2.1.13: Extensionality
Extensionality, sometimes called relevance, can be expressed informally as follows: the only factors that bears upon the satisfaction of formula \(A\) in a structure \(\Struct M\) relative to a variable assignment \(s\) , are the size of the domain and the assignments made by \(\Struct M\) and \(s\) to the elements of the language that actually appear in \(A\) .
One immediate consequence of extensionality is that where two structures \(\Struct M\) and \(\Struct M'\) agree on all the elements of the language appearing in a sentence \(A\) and have the same domain, \(\Struct M\) and \(\Struct M'\) must also agree on whether or not \(A\) itself is true.
Let \(A\) be a formula, and \(\Struct M_1\) and \(\Struct M_2\) be structures with \(\Domain{M_1} = \Domain{M_2}\) , and \(s\) a variable assignment on \(\Domain{M_1} = \Domain{M_2}\) . If \(\Assign{c}{M_1} = \Assign{c}{M_2}\) , \(\Assign{R}{M_1}=\Assign{R}{M_2}\) , and \(\Assign{f}{M_1} = \Assign{f}{M_2}\) for every constant symbol \(c\) , relation symbol \(R\) , and function symbol \(f\) occurring in \(A\) , then \(\Sat[,s]{M_1}{A}\) iff \(\Sat[,s]{M_2}{A}\) .
Proof. First prove (by induction on \(t\) ) that for every term, \(\Value[s]{t}{M_1} = \Value[s]{t}{M_2}\) . Then prove the proposition by induction on \(A\) , making use of the claim just proved for the induction basis (where \(A\) is atomic). ◻
Carry out the proof of Proposition \(\PageIndex{1}\) in detail.
Let \(A\) be a sentence and \(\Struct{M_1}\) , \(\Struct{M_2}\) as in Proposition \(\PageIndex{1}\) . Then \(\Sat{M_1}{A}\) iff \(\Sat{M_2}{A}\) .
Proof. Follows from Proposition \(\PageIndex{1}\) by Corollary 5.12.1 . ◻
Moreover, the value of a term, and whether or not a structure satisfies a formula, only depends on the values of its subterms.
Let \(\Struct M\) be a structure, \(t\) and \(t'\) terms, and \(s\) a variable assignment. Let \(\varAssign{s'}{s}{x}\) be the \(x\) -variant of \(s\) given by \(s'(x) = \Value[s]{t'}{M}\) . Then \(\Value[s]{\Subst{t}{t'}{x}}{M} = \Value[s']{t}{M}\) .
Proof. By induction on \(t\) .
- If \(t\) is a constant, say, \(t\ident c\) , then \(\Subst{t}{t'}{x} = c\) , and \(\Value[s]{c}{M} = \Assign{c}{M} = \Value[s']{c}{M}\) .
- If \(t\) is a variable other than \(x\) , say, \(t \ident y\) , then \(\Subst{t}{t'}{x} = y\) , and \(\Value[s]{y}{M} = \Value[s']{y}{M}\) since \(\varAssign{s'}{s}{x}\) .
- If \(t \ident x\) , then \(\Subst{t}{t'}{x} = t'\) . But \(\Value[s']{x}{M} = \Value[s]{t'}{M}\) by definition of \(s'\) .
- If \(t \ident \Atom{f}{t_1,\dots,t_n}\) then we have: \[\begin{gathered} \begin{aligned}[b] \Value[s]{\Subst{t}{t'}{x}}{M} & = \Value[s]{\Atom{f}{\Subst{t_1}{t'}{x}, \dots, \Subst{t_n}{t'}{x}}}{M}\\ & \qquad \text{ by definition of $\Subst{t}{t'}{x}$}\\ & = \Assign{f}{M}(\Value[s]{\Subst{t_1}{t'}{x}}{M}, \dots, \Value[s]{\Subst{t_n}{t'}{x}}{M})\\ & \qquad \text{ by definition of $\Value[s]{\Atom{f}{\dots}}{M}$}\\ & = \Assign{f}{M}(\Value[s']{t_1}{M}, \dots, \Value[s']{t_n}{M})\\ & \qquad \text{ by induction hypothesis}\\ & = \Value[s']{t}{M} \text{ by definition of $\Value[s']{\Atom{f}{\dots}}{M}$} \end{aligned}\end{gathered}\]
◻
Let \(\Struct M\) be a structure, \(A\) a formula, \(t\) a term, and \(s\) a variable assignment. Let \(\varAssign{s'}{s}{x}\) be the \(x\) -variant of \(s\) given by \(s'(x) = \Value[s]{t}{M}\) . Then \(\Sat[,s]{M}{\Subst{A}{t}{x}}\) iff \(\Sat[,s']{M}{A}\) .
Proof. Exercise. ◻
Prove Proposition \(\PageIndex{3}\).