It doesn't need to have a particular definition – it's just a predicate. — Snakes Alive
Sometimes, the exclamation point signals unique existence — Snakes Alive
The unique existence predicate is second-order — Snakes Alive
it has a different syntax, and occurs alongside a variable and a formula — Snakes Alive
It's like an existential quantifier, except only one individual in the domain is allowed to satisfy the formula. — Snakes Alive
E!v[p] is true at w iff there is exactly one individual x in the domain such that p is true at w on any assignment that maps v to x — Snakes Alive
Quantifiers are predicates of formulae. — Snakes Alive
Not to say you couldn't construct such a notion, of unique existence — Snakes Alive
Do you have any thoughts on my question: How can we have a method of models in which, for certain models, there are constant symbols such that the model does not assign a member of the domain of the model? It throws off the way we evaluate satisfaction and truth in models. — TonesInDeepFreeze
In fact, on a classical Kripkean treatment, existence is always necessary existence, since if there is some individual x identical to a in the domain of individuals, then there will be at any world, since the domain of individuals and the domain of worlds are simply separate. — Snakes Alive
he point of a non-logical constant is that its value is invariant across worlds. — Snakes Alive
You could, of course, have a modal logic where individual terms like constants have different denotations relative to different worlds. — Snakes Alive
And you could then allow that they refer to 'nothing,' say, at worlds where the relevant individual doesn't exist. — Snakes Alive
How you want to represent this formally is up to you – one old formal trick is to use a dummy object, say *, to which the value of all terms that have nothing satisfying them at the world map to. — Snakes Alive
You would then need to make a semantics that deals with the dummy object — Snakes Alive
I'm referring to constant symbols. It is not the case that the point is to have the value for constant symbol invariant across models. A model assigns a member of the universe of the model to a constant symbol. There is no requirement that all models agree on what they assign to the constant symbol. Not all models have the same universe, so it's not even possible that they all agree on what they assign to a constant symbol. — TonesInDeepFreeze
Anyway, the question I have is not what happens when a definite description fails, but rather, how do we do we reconcile ordinary semantics for either predicate logic or modal predicate logic with dangling non-denotating constants? — TonesInDeepFreeze
And I don't know what 'the relevant individual' refers to in your remark. — TonesInDeepFreeze
The method requires a theory in which at least one constant symbol 's' is either primitive or already defined: — TonesInDeepFreeze
I can specify world in which Donovan doesn't exist — Banno
Across worlds, not models. A model has a set of worlds, in its frame. — Snakes Alive
What do you mean, 'ordinary?' — Snakes Alive
Obviously in a strict sense you cannot reconcile them, since ordinary predicate logic has no notion of a constant that doesn't refer to anything. There's just a domain, and then the interpretation function maps each constant to a member of that domain. — Snakes Alive
So if 'b' refers to Bob, then it might refer to Bob in all worlds where Bob exists, but to * in all worlds in which he doesn't exist. — Snakes Alive
No, it just requires some distinguished object in the domain, like say *. — Snakes Alive
if a = b in w, then in all w', a = b. — Snakes Alive
∃x[x = a], to mean 'a exists,' — Snakes Alive
Then if you have '∃' quantifying over the domain of individuals, independent of the domain of worlds, then it will have the same value at any world – either it will be necessarily true, or necessarily false. — Snakes Alive
'∃x[x = a]' is true in w iff there is an individual x in the sub-domain associated with w that is identical to a. — Snakes Alive
In S4 or S5, or a derivative therefrom, can an individual exist in every possible world without contradiction? — Banno
The next question is, must there be an individual which exists in every possible world? — Banno
Does an entity necessarily exist in all possible worlds? Set out a set of possible worlds and an accessibility relation and we'll talk. — fdrake
Is logic constructed by fiat? — Amalac
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