What is the difference between ∃ (existence) and A (actuality)?



The existential quantification can, it seems, be used with modal logic. Now excuse my naivety, but:

  • if so, what is the difference between being actual and existential quantification?

I'm just asking cos I wondered if it makes sense to talk about worlds which aren't this one being real.


Posted 2016-04-16T19:57:03.043


1Do you mean like in first order modal logic? – Eliran – 2016-04-16T20:19:01.137

i don't know, i will google it – None – 2016-04-16T20:22:40.917


Yes, in what is called quantified modal logic http://plato.stanford.edu/entries/logic-modal/#QuaModLog That is the one that Quine had misgivings about, which were formally resolved by Kripke. Interpretational issues with quantification into modal contexts remain controversial however, Kripke's approach requires undesirable ontological commitments (essentialism).

– Conifold – 2016-04-16T20:30:55.880

@Conifold - In that article doesn't "A" just stand for some arbitrary proposition, rather than being a specifically modal logic related symbol? The list of modal logic symbols in part 1 at the very top of the article doesn't give "A" as a symbol. Or is it possible MATHEMETICIAN was talking about the upside-down A symbol? – Hypnosifl – 2016-04-16T22:30:21.180

@Hypnosifl it's mentioned about half way down ? A similar phenomenon arises in modal logics with an actuality operator A (read ‘it is actually the case that’). – None – 2016-04-16T22:54:35.797

Ah, I was just looking in section 14 which Conifold linked to, while the sentence you quote is from section 10. As far as I can tell the rest of the article doesn't give any examples of how the actuality operator is used though...aside from that one mention, it seems like when they use the symbol "A" they're generally just using it to represent some arbitrary proposition in predicate logic. So I suppose one would have to look elsewhere to learn more about it, or how it might relate to the quantified modal logic that Conifold brought up. – Hypnosifl – 2016-04-17T00:30:42.683



The actuality operator is usually not interpreted as a quantifier, it indicates that what follows belongs to the privileged word, the actual world. The existential quantifier, on the other hand, quantifies, and over things in whatever possible world. So ☐(∃x Px) for example says that there exist objects with property P in every possible world, i.e. they exist necessarily. In some theories ∃x is even used unrestrictedly, i.e. quantifies over all possible worlds, as in Lewis's counterpart theory. Then one can combine the two as in ∃x Ax, which says that the actual world has things. In some non-modal theories there is a similar distinction between the existence predicate and the existential quantifier, when people (Meinongians) want to quantify over non-existent things but without committing themselves to their existence. So ∃x(¬Ex) says that some things do not exists like ∃x(¬Ax) says that some things are not actual.

However, Hazen in Actuality and Quantification does introduce actuality quantifiers:"Ordinary, world-restricted, quantifiers are interpreted as ranging over existents; their logic is formalized by putting existence premisses/hypotheses into the familiar rules. Actuality quantifiers are interpreted as ranging over things that actually exist (actual existents)". This is to satisfy actualists, who would only allow quantification over actual objects, not modal abstractions.


Posted 2016-04-16T19:57:03.043

Reputation: 38 006

Conifold so it seems i can say that something can possibly exist. my follow up comment / question, and please excuse my clumsiness etc., is whether i can say that it is physically possible for something to exist but in reality it actually does exist? i think that's what i mean... – None – 2016-04-17T18:23:46.377

@MATHEMETICIAN Well, your setup of possible worlds reflects what kind of possibility you have in mind, physical, metaphysical, epistemic, etc. If it is physical then ◊(∃x Px) says that something with property P is physically possible. If you want to say that it exists in actuality then you write something like ∃x (Px ∧ Ax). – Conifold – 2016-04-17T19:35:40.270