# Why Existence is NOT a predicate

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First some definitions (or the outline of definitions). We have two quantifiers in first and second order logic. We have the existential quantifier E and the universal quantifier A. ExP(x) means there exists x such that predicate P is true of x. AxP(x) means for all x the predicate P(x) is true. A is really logical conjunction (AND) on steroids. Suppose the domain of individuals is well ordered and written thus (t1, t2, ....) where t1, t2 are all the individuals. Then AxP(x) is essentially the extended -conjunction- P(t1) AND P(t2) AND .... . Likewise E is really logical disjunction (OR) on steroids. Enumerating the domain of individuals as above, ExP(x) means

P(t1) OR P(t2) OR ... . Where OR is the inclusive OR. So ExP(x) means for at least one individual tN P(tN) holds.

Because A and E are conjunctions and disjunctions, respectively, de Morgans law holds. Let - mean NOT (or denial). -ExP(x) is equivalent to Ax[-P(x)] and -Ax[P(x)] is equivalent to Ex[-P(x)]. and finally -Ax[-P(x)] is precisely ExP(x) and -Ex[-p(x)] is precisely AxPx

Now suppose in addition to being a quantifier existence were also a -predicate-, i.e. a property. Let e(x) mean x exists. This would follow if existence (denoted e) were a property. Given what we intend by existence one would have to assume -Ex[-e(x)], which is to say there does not exist an individual x which lacks the property of existence. That makes sense doesn't it? Or does it? By de Morgan -Ex[-e(x)] is precisely Ax[e(x)] which is to say for all x, x exists. Uh, uh. Unicorns don't exist. Five sided triangles don't exist, etc. etc..

And -that- is why existence cannot be a property or predicate.

Ba'al Chatzaf

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ExP(x) means there exists x such that predicate P is true of x.

Now suppose in addition to being a quantifier existence were also a -predicate-, i.e. a property. Let e(x) mean x exists.

Ba'al Chatzaf

If you look at your problem, it will answer its self. You cannot talk about x without it existing (the first sentence that I left of your quote)

Since x must exist to talk about x, then e(x) is a illogical.

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ExP(x) means there exists x such that predicate P is true of x.

Now suppose in addition to being a quantifier existence were also a -predicate-, i.e. a property. Let e(x) mean x exists.

Ba'al Chatzaf

If you look at your problem, it will answer its self. You cannot talk about x without it existing (the first sentence that I left of your quote)

Since x must exist to talk about x, then e(x) is a illogical.

Would you like to talk about my pet Unicorn? I would. Rethink what you said. One can easily talk about five sided triangles. See! I just did. But there aren't any. As long as the empty set exists (it does, in the same way as non-empty sets) one can talk about thinks that do not exist. They are elements of the empty set.

Ba'al Chatzaf

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I took Logic in university, under the department of philosophy, and I can see that Kolker doesn't know what he is talking about when he tries to apply it to basic philosophic questions. He may be a good scientist (I don't know) but he just does not grasp philosophy.

My grade was 100% by the way.

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I took Logic in university, under the department of philosophy, and I can see that Kolker doesn't know what he is talking about when he tries to apply it to basic philosophic questions. He may be a good scientist (I don't know) but he just does not grasp philosophy.

My grade was 100% by the way.

Point out what was wrong with my presentation. The predicate e(x) for x exists has a problem. If one assume there does not exists any x such that -e(x) then one concludes for all x e(x). In short everything exists, including my pet Unicorn. This is standard first order logic. Point out my error in applying the de Morgan laws or hold you peace.

Ba'al Chatzaf

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I will hold my peace.

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ExP(x) means there exists x such that predicate P is true of x.

Now suppose in addition to being a quantifier existence were also a -predicate-, i.e. a property. Let e(x) mean x exists.

Ba'al Chatzaf

If you look at your problem, it will answer its self. You cannot talk about x without it existing (the first sentence that I left of your quote)

Since x must exist to talk about x, then e(x) is a illogical.

Would you like to talk about my pet Unicorn? I would. Rethink what you said. One can easily talk about five sided triangles. See! I just did. But there aren't any. As long as the empty set exists (it does, in the same way as non-empty sets) one can talk about thinks that do not exist. They are elements of the empty set.

Ba'al Chatzaf

You can talk about five sided triangles and your pet unicorn all you want, but they are only ideas (your idea exist as a thought) but they are not objects.

Example:

If x is my computer, then x exist as an object sitting on my desk. You cannot add or take away existence.

If x is your pet unicorn, then x exist as a thought in your head, but not as an object in you yard. You cannot add or take away existence.

They are two different things.

--Dustan

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By the way Ba'al I agree with your conclusion. Existence is not a predicate. Objects exist, it is self evident. To say x is to say x exist as something (whether that is an object or idea)

Dustan

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ExP(x) means there exists x such that predicate P is true of x.

Now suppose in addition to being a quantifier existence were also a -predicate-, i.e. a property. Let e(x) mean x exists.

Ba'al Chatzaf

If you look at your problem, it will answer its self. You cannot talk about x without it existing (the first sentence that I left of your quote)

Since x must exist to talk about x, then e(x) is a illogical.

Really. Then why do Objectivists talk so much about John Galt and Dagney Taggart. Neither exist.

Ba'al Chatzaf

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ExP(x) means there exists x such that predicate P is true of x.

Now suppose in addition to being a quantifier existence were also a -predicate-, i.e. a property. Let e(x) mean x exists.

Ba'al Chatzaf

If you look at your problem, it will answer its self. You cannot talk about x without it existing (the first sentence that I left of your quote)

Since x must exist to talk about x, then e(x) is a illogical.

Really. Then why do Objectivists talk so much about John Galt and Dagney Taggart. Neither exist.

Ba'al Chatzaf

If you read my response you will see the answer to that, they exist as ideas, not people.

Edit-- They also exist as characters in a book written by the philosopher Ayn Rand

Dustan

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If you read my response you will see the answer to that, they exist as ideas, not people.

Edit-- They also exist as characters in a book written by the philosopher Ayn Rand

Dustan

And unicorns exist in fairy tales along with dragons and wicked witches who cast spells.

Ba'al Chatzaf

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If you read my response you will see the answer to that, they exist as ideas, not people.

Edit-- They also exist as characters in a book written by the philosopher Ayn Rand

Dustan

And unicorns exist in fairy tales along with dragons and wicked witches who cast spells.

Ba'al Chatzaf

Then we agree.

--Dustan

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• 2 months later...

~ My problem with the...er...'concept'...of an empty set as being meaningful is that the very idea of 'set' is supposedly to be regarded as some kind of mental 'box', like "category", but, if there's no contents (so to speak), there's no way of telling one empty (can we say 'folder'?) from any other; in which case, the 'set' of blue-unicorns is no different from the 'set' of square-triangles.

~ Yet, it seems there ought to be a way to distinguish the two, apart from the labelling, no?

LLAP

J:D

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~ My problem with the...er...'concept'...of an empty set as being meaningful is that the very idea of 'set' is supposedly to be regarded as some kind of mental 'box', like "category", but, if there's no contents (so to speak), there's no way of telling one empty (can we say 'folder'?) from any other; in which case, the 'set' of blue-unicorns is no different from the 'set' of square-triangles.

~ Yet, it seems there ought to be a way to distinguish the two, apart from the labelling, no?

LLAP

J:D

Why the scare quote around the word concept? The empty set is the minimal element of a Boolean complemented lattice. It is a perfectly legitimate mathematical idea. Tell me, do you know -any- mathematics at all? Do you know anything about set theory? Do you know any abstract algebra? If the answer is not yes to these questions I suggest you learn something before using quotes in a snide fashion. If you do know something about the aforementioned, then shame on you!

Ba'al Chatzaf

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John,

There is a passage from the epistemology workshops you might find applicaple (ITOE, 2nd. Ed., pp. 304-306):

Prof. C: An imaginary number is supposed to be that number which when multiplied by itself is equal to minus one. There is, in fact, no real number which when multiplied by itself gives you minus one.

AR: What is its purpose?

Prof. C: It turns out that it has a great usefulness as a device mathematically for solving problems of a real kind—for instance, problems involving electrical circuits. But I personally do not see the validity of this concept. There is nothing in reality to which it corresponds. Nothing is measured except by real numbers.

AR: But here there is a certain contradiction in your theoretical presentation. If you say that these imaginary numbers do serve a certain function in measurement, then—

Prof. C: Excuse me, not in measurements of anything, but in computation—in solving an equation.

AR: The main question is: do they really serve that purpose?

Prof. C: In practice, yes.

AR: If they serve that purpose, then they have a valid meaning—only then they are not concepts of entities, they are concepts of method. If they have a use which you can apply to actual reality, but they do not correspond to any actual numbers, it is clearly a concept pertaining to method. It is an epistemological device to establish certain relationships. But then it has validity. All concepts of this kind are concepts of method and have to be clearly differentiated as such.

Whenever in doubt, incidentally, about the standing of any concept, you can do what I have done in this discussion right now. I asked you, "What, in reality, does that concept refer to?" If you tell me that the concept, let's say, of an imaginary number doesn't do anything in reality, but somebody builds a theory on it, then I would say it is an invalid concept. But if you tell me, yes, this particular concept, although it doesn't correspond to anything real, does achieve certain ends in computations, then clearly you can classify it: it is a concept of method, and it acquires meaning only in the context of a certain process of computation.

Therefore, when in doubt about the classification or nature of a concept, always refer ultimately to reality. What in reality gives rise to that concept? Does it correspond to anything real? Does it achieve anything real? Or is it just somebody's arbitrary theory?

Those who habitually bash Rand's knowledge of math usually ignore this side of her (and I suspect they wish it did not exist so their criticisms could be valid). They have been so used to calling "concepts of method" "a priori knowledge," "analytic truth" and other such terms that they simply pull up a blank with this one. They think Rand knew nothing about these things and made up some arbitrary rules and observations to deal with them.

Michael

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Those who habitually bash Rand's knowledge of math usually ignore this side of her (and I suspect they wish it did not exist so their criticisms could be valid). They have been so used to calling "concepts of method" "a priori knowledge," "analytic truth" and other such terms that they simply pull up a blank with this one. They think Rand knew nothing about these things and made up some arbitrary rules and observations to deal with them.

Instead of continuously psychologizing OL members, telling us what they think and what they wish, you'd better check your facts. And calling any criticism of Rand that you don't agree with "bashing" isn't very productive either. In the past I've extensively written about this passage, so I haven't ignored it at all. See for example here (and also the following discussion) and here.

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Dragonfly,

By "side of Rand" I was referring to "concepts of method," not the specific excerpt from ITOE. Incidentally, there is an equivocation you made that jumped out at me. You said:

Now Rand accepted the validity of the concept of imaginary numbers when she was told that they could be used for practical purposes (for example in electrical circuits).

The practical purpose she referred to was given by Prof C, i.e., "computation—in solving an equation," and not merely "electrical circuits." Solving equations is what she meant by "apply to actual reality" in that specific instance.

Michael

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By "side of Rand" I was referring to "concepts of method," not the specific excerpt from ITOE.

Well, this excerpt and a great part of the discussion were about the concepts of method, so I don't see what hairs you're trying to split here.

Now Rand accepted the validity of the concept of imaginary numbers when she was told that they could be used for practical purposes (for example in electrical circuits).

The practical purpose she referred to was given by Prof C, i.e., "computation—in solving an equation," and not merely "electrical circuits." Solving equations is what she meant by "apply to actual reality" in that specific instance.

Oh no, you can solve lots of equations that have no application at all to actual reality (you aren't going to tell me that Rand didn't know that, are you?). She wrote: If they have a use which you can apply to actual reality, but they do not correspond to any actual numbers, it is clearly a concept pertaining to method. From the context it's obvious that she referred to the electrical circuits mentioned by "prof." C. Rand: "If you tell me that the concept, let's say, of an imaginary number doesn't do anything in reality, but somebody builds a theory on it, then I would say it is an invalid concept." Seems crystal clear to me.

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Dragonfly,

This is exactly what I referred to by "pull up a blank." You are misinterpreting Rand and insisting on the misinterpretation.

You are being thrown by Prof. C's random example because he used the phrase "problems of a real kind—for instance, problems involving electrical circuits," and ignoring what followed (like "theoretical presentation," etc.).

Rand was referring to the act of computing. The act of computing happens in reality. Her phrase "which you can apply to actual reality" means something happening in actual reality, which can mean "computing an equation." She is not referring to what the equation is being used for. Acts are part of reality.

A counter-example would be something like numerology, where numbers are used for mystical purposes. Using imaginary numbers in that kind of context would not be a valid act except as a flight of fancy.

You could probably form some kind of "junk concept" with this usage, but not a valid mathematical concept if your standard is reality. In mysticism, 3 can be 1 (literally) and 0 can mean all. This kind of usage is not valid math, neither in reality nor in method.

Michael

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You are being thrown by Prof. C's random example because he used the phrase "problems of a real kind—for instance, problems involving electrical circuits," and ignoring what followed (like "theoretical presentation," etc.).

The "theoretical presentation" obviously refers to "prof." C's argument as a whole (i.e. that imaginary numbers do not measure anything in reality, but that they are nevertheless useful in practical applications, although he thinks they are not a valid concept).

Rand was referring to the act of computing. The act of computing happens in reality. Her phrase "which you can apply to actual reality" means something happening in actual reality, which can mean "computing an equation." She is not referring to what the equation is being used for. Acts are part of reality.

Solving equations is not the same as computing (although some equations may of course be used to compute something). Rand seems to have thought that mathematics is only about crunching numbers (she even called it "the science of measurement"!), which is completely false. It is very well possible to solve equations that have no application to reality, they are "mere theory", in which case Rand says that no valid concepts are used. Building a completely abstract theory and deriving theorems which have no application to reality (or even constructing a completely nonsensical theory) also "happen in reality", but that doesn't imply that they refer to physical reality.

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It is very well possible to solve equations that have no application to reality, they are "mere theory", in which case Rand says that no valid concepts are used. Building a completely abstract theory and deriving theorems which have no application to reality (or even constructing a completely nonsensical theory) also "happen in reality", but that doesn't imply that they refer to physical reality.

The first sentence is not accurate. I believe Rand would say that the equation is not a valid equation or something like that. Even she wrote about unicorns and did not say that no valid concepts were involved.

The second sentence is correct. However, your definition of "happen in reality" appears to be vastly different to hers, and I attribute your blind spot to this.

To her, "concepts of method" are ultimately tied to things like the law of identity and the rules derive from that base. To you, from what I have been able to gather, "analytic knowledge" is merely an accumulation of rules that only happens in our minds.

Michael

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The first sentence is not accurate. I believe Rand would say that the equation is not a valid equation or something like that.

Whatever. It's equally false.

Even she wrote about unicorns and did not say that no valid concepts were involved.

I don't know what she wrote about unicorns, but I doubt that she found it a valid concept.

The second sentence is correct. However, your definition of "happen in reality" appears to be vastly different to hers, and I attribute your blind spot to this.

Where does Rand define "happen in reality"? I was referring to your "the act of computing happens in reality". So does the act of creating nonsensical theories. I see nowhere in Rand's text that while the act of doing something is happening in reality, what you're doing (for example creating some abstract theory or writing some metaphysical speculations) involves valid concepts. She's always talking about what the concepts are referring to in reality. In this case she says that imaginary numbers are a valid concept, not while you can do just anything with them in reality (like reciting the equations), but while you can use them for doing something quite specific, namely calculating values of certain variables (like voltages in an electronic circuit) that can be measured. Those numbers do not refer directly to those values, they are part of a method for obtaining those values, and therefore she calls them a concept of method. (It will be obvious that I don't agree with her definition of the validity of a concept; see my earlier discussions about this subject.)

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Dragonflty,

This part you wrote is exactly right: "I was referring to your 'the act of computing happens in reality.'

Where the waters get parted is with respect to what "computing" means. If you claim that "computing" entails making up a structure where 7 equals three units one time, six units another time, 26789 units another time, there is no way on earth you can establish any valid concept from this. Yet the person is acting if he is going though the motions of drawing equations on paper and the integer 7 is present (with those meanings). So it is not just the act of moving a pencil or pen across paper (or using a calculator or whatever) Rand was referring to. She was referring to "the act of computing," meaning that in that act, the numbers behaved in manners consonant with the rules of math (as derived from the law of identity where 1 is 1 and 2 is 2, etc.).

When a person is calculating according to that method (consonant with the rules of math), he is performing an "act of computing" in reality. Otherwise he is performing some kind of act, but not computing. Within that act of computing, if an apparently "invalid concept" like an imaginary number were to be used because it works and is useful, she claims that the concept of imaginary number is valid, even though it does not directly refer to anything within reality. This is because it is used in a legitimate computation, which is a part of reality—the method part.

If you read the excerpt with an open mind, starting from zero, and set aside the preconceived idea of proving that Rand was a math idiot, you will see it.

She was NOT referring to what the math was to be used for in that example, but to the method—to the math itself. Prof. C's example was a random example. It could have been "math problem" instead of "problems involving electrical circuits." That is why the imaginary number can be called a "concept of method." I am repeating myself, so let's just leave it for another time.

Michael

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She was NOT referring to what the math was to be used for in that example, but to the method—to the math itself. Prof. C's example was a random example. It could have been "math problem" instead of "problems involving electrical circuits." That is why the imaginary number can be called a "concept of method." I am repeating myself, so let's just leave it for another time.

Michael

Then EVERY number is a concept of method. The number 1 has no more physical existence than the imaginary number i, the positive square root of -1.

Bob Kolker

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Bob,

Does the law of identity operate with math?

Michael