Logic of Predicates
Let's begin with the simple,
1. J = Biden exists = [math](\exists x)(x = b)[/math] where b = Biden. Eb = Biden exists (existence, E, as a predicate) isn't the way J is translated in to symbolic predicate logic.
Either existence is not a predicate (Eb is disallowed) or [math](\exists x)[/math] is to be treated as one, I'm not sure.
2. D = All dogs exist. How does one translate D into symbolic predicate logic?
Consider M = All dogs are mammals = [math](\forall x)(Cx \rightarrow Mx)[/math] where Cx = x is a dog and Mx = x is a mammal.
In categorical logic D can be translated as follows:
All dogs are existent things. The category "existent things" is a valid predicate (in categorical logic).
So, is it that D = [math](\forall x)(Cx \rightarrow Ex)[/math] where Cx = x is a dog and Ex = x exists (existence as a predicate)?
The negation of D is S = some dogs don't exist = [math]\neg (\forall x)(Cx \rightarrow Ex) = (\exists)(Cx \land \neg Ex)[/math].
Notice how [math](\exists)(Cx \land \neg Ex)[/math] is self-contradictory (there exists a dog that does not exist).
Salient points
1. We know some dogs don't exist (S is true) but S in symbolic predicate logic is a contradiction and that happened because we used existence as a predicate (Ex). How do we translate S into symbolic predicate logic in the proper way?
2. The existential import of "some" or [math]\exists x[/math] is getting in the way.
3. Perhaps the way to translate D = all dogs exist is to first identify all dogs. For simplicity, say there are only 3 dogs, Timmy (t), Fang (f) and Yeller (e). Then we could translate D like so: [math]((\exists x)(x = t)) \land ((\exists x)(x = f)) \land ((\exists x)(x = e))[/math]
Comments...
1. J = Biden exists = [math](\exists x)(x = b)[/math] where b = Biden. Eb = Biden exists (existence, E, as a predicate) isn't the way J is translated in to symbolic predicate logic.
Either existence is not a predicate (Eb is disallowed) or [math](\exists x)[/math] is to be treated as one, I'm not sure.
2. D = All dogs exist. How does one translate D into symbolic predicate logic?
Consider M = All dogs are mammals = [math](\forall x)(Cx \rightarrow Mx)[/math] where Cx = x is a dog and Mx = x is a mammal.
In categorical logic D can be translated as follows:
All dogs are existent things. The category "existent things" is a valid predicate (in categorical logic).
So, is it that D = [math](\forall x)(Cx \rightarrow Ex)[/math] where Cx = x is a dog and Ex = x exists (existence as a predicate)?
The negation of D is S = some dogs don't exist = [math]\neg (\forall x)(Cx \rightarrow Ex) = (\exists)(Cx \land \neg Ex)[/math].
Notice how [math](\exists)(Cx \land \neg Ex)[/math] is self-contradictory (there exists a dog that does not exist).
Salient points
1. We know some dogs don't exist (S is true) but S in symbolic predicate logic is a contradiction and that happened because we used existence as a predicate (Ex). How do we translate S into symbolic predicate logic in the proper way?
2. The existential import of "some" or [math]\exists x[/math] is getting in the way.
3. Perhaps the way to translate D = all dogs exist is to first identify all dogs. For simplicity, say there are only 3 dogs, Timmy (t), Fang (f) and Yeller (e). Then we could translate D like so: [math]((\exists x)(x = t)) \land ((\exists x)(x = f)) \land ((\exists x)(x = e))[/math]
Comments...
Comments (12)
Quoting Agent Smith
It would depend on how you take the existential quantifier, for 'existence' is ambiguous, but is treated as unambiguous in the symbolism. See how the entire equation is is problematized by this ambiguity. Existence can only be a predicate if it is possible for something not to exist; such is the case for all predicates: their opposites have to make sense. "The snow is white" makes sense only if it is possible for snow to be other than white. "Snow has a color" is not a predication, it is analytic, for it is impossible for snow not to have a color--apodictic that all things have a color; can't imagine a thing and no color in the same object. If you treat existence like color, then the predication is really a tautology, and "all dogs exist" is merely tautologically true. But if existence can be defined as synthetic (in Kant's terms) and some things do not exist (unicorns?) then "X exists" is a predication. But it depends.
Best I can do.
Vladimir Putin exists. Where p = Vladimir Putin, [math](\exists x)(x = p)[/math]
Sherlock Holmes doesn't exist. Where s = Sherlock Holmes, [math](\forall x) \neg (x = s)[/math] = [math]\neg (\exists x) (x = s)[/math]
As I thought, in predicate logic, predication is only possible for existent things. You can't talk about particular nonexistent objects while you can about them as a class:
1. Some unicorns are white = [math](\exists x)(Ux \land Wx)[/math] where Ux = x is a unicorn and Wx = x is white. Clearly this is false because "some"/[math]\exists x[/math] has an existential import.
2. All unicorns are white = [math](\forall x)(Ux \to Wx)[/math]. The "all" lacks existential import and so this can be true despite the fact that unicorns exist.
So, we can talk about an entire class of imaginary entities but not of individuals in that class.
Come now to Anslem's ontological proof. God is the maximally great being.
A maximally great being exists. As you can see, Anselm is usimg existence as a predicate i.e. [math](\forall x)(Mx \to Ex)[/math] where Mx = x is a maximally great being and Ex = x exists. We can see where Anselm goofs up. All maximally great beings are existent things (IF x is maximally great being THEN x exists). The class of maximally great beings can be an empty set, but then the consequent claims there's a member in that set.
I suppose the matter comes down to soundness and validity. We can talk about anything at all if we like, and if the logical form of what we say is in tact, no contradictions, then we can say the talk has logical validity, of course. But if the talk is just scrabbled eggs in the observed world, then it lack soundness. So if you say, "there is a man named Sherlock Holmes, and he does not exist, you are running a contradiction, saying he is and is not at once. But note the language: There IS a man named SH. What do we mean by this? This is ambiguous in the symbolic representation if by "is" what is meant is "exists". Once you disambiguate, and qualify "is" as fictional, speculative, imaginative or the like, then the logic doesn't produce an absurdity.
So, I can talk about particular nonexistent objects, as is done in the novels. But the context of what is "real" and not, itself is fictional; the standard is simply different.
Quoting Agent Smith
On the other hand, whether or not this can be an empty set is still entirely at issue. The claim is the set cannot be empty since it is analytically true that the greatest possible being exists. Taken as logical construction only, the matter rests on the definitions: does existence have to be included in the GPB because what a GPB IS, is existence. It would not be the GPB if it did not exist, just as a body would not be a body without extension. This goes to essence or definition, which goes to the term "greatest". There is that notorious response by Gaunilo's greatest possible island. But greatness is a contingent term and is built out of the terms that apply: a great couch is not a great telescope. So then, what kind of greatness are we talking about when we are talking about God (capital 'G')? The standard omni this and omni that are just arbitrary. Being absolutely knowledgeable is just a vacuous extension of the way we talk about ourselves, e.g. And omnipotence begs the question: what good is this? Only one thing can fit this bill, and this is goodness: what is GOODNESS? Goodness is not arbitrary or question begging in the description of God.
Conclusion: It has to be understood that logic will only produce more logic. Existence is, well, existential, and this makes the appeal to the world. God's existence is reducible to the existence of what must exist, and it is not some guy sitting on a cloud. What must exist is goodness. The GPB of the world concept comes to this. A highly disputable proposition.
If x is a dog then there exists a y such that y = x.
Nice! Simpler than my formulation which isn't wrong per se, but is cumbersome.
If god is the maximally great being then god exists = [math]Mg \to (\exists y)(y = g)[/math]
What's the difference between [math](\exists x)(Gx)[/math] (God exists) and [math](\exists x)(x = g)[/math] (there exists something and that something is god) where g is God?
That's wrong. This is correct:
Mx for "x is maximally great"
ExMx for "there is an x such that x is maximally great", which is to say "a maximally great being exists".
Quoting Agent Smith
That's wrong. This is correct:
g for "god"
Meanwhile, it is merely a logical truth that Ey y=g. Whatever 'g' stands for, it is something that exists. If 'g' stands for god, then it is not required to argue that god exists, since existence is given by the mere fact that 'g' stands for some existent which we have stipulated to be god. But a more meaningful assertion is:
Gx for "x is a god", or with monotheism, "x is a god and nothing else is a god". Then ExGx for "there is a god", or with monotheism, "there is a god and only one god".
That is, if you have a constant symbol such as 'g', then it is a given that 'g' refers to something. So Ey y = g
(In modal logic, we can have an existence predicate, but that is another subject, more complicated, and requires specification of the modal logic.)
Quoting Agent Smith
ExGx does not say "god exits". Rather, 'G' is a predicate symbol, so that Gx may be taken as saying "x has the property of being a god", or, for monotheism, "x has the property of being the one and only god". Then ExGx says "there is an x such that x has the property of being a god", or with monotheism, "there is an x such that x as the property of being a god and nothing else has that property'.
Again, Ex x = g is merely a logical truth. 'g' is a constant' and so there is an x such that x = g.
In sum, there is a difference between a name 'g' for some object and a predicate 'G' for a property. When we use a name 'g' then it names something. When we state 'Gx' we assert that x has the property G.
That's wrong. This is correct:
Hx for "there is an x such that x is a Scotland Yard detective named 'Sherlock Holmes'".
~ExHx for "there does not exist a Scotland Yard detective named 'Sherlock Holmes'".
Also, the '=' sign is a 2-place predicate symbol that goes between terms, not between formulas. Using '=' between formulas is not syntactical. For equivalence of formulas, use the biconditional '<->' that is a 2-place conncective.
How would you translate into logic the following statements?
1. Joe Biden exists.
2. Peter Parker doesn't exist.
3. Some dogs exist.
4. All apples exist.
As I mentioned for other examples, your question depends on whether we are working with predicate logic or modal predicate logic. With predicate logic, there is no existence predicate. With modal predicate logic, there is an existence predicate but it's complicated to explain. I'll address the question with predicate logic and say that modal predicate logic is better suited.
In either case, the first crucial distinction you need to be very clear upon is a distinction I've mentioned: names vs. predicates.
'Joe Biden' and 'Peter Parker' are names.
'is a dog' and 'is an apple' are predicates.
In predicate logic (at least in certain airtight treatments, such with the Fregean method) every term (variable, constant or compound naming term) evaluates to some member of the domain of discourse.
Let 'b' stand for Joe Biden. 'b' is a constant (a name).
Let 'k' stand for Peter Parker. 'k' is a constant (a name).
Let 'D' stand for 'is a dog. 'D' is a predicate.
Let 'L' stand for 'is an apple'. 'L' is a predicate.
Let 'E' stand for 'there exists'. 'E' is a quantifier, and it is not a predicate.
/
Eb (intending to mean "there exists Joe Biden" or "Joe Biden exists") is not well-formed. It is not a formula. It is not considered.
So "Joe Biden exists" is not really translatable into predicate logic at face value in way that you might wish. Sure, you might say Ex x=b. But that's just a logical truth, a theorem of identity theory. It doesn't say much. It has nothing to do with the fact that there exists a person who is Joe Biden. No matter what name we choose and symbolize with 'b', we still have the trivial theorem Ex x=b.
Perhaps the best workaround would be to have a predicate 'B' that stands for the conjunction of all the required properties for being Joe Biden. Then we would say:
ExBx.
And that is well-formed and is meaningful.
/
But the Peter Parker example is trickier (when, for sake of discussion, we dismiss that the name 'Peter Parker' is taken as otherwise referring to a fictional character, and we are only interested in the sense of names being for objects in the real world). It's tricky because I can't say "the conjunction of all the required properties for being Peter Parker" since we agreed that there is no such conjunction of properties.
The best I can offer is to have a predicate 'K' that stands for the conjunction of all the required properties of being Peter Parker if there were a real person just like the fictional character. Then maybe (I'm not sure because of that subjunctive 'if there were' bit):
~ExKx
/
"Some dogs exist" is not problematic:
ExDx
/
"All apples exist". It's not clear what that is supposed to mean. Is it supposed to mean that everything that is an apple is something that exists? But see my remarks earlier in this post about existence as a predicate. I don't think I have a satisfactory symbolization other than:
Ax(Lx -> Ey y=x)
But, if I recall, that is, mutatis mutandis, what you came up with one or your own formulas. And my point stands that it doesn't really say much.
/
So for three of your examples, modal predicate logic is much better suited than mere predicate logic.