1

According to wiki a theory (im instered in FOL) is

is a set of sentences in a formal language. Usually a deductive system is understood from context

It's clearly specificied that we are considering only sentences ( = closed formulas) but we can't build a theory using open formulas. But why ???

Qwerto
  • 969
  • 6
  • 20

1 Answers1

1

We can, but open axioms must be read as universally quantified.

See the well-known example of Tarski's axioms for (elementary) geometry, formulathed in first-order logic with identity, and requiring no set theory (1959):

the axioms should be read as universal closures; hence any free variables should be taken as tacitly universally quantified.

See examples:

Congruence axioms: Reflexivity of Congruence

<p>$xy\equiv yx$,</p>

and so on.

Mauro ALLEGRANZA
  • 94,169
  • Nice. But why they must be universally quantified ? – Qwerto Mar 01 '18 at 10:46
  • @Qwerto - because if you want to "read" them existentially quantified, you have to explicitly write it. $(x=0)$ is true if we read it as $\exists x (x=0)$ but it is false if we read it $\forall x (x=0)$. Thus, we can omit only one of the quantifier: the "usual convention" is to omit $\forall$. – Mauro ALLEGRANZA Mar 01 '18 at 10:52