is a formula, $\mathcal{A} \vDash \psi (f)$ for all evaluations $f$

$\bullet$ If $\phi$ is $\exists x_i \psi$, where $x_i$ is a variable and $\psi$ a formula, $\mathcal{A} \vDash \psi (f)$ for some evaluation $f$

Note that the definition is recursive in the complexity of formulas and it depends not only in an interpretation of the symbols, but also in an evaluation of the variables. A structure along with an evaluation in predicate calculus is the equivalent of a truth assignment in propositional calculus, in the sense that it is way we can evaluate the truth value of a formula - $\mathcal{A} \vDash \phi(f)$ means that the value of $\phi$ is $V$, and not $\mathcal{A} \vDash \phi(f)$ means that the value of $\phi$ if $F$.