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Reading John Nolt's "Free Logic" article I see the symbol "E!" in the "2.1 Axiom Systems".

The following axioms are special for free logics.

(A4)   ∀xA→(E!t→A(t/x))
(A5)   ∀xE!x

(A4) modifies the classical principle:

∀xA→A(t/x)(A4c)

by using ‘E!’ to restrict specification. (A4) stipulates in effect that the quantifiers range over all objects that satisfy ‘E!’, (A5) that they range only over objects that satisfy ‘E!’. Omitting (A5) and replacing (A4) with (A4c) yields classical logic.

Could someone explain what this specification restriction is? An alternate reference would also be helpful.

I am familiar with the uniqueness quantifier "∃!", but I don't think that is what is involved here.


Nolt, John, "Free Logic", The Stanford Encyclopedia of Philosophy (Fall 2018 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/fall2018/entries/logic-free/.

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    Thank you for asking this question, I already thought of the same question this morning as another question here in philosophy SE uses the same notation : philosophy.stackexchange.com/questions/59823/… – SmootQ Jan 25 at 15:09
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    @SmootQ It is questions such as the one you linked where I first saw the symbol wondering as well what it meant. I am now looking for some proof examples using this existence predicate in natural deducdtion from Nolt's article. – Frank Hubeny Jan 25 at 15:46
  • Yes, that got my curiosity too. I am currently looking at the page you mentioned about Free Logics, I googled for that notation but didn't find it this morning. thanks again. – SmootQ Jan 25 at 15:50
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It says so a couple of paragraphs above:

To distinguish terms that denote members of D from those that do not, free logic often employs the one-place “existence” predicate, E! (sometimes written simply as E). For any singular term t, E!t is true if t denotes a member of D, false otherwise.

Here D is the domain of the objects that "really" exist.

Specification is the principle

∀xA→A(t/x).

Its validity in FL is restricted to the terms that denote, which explains the terminology.

If I remember correctly, van Fraassen had a good introductory paper on FL. Van Fraassen was a student of Lambert, who introduced this kind of stuff.

  • I assume anything in D is in the classical domain and these objects exist. – Frank Hubeny Jan 25 at 1:23
  • Yes, this is the case at least for the "negative semantics" that Nolt constructs. There is a small passage in the section on "positive semantics" that explains how one can have models where D is just the extension of E!, and the domain of the model is a superset of D. – Jishin Noben Jan 25 at 1:32

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