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The Empty Set (or lack thereof)

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Jake

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You need to say the empty set is a subset of every set for consistency's sake.
I woudn't put it that way at all. We (editorial 'we') hope (or believe) that Z set theories are consistent. But that 0 is a subset of every set is merely a consequence of the definition of 0. There's no adjustment for consistency needed, other than it would be simply inconsistent (mathematically speaking) to assert that there is a set that the empty set is not a subset of. Edited by Hodge'sPodges
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For a symbol to be defined, it must have a referent (take it or leave it)
No, it must have a meaning (which in this context implies rules of combination and transformation into other symbols). Symbols in a formal system dont need to correspond to anything outside the system - this correspondence is set up later when you create a particular interpretation of the system by assigning symbols to non-system entities.

This is what I was referring to previously when I said that your interest in formal systems is their use in describing natural language, which is something I dont care about.

First, one of us has to be wrong, because there is only one reality and it is non-contradictory.
No. Its similar to me wanting to notate a spoken language by using a roman-style alphabet, and you wanting to use chinese-style pictograms. Its a purely methodological decision on which nothing hangs (other than pragmatic concerns) and there is no 'correct' way of doing it.

Do you want me to bitch-slap you for devaluing proper methods of reasoning? It's what I do.

It has nothing to do with correct reasoning, unless you think that correct reasoning requires formalization. Natural language is too complex to be captured in a formal system and any attempt to do so is going to be filled with arbitrariness like this.

Edited by eriatarka
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News to me. Can you reproduce the proof or at least direct me to where it was published? It's an accurate description of the meaning of "all", but not the properties definitionally attributed to ∀.
By Kleene's book, I take it you mean the book now published by Dover. I think you might find it there. If not, it's easy to prove:

Suppose AxF. Then, by universal instantiation, we have F. Then by existential generalization we have ExF.

In the particular instance where 'F' is 'Px', we have

Suppose AxPx. Then, by universal instantiation, we have Px. Then by existential generalization we have ExPx.

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the union C is the set which contains all elements that are in A and all elements that are in B. Since Ø is not an element, it is not in A or B, so it is not in C either.
Wait. 0 is an element of certain sets. 0 has no elements in it, but 0 may be an element itself of certain sets. For example 0 is an element of {0}. So, unless we are told more about A and B are, we can't infer whether 0 is or is not an element of either A or B.

Thus for any set C which is the union of two sets A, B, the empty set is not in C. Except that since 0 i.e. {} has no elements, and since C is all of the elements of the two component sets, then C does not contain Ø.
It depends on what A and B are. If 0 is a member of either A or B, then 0 is a member of AuB.

Anyway, I don't know what that has to do with 0 being a subset of every set. By saying that 0 is a subset of every set, we're not saying anything about whether 0 ITSELF is a MEMBER of any particular set.

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By Kleene's book, I take it you mean the book now published by Dover. I think you might find it there. If not, it's easy to prove:

Suppose AxF. Then, by universal instantiation, we have F. Then by existential generalization we have ExF.

No, you can only use existential generalization here if x is a bound variable, which it isnt.

edit: to clarify, AxF means that if x is an element in the universe of discourse, then Fx holds. But in the case where the universe of discourse is empty, you cant use universal instantiation to get Fx, so x here is a free variable rather than being bound.

Edited by eriatarka
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No, you can only use existential generalization here if x is a bound variable, which it isnt.
No, existential generalization (like universal generalization) is a rule that permits us to move from a formula in which a certain variable is free to a formula in which that variable is then bound (by an existential quantifier or universal quantifier, respectively). To require that existential generalization apply only to bound variables would make the rule of existential generalization otiose. If x is ALREADY bound in F, then there is no need to apply a vacuous existential quantifier to the front of F.

The rule of existential generalization may be stated:

For any term t, any variable x, and any formula F, if t is free for x in F, then

from F[t|x], we may infer ExF

where 'F[t|x]' stands for the result of replacing t for all free occurrences of x in F.

Thus, since x is always free for x in any formula, and since F[x|x] is F, we have that, for any formula F and any variable x, from F we may infer ExF. In particular, from Px we may infer ExPx.

Ask any logician anywhere in the world. Print my post, hold it up to any logician anywhere in the world. He or she will tell you that I am correct.

edit: to clarify, AxF means that if x is an element in the universe of discourse, then Fx holds.
Informally, that's an okay way of putting it.

But in the case where the universe of discourse is empty,
In classical mathematics we do not allow empty domains of discourse. In particular, the predicate calculus is designed so that it works with semantics in which the domain of discourse is never empty. Thus the predicate calculus DOES permit us to infer ExF from AxF.

you cant use universal instantiation to get Fx, so x here is a free variable rather than being bound.
Your initial comment was that my use of existential generalization was incorrect. But you're incorrect on that point. Now you're next remark is as to universal instantiation. But my universal instantiation was as SIMPLY correct as can be: From any formula AxF we may infer F. In particular, from AxPx we may infer Px. That the result of the inference in this case gives a formula with a free variable is not a problem. Open formulas (those with free variables) occur quite commonly as lines in proofs, and indeed it is the part of our PURPOSE in the rule of universal generalization that it permit moving from a formula in which a certain variable is bound to a formula in which that variable is free.

The rule of universal instantiation may be stated as:

For any term t, any variable x, and any formula F, if t is free for x in F, then

from AxF, we may infer F[t|x]

where 'F[t|x]' stands for the result of replacing t for all free occurrences of x in F.

Thus, since x is always free for x in any formula, and since F[x|x] is F, we have that, for any formula F and any variable x, from AxF we may infer F. In particular, from AxPx we may infer Px.

So you see that universal instantiation and existential generalization are like "two sides of a coin".

Anyway, all told:

1 AxPx ... supposition

2 Px ... from 1 by universal instantiation (set of lines charged: {1})

3 ExFx ... from 2 by existential generalization (set of lines charged: {1})

4 AxPx -> ExPx ... from 1, 3, by conditional proof (set of lines charged: { })

is a perfectly correct proof and with annotation.

Ask any logician anywhere in the world.

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Huh? You can quantify over empty domains.

Let S be the set of all natural numbers which are smaller than 0. Then we can quantify over S to form statements like ∀(x∈S) (x is divisible by 2). And this statement would be true, since there is no element of S which isnt divisible by 2. But the point is that S here is the empty set, so it isnt true that ∃x(x∈S).

More generally, its true that ∀(x∈∅)(Fx) for any property F, but this doesnt imply that ∃x(Fx) or ∃x(x∈∅)

Similarly, we define A ⊆ B to be true if ∀x(x∈A => x∈B). So if A is the empty set, this condition holds trivially for any set B. But this doesnt imply that ∃x(x∈A), otherwise youd be saying that the empty set had members.

Unless I'm misunderstanding what youre saying.

Edited by eriatarka
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using Ax for "for all x" Ex for "there exists x", and S(x) for "x is an element of S" -

1. We define a set S' to be a subset of a set S iff Ax S'(x) -> S(x)

2. So a set S'' is not a subset of S iff Ex ~(S''(x) -> S(x))

3. That is S'' is not a subset of S iff Ex S''(x) & ~ S(x)

You meant Ex(S''(x) & ~S(x)).

4. We define the empty set 0 to be the set with no elements, ie Ax ~0(x)

5. Suppose 0 is not a subset of a nonempty set S

6. Then Ex 0(x) & ~S(x), call this x 'y'

7. So 0(y)

8. But by 4 we have ~0(y)

9. =><=

10. Therefore 0 is a subset of S

That is quite fine. Is anyone disputing it?

Though, in my notation ('e' stands for 'is a member of') it would be:

1. Axy(x subset of y <-> Az(zex -> zey)) ... definition

2. 0 subset of y <-> Az(ze0 -> zey) ... from 1 by universal instantiation twice

3. Suppose ~Ay 0 subset of y ... supposition toward contradiction

4. Ey~0 subset of y ... from 3 by quantifier exchange

5. ~0 subset of y ... from 5 by existential instantiation

6. ~Az(ze0 -> zey) ... from 2 and 5 by sentential logic

7. Ez(ze0 & ~zey) .... from 6 by quantifier exchange and sentential logic

8. ze0 & ~zey ... from 7 by existential instantiation

9. ze0 ... from 8 by sentential logic

10. Az~ze0 ... from definition of '0'

10. ~ze0 ... from 10 by universal instantiation

11. Ay 0 subset of y ... from 3, 9, 10 by reductio ad absurdum

Note: Some people are not familiar with using the variables y and z in the existential instantiations at lines 5 and 8, respectively, as I did. Though what I did is correct, if you're not familiar with such use, just make those instantiations go to whatever constants you wish and adjust the rest of the proof accordingly. You should be able to see that it works quite fine.

However, this is much simpler:

1. Axy(x subset of y <-> Az(zex -> zey)) ... definition

2. 0 subset of y <-> Az(ze0 -> zey) ... from 1 by universal instantiation twice

3. Az ~ze0 ... from definition of '0'

4. ~ze0 ... from 3 by universal instantiation

5. ze0 -> zey ... from 4 by sentential logic

6. Az(ze0 -> zey) ... from 5 by universal generalization

7. 0 subset of y ... from 2 and 6 by sentential logic

8. Ay 0 subset of y ... from 7 by universal generalization

And all of that is just a laborious way of saying an argument found in virtually any textbook in set theory:

Theorem: Ay 0 subset of y.

Proof: For all z, vacuously, if ze0 then zey. So 0 subset of y.

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Huh? You can quantify over empty domains.
I didn't use the word "quantify over". What I said is correct. I'll state it even more precisely. With classical first order logic, when we give an interpretation (structure, model, whatever) for a language, we always give a NON-empty domain of discourse. Thus, the logical axioms and inference rules are set up to adhere to that sematntical restriction, and thus the logical axioms and inference ruels permit inferring the existence of at least one object. This can be done in various ways:

Let F be any predicate symbol of arity n (every language has at least one predicate symbol). Then we have the theorem:

Ex(Fx1...xn -> Fx1...xn)

Or just with equality, we have:

Ex x=x

And moreover the rules do permit inferring ExF from AxF, just as I showed in my earlier post.

Let S be the set of all natural numbers which are smaller than 0. Then we can quantify over S to form statements like ∀(x∈S) (x is divisible by 2).

Your math symbols are showing up as square boxes for me, so I don't know what formula you have in mind.

In any case, what I have said - that the domain of discourse is never empty and that we can always infer ExF from AxF does not contradict that - for any formula F, such sentences as Ax(xe0 -> F) are theorems of set theory. Don't forget that saying that the domain of discourse is not empty is NOT to say that the empty set is not a MEMBER of the domain of discourse. So, yes, of course, if the empty set is a member of the domain of discourse, then the universal quantifier ranges over the empty set just as the universal quantifier ranges over all members of the domain of discourse. But, for any given language and interpretation of that language, the domain of discourse ITSELF is not (in classical mathematics) empty.

Edited by Hodge'sPodges
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{} or Ø, like 0, is not an object, it's a notation representing said metaphysical absence.
You may have that view, however, in ordinary mathematics the empty set is a mathematical object. 'the empty set' doesn't mean "nothingness" or whatver metaphysical description. It just means that it is the object that doesn't have any members in it. Going with your sack analogy, you can think of the empty set as being the empty sack. Call this "sack theory". Then we say that two sacks of things are the same sack of things if and only if they have the exact same members in them. Then the empty sack is that unique sack that has nothing in it.

The newer books are better than what? I'm wondering: if you haven't read Kleene, how you can determine the relative worth of the newer books to Kleene?
I didn't make the remark about the books, but I do say that though Kleene is one of the great logicians and mathematicians, and though his book (the one published by Dover) is chock full of great stuff, it is not as well organized or pedagogically helpful as other books on the subject. Indeed, for the wealth of information in the book, it's fairly difficult to navigate through all of it. On the other hand, the Enderton book that was mentioned is pedagogically wonderful. Edited by Hodge'sPodges
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By the Law of Excluded Middle, the following is a true statement: "The lights in the hallway are either on or off." What if there are no lights in the hallway? Which is the case?
This would depend on the exact manner in which the particular system of formal logic you're working in handles the issue of definite descriptions (particualy definite descriptions that don't properly describe, such as 'the light in the hall' when there is no light in the hall). That subject is taken up toward the end of a course in symbolic logic.
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I know this is off topic, but exactly which combination of premises make it so that you cannot have a universal set, IE a set for which every possible set is a subset? Most people say it's a result of the Axiom of Choice, but I think, more fundamentally, it is the consequence of the requirement of strict heiarchy that exists as one of the axioms in typical set theory. Is there any proof of this, either positive or negative?
It has nothing to do with the axiom of choice.

For any relation R, it is a theorem of pure logic:

~ExAy(Ryx <-> ~Ryy)

That is, there is not an object x such that for all objects y, y bears the relation R to x if and only if y does not bear the relation R to itself.

No matter what relation R is, we have that theorem. (I can explain the proof of the theorem if you like.)

Now, let that relation R be 'is a member of', so instead of writing 'Rxy' we write 'xey'. That is, x bears the relation to y of being a member of y. So we have:

~ExAy(yex <-> ~yey)

And if we let x and y range over sets, that reads:

There is not a set x such that for all sets y, y in x if and only if y not in y.

Now, if we had a set z that had all sets as members, symbolized:

EzAy yez

then we could take a subset of z (using the axiom schema of separation that allows asserting the existence of subsets of a given set, as those subsets are defined by a formula; or using a similar axiom, depending on the exact system of set theory) that it is the set of all sets that are not members of themselves.

That is, we'd have

ExAy(yex <-> ~yey)

I.e, we would have that there exists a set x such that for all y, y is a member of x if and only if y is not a member of itself, which contradicts the theorem:

~ExAy(yex <-> ~yey)

So we conclude ~EzAy yez.

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Effectively all the expressions are defined in such away that you can test every object in the universe to see if it holds.
It would be better to say that every well formed formula is such that it is meaningful whether that formula holds for a given object in the domain of discourse. But, with set theory, it is not the case yhat there is a mechanical test (effective procedure) for the decision as to whether a any given formula holds for any given object.

No, as above, if we require the Law of the Excluded Middle then every expression must be either true or false.
Every SENTENCE must be true or false.

Which is fine, there is a logic without that Law, namely Intuitionistic logic.
However, the proof I gave that the empty set is a subset of every set holds intutionisitcally too. Indeed, the principle of vacuous truth holds intuitionistically, in this sense:

Let 'f' stand for falsehood.

For any formula Q,

f -> Q.

Edited by Hodge'sPodges
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Because if you call that set S then the power-set of S is also a possible set, and by definition must be a subset of S. But the power-set of any given set X has a strictly greater cardinality than X and hence cant be a subset, so you have a contradiction.

(This has nothing to do with the axiom of choice)

That works. But it seems to me to be more common to show the result even more basically using just the axiom schema of separation.
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Under the standard formalization, the statement is false. The definite article asserts existence and uniqueness w.r.t. a set, and the existential part is false. An indefinite article similarly has an existential assertion, so if there is no light, the statement is false. The focus of the sentence seems to be the mutual-exclusiveness of on/off, but it carries a presupposition which coupled with your factual claim about no light makes the statement a contradiction (specifically, "The lights in the hallway are either on or off and there are no lights in the hallway" is a contradiction). You can avoid the contradiction by saying "If there are lights in the hallway they are are either on or off and there are no lights in the hallway", which is simply stupid, not contradictory.
For formal mathematical systems there are other approaches also. Most particularly, the Fregean method, which should have no pretense of matching natural language usage but is still a quite elegant way of handling the situation formally, and may even be regarded not as being in conflict with natural language usage but instead may be cast in a form of stipulating quite reasonable rules for mathematical definitions.
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I don't think any of us implied that about set theory. The issue, to me, is that set theory depends upon a certain kind of logic to 'prove' that the empty set is a subset of all sets. I have a problem with that kind of logic, namely, its artificial necessity for everything to have a truth value.
Not everything has a truth value. Every SENTENCE of the language has a truth value. But still, even confined to sentences, your objections will apply. Indeed if you disagree with the logic of classical mathematics, then you'll be at constant odds studying it. I suggest first learning the first order predicate calculus from a book such as Kalish/Montague/Mar. If you find that after that point you still don't accept the logic, then, if you wish to study such things as set theory (even just upper division mathematics in general) then you'll have to do it always holding a grain of salt or an ongoing objection, or you might also branch out to study alternative logic systems (perhaps at least one that better suits your philosophy) and see what mathematics can be developed.

I daresay that if the authors of textbooks on Discrete Mathematics would just admit this outright, instead of hiding behind the notion of vacuous proofs as their justification, then this entire conversation would have been unnecessary.
The notion of something holding vacuously is quite common in mathematics. The authors of the textbook are not at fault for referring to it. It might be the case that the authors don't make clear enough in the preface (or wherever) what mathematical background is required. But merely referring to something holding vacuously is not in itself a defect. Edited by Hodge'sPodges
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Words

Thanks for the book advice. Luckily, I have a solid undergraduate Math background to work from, including a year-long Analysis course. Unfortunately, I learned most of it 12-13 years ago, so I'm pretty rusty. My recently obtained Engineering B.S. didn't involve any upper-level math courses.

Are any of the more frequently applied aspects of Calculus, Linear Algebra, or Differential Equations dependent upon vacuous reasoning?

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No, it must have a meaning (which in this context implies rules of combination and transformation into other symbols).
You have yet to establish what a "meaning" is in your meaningless system. I'm simply saying that a symbol needs to be well-defined. And as you presumably know, this is impossible for propositional variables, staying within "the system". Anyhow, take it or leave it means that if you deny the requirement that a well-formed formula has to be composed of well-defined combinations of well-defined symbols, then you have a completely different and mystical notion of "statement", different from what you gave before in #39 and anything that bears a resemblance to a "formal system". Operating in terms of undefined shapes isn't "formal" in any universe that exists.
Symbols in a formal system dont need to correspond to anything outside the system - this correspondence is set up later when you create a particular interpretation of the system by assigning symbols to non-system entities.
I assume that you accept that (AףD). And you would have no formal problem with ≡9cae4437756a15b8e44ec23e07fb1f65.pngAb71edd70fcad670e99a9912ba5e55d77.pngx∀¬xD), and I would assume that you accept that A0f2c04f82a1eb8e3e371366214579f5b.pngB≡¬B∨A (that one is hard: don't feel bad if you don't get it). Clearly, A¬C is a tautology.
Huh? You can quantify over empty domains.
Only if you have a stipulation to that effect. If you reify "domain" no proble, but you haven't given any reason to hold that one should reify domains.
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Any formalization of the definite article that I've heard of does attribute to it a non-conditional existential formalization.
That is not the case for the Fregean approach.

but the ostensibly set-theoretic "proofs" are being carried out in natural language, not symbolic logic, so it is important to be clear on the point that for example ≡ is not the same as "just in case", or that natural language "all" is not the same as ∀.
Set theoretic proofs are usually given in a mix of natural language and symbols. But in just about any context, such arguments are meant to withstand strict formalization. In that sense, ZFC (or whatever classical first order set theory) is the set of formal sentences in a given formal language that are entailed, by formal first order predicate calculus, from a set of formal axioms. Often authors do not go to the extent of spelling out all of that pedantic stuff I just said, but it should be clear enough from either less explicit mention, from context, or especially from the literature that is concerned with stating exactly what ZFC is as a formal theory.
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I assume that you accept that (AףD). And you would have no formal problem with ≡9cae4437756a15b8e44ec23e07fb1f65.pngAb71edd70fcad670e99a9912ba5e55d77.pngx∀¬xD), and I would assume that you accept that A0f2c04f82a1eb8e3e371366214579f5b.pngB≡¬B∨A (that one is hard: don't feel bad if you don't get it). Clearly, A¬C is a tautology.
What do you mean 'I accept them'? If these are strings derivable from the axioms of some formal system then they are valid and provable within that system. If you want to use them to say anything interesting about the world then youll have to start associating the symbols with 'things' (giving them meaning, as you call it), but this is a separate process from the formal definition of the system. Syntax, semantics, and interpretation ('models') are all different things, but you seem to be conflating them.

Here is a (very informal) description of a basic formal system

allowed symbols: {X,Y}

axiom 1: 'X' is a well-formed formula
axiom 2: if F is a well-formed formula, so is FY[/code]

from these axioms you can show that XY, XYYYYYYYYY, etc are provable within the system (ie they can be derived from axioms) whereas XXYY and XXX cant. But none of these symbols have 'meaning' yet (we havent yet created a model of the system).

Edited by eriatarka
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Every SENTENCE of the language has a truth value.
Every declarative sentence.Do you understand? Think about it! Besides, there are massive problems with that claim, even for declaratives. What is the truth value of the sentence "My mother is taller than your oldest sister"? I have infinitely many more like that if you need further examples.
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No, generally we have the theorem ∀x(Fx) -> ~∃x(~Fx) ('there is no x for which Fx fails to hold'), which doesnt imply ∃x(Fx) ('there exists an x such that Fx holds').
Again, your math symbols (your quantifiers here) are showing up for me as square boxes, so I might not be accurately gleaning what you wrote.

Anway, I think you wrote:

AxFx -> ~Ex~Fx.

Of course that is correct.

But I've already given you an exact derivation of

ExF from AxF (where 'F' is any formula), or of ExPx from AxPx, or now we can say ExFx from AxFx.

And your remarks about my instantiations were incorrect, as I explained.

If what you were saying were true, then the empty set wouldnt be a subset of every set if we were using the normal definition of subset (that A is a subset of B iff every element of A is an element of B).
No, that is incorrect. I gave a proof that the empty set is a subset of every set. And it does not contradict that for any formula F, we have

AxF -> ExF

as a theorem of the first order predicate calculus.

But of course, theres nothing stopping you making a logic in which ∀x(Fx) -> ∃x(Fx), it would just have different properties from standard predicate calculus.
No, on the contrary, AxFx -> ExFx is a theorem of the first order predicate calculus. I proved it as well as I explained in detail your misconceptions about instantiations and domains of discourse. As well, even the pure logical matter has been shown to you, so that appeal to authority would only be icing, you may ask any logician anywhere in the world to see that

AxFx -> ExFx is a theorem of first order predicate calculus.

I'm wondering whether you're conflating with a different matter. It is true that

Ax(Fx -> Gx) -> ExFx is NOT a theorem of the predicate calculus.

That is, from "Everything that is an F is a G" does not entail that there is something that is an F, but AxFx -> ExFx is a theorem of the first order predicate calculus.

That is "If everything is an F, then there exists something that is an F".

That is correct, since the standard first order predicate calculus requires that any domain of discourse be non-empty, so if everything in a NON-empty domain has a certain property, then there is at least one thing that has that property (since the domain has at least one thing in it). And thus the rules of inference permit inferring ExFx from AxFx.

Please, refer to any good book on the first order predicate calculus or on beginning mathematical logic.

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Here is a (very informal) description of a basic formal system
Well, finally, some progress. Notice that ¬,A,0f2c04f82a1eb8e3e371366214579f5b.png,B,≡,∨,P are not allowed symbols, so your logic doesn't allow you to say anything in set theory, or anything else of interest. Just XY*.

Do you have a shred of evidence that, aside from order, there is a valid distinction between syntax and semantics? The choice xPy, P(x,y) and the little-used (x,y)P is fundamantally arbitrary, and I have never hear of anyone deriving anything substantive from the syntax of xPy versus P(x,y). So what is your refense that there exists a "syntax" to formal systems which is not a notational variant of the semantics?

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Every declarative sentence.Do you understand? Think about it! Besides, there are massive problems with that claim, even for declaratives. What is the truth value of the sentence "My mother is taller than your oldest sister"? I have infinitely many more like that if you need further examples.
I am talking about formal first order languages such as for set theory. I'm not opining as to natural languages. For any given structure for the formal language of set theory, every sentence (as, for a given langauage, 'sentence' is defined by recursive definition) of the language is either true or false and not both per that structure.
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