It is exact in the a priori intensional sense of being defined as an equation or algorithm with instantly recognizable form. — sime

Set theoretically, it is not defined "as an equation or algorithm". Rather the constant symbol is defined by the ordinary definitional method of mathematics, which is to state a given property that is had by a certain object and no other object.

a sequence of rational numbers — sime

It is not a sequence of rational numbers. It is an equivalence class of Cauchy sequences of rational numbers.

pi as a constant is ambiguous — sime

It is not ambiguous. The constant refers to exactly one object.

Buh bye.

You're trolling. What was uninformed is the post::

I merely stated the needed corrections to your uninformed argument.

You just went right past what I wrote.

* 'infinity' as a noun does not ordinarily have a mathematical definition, though 'is infinite' does. A mathematical definition is never circular nor a tautology.

* The notion of 'potentially infinite' is of course central to important alternatives to classical mathematics. However, as far as I know, formalization of the notion is not nearly as simple as the classical formalization of 'infinite'. Therefore, if one is concerned with truly rigorous foundations, when one asserts that the notion of 'potentially infinite' does better than that of 'infinite' one should be prepared to accept the greater complexities and offer a particular formalization without taking it on faith that such formalizations are heuristically desirable, as we keep in mind that ordinary mathematical application to science and engineering uses the simplicity of classical mathematics as one first witnesses in Calculus 1.

* What writings by intuitionists are fairly rendered as describing potentially infinite sets or sequences as "finite entities of a priori indefinite size" or as "finite entities" of any kind?
.
* One may take mathematics as a priori true without commitment to infinite sets.

* There are rigorous definitions of 'limit', in various contexts, in mathematics.

* To my knowledge, there is no general mathematical definition of 'is a number'. However 'ordinal number', 'cardinal number' and the predicate 'is infinite' have rigorous mathematical definitions, and there are proofs that there are ordinal numbers and cardinal numbers that are infinite.

* The lemniscate [here I'll use 'inf'] does not ordinarily denote a particular object. Probably its two most salient uses are for (1) points on the extended number line and (2) in expressions such as "the SUM[n = 1 to inf] 1/(2^n). With (1), inf and -inf can be any arbitrary objects (they don't even have to be infinite) that are not real numbers, serving as points for the purpose of a system. With (2), 'inf' is eliminable as it is merely convenient verbiage that can be reduced to notation in which it does not occur.

So your examples of arithmetic involving inf are not meaningful. However, there are rigorously defined operations of ordinal addition, subtraction, multiplication, and division.

* I agree that you in particular are better off not speaking on 0!, which also has a rigorous mathematical treatment.

Brouwer's philosophy of Intuitionism, in which ' x1,x2,... ' is interpreted as referring to partially defined finite sequence of unstated finite length, rather than as referring to an exactly defined sequence of actually infinite length. In other words, x1,x2,... is interpreted as referring to a potentially infinite sequence whose length is unbounded a priori, but whose length is eventually finitely bounded a posteriori at some unknown future date. — sime

What passages in the writings of Brouwer (or in writings about him) do you believe are fairly rendered that way?

It's simply impossible! — Agent Smith

It's impossible for you to understand?

What is the first sentence in my previous post that you don't understand?

Try it this way:

Some Americans are Brainy.
All Statisticians are Brainy.

We want to prove:

No American that is not Brainy is a Statistician.

But "No American that is not Brainy is a Statistician" means the same as "If something is an American and not Brainy, then it is not a Statistician."

Now, since, all Statisticians are Brainy, it follows that if something is not Brainy then it is not a Statistician. So, perforce, if something is an American and not Brainy, then it is not a Statistician. QED.

But what about the premise "Some Americans are Brainy"? Well, we never used it. We didn't need to. Which is fine. If a statement (such as "All Statisticians are Brainy") proves a conclusion, then that statement plus any other extra unneeded statement (such as "Some Americans are Brainy") still proves the conclusion (this is the principle of Monotonicity of Entailment).

I was trying to see if the two statements could be used to form a classic syllogism — Agent Smith

Yes, we went through that with tim wood. Anyway, glad that you see now that (D) is the answer.

:roll: — jgill

I don't know what to make of people who still can't see it after it's been explained six ways to Sunday.

there's no need to rack our brains on such a simple matter. — Agent Smith

Yes, it is a simple matter that (D) is the correct answer.

1. Some Ayes are Bees.
2. All Seas are Bees.

No conclusion follows. — Agent Smith

No, many conclusions follow. And one of them is:

No Ayes that are not Bees are Seas.

Moreover, it follows from "All Seas are Beas" alone.

And one shouldn't have to rack one's brain to see that, except you still haven't racked your brain enough.

If some ayes are bees, and seas are bees, then some ayes are sees. Unless not all bees are sees, which isn't given. — John McMannis

Yes, "Not all Beas are Seas" is not a premise. But "All Beas are Seas" is also not a premise. So you don't get to use either in the inference.

An inference is not valid when there is an example in which the premises are true and the conclusion is false. Here's an example.

Let the set of Ayes be {Jack}

Let the set of Beas be {Jack, Lucy}

Let the set of Seas be {Lucy}

So:

"Some Ayes are Bees" True
"All Seas are Bees" True

"Some Ayes are Seas" False

So the inference is invalid.

Because you never said you gave it up; and your next post seemed to still be trying to connect existential import to what we had been discussing. Granted, it is also reasonable that you were not trying to make the earlier connection, in which case I would grant my previous post would have been beating a dead horse.

This goes to the existential problem — tim wood

Yes,that matter hinges on existential import. But the problem in the first post of this thread does not. Nor does your other concern about undistributed middle.

"The presupposition [...] contradictory relation." — tim wood

All of that quote seems correct to me and it in no way vitiates anything I've said, and it in no way supports your notion that the question of this thread hinges on existential import or undistributed middle.

house rules — tim wood

Of course, discussions about drawing inferences need to be in context of what principles of logic we have in mind. But the question in this thread has been answered according to everyday principles of reasoning, which also are formalized. And those particular principles do not hinge in any way on matters of existence or vacuity. I have explained exactly why that is in this case. I don't know why you continue to ignore it.

You mentioned cutting a knot with a knife. I rebutted that analogy already. But with your fixation that existential import plays a role in the particular question of this thread, you remind me of the saying that if a person has only a hammer then everything looks like a nail.

Equivalences:

"It is not the case that some cats are mammals"
is equivalent to
'No cats are mammals"

"It is not the case that all cats are mammals"
is equivalent to
"Some cats are not mammals"

"It is not the case that no cats are animals"
is equivalent to
"Some cats are mammals"

"It is not the case that some cats are not mammals"
is equivalent to
"All cats are mammals"

In basic logic:

"Some cats are mammals"
means
"There is a thing that is both a cat and a mammal"

"All cats are mammals"
means
"If a thing is a cat then it is a mammal"

"No cats are mammals"
means
"If a thing is a cat then it is not a mammal"
means
"All cats are not mammals"

"Some cats are not mammals"
means
"There is a thing that is a cat and is not a mammal"

There is no ambiguity.

/

Moreover:

"Some cats are mammals" does NOT imply "Some cats are not mammals".

"All cats are mammals" does NOT imply 'Some cats are mammals" (because, if there are no cats, then "All cats are mammals" is vacuously true but "Some cats are mammals" is false). (Though I don't recall what, if anything, Aristotle said about that; and, while the notion of vacuous truth is basic in usual formal logic, it is not ordinarily used in everyday logic.)

If you have a dollar in your pants pocket, do you (not) have also 32 cents?
— tim wood
No, you don't. A dollar is a dollar and cents are cents. Also, you cannot use some vending, gambling etc. machines if you don't have the exact amount of cents. — Alkis Piskas

That is ridiculous captiousness. The example is not vitiated by quibbles about the difference between coins and bills. The point of the example is that you can have Some and also All.

Again, I pointed out to you that in the context of basic logic, 'some' means 'at least one' and doesn't mean 'some but not all'.

it is true because its inference is valid — Alkis Piskas

To be clear, a valid inference does not ensure the truth of the conclusion. A valid inference does ensure the truth of the conclusion when either (1) the premises are true or (2) the conclusion is logically true anyway.

A) Some animals are cats: True, since mammals are animals (based on the first premise) and cats are mammals
— Alkis Piskas
You can infer this adding additional information, but you cannot from the premises given validly conclude it.
— tim wood
You are right that you have to infer it, i.e. we don't know that directly, but it is true because its inference is valid, — Alkis Piskas

No, your inference is not valid. It is true that some animals are cats, but it does not follow from your premises.

Mammals are a subset of animals. Cats are a subset of mammals. That is, cats are a subset of a subset of animals. — Alkis Piskas

Yes, that is valid. But that is different from your original argument.

It is true that some animals are cats. But it is not entailed by your premises.
— TonesInDeepFreeze
If it is true, well, it is True! That's what I said! — Alkis Piskas

No, what you said is that it follows from your premises.

Saying that "some cats are mammals" suggests that there are some cats that are not mammals. — Alkis Piskas

In certain everyday contexts, yes, 'some' may mean or at least suggest 'not all'. But not in the study of basic logic. I'll say it again: In ordinary basic logic [also, in certain other everyday contexts]
'some' means 'at least one' and it doesn't mean 'some but not all'.

(A) is true, but it is not entailed by your premises.
— TonesInDeepFreeze
Yes, you have already said that! — Alkis Piskas

Yet you made the same mistake in a subsequent post!

B) Some cats have four legs: False, since we know that "All cats have four legs" (and not only some) — Alkis Piskas

Wrong. For the fourth time, I'm telling you that in basic logic 'some' means 'at least one' and not 'some but not all'.

"No animals" is ambiguous — Alkis Piskas

I pointed out before that that is wrong. You merely persist in claiming again what has already been explained to you to be incorrect.

Drawn from what? The premises. And what is to be drawn from the premises? A conclusion. — tim wood

Yes, so what?

you have made it clear that your methods are not those of the problem. — tim wood

False. I already addressed that. Please read again:

The question was "which conclusion can be drawn?" The question was not "which conclusion can be drawn by the method of Aristotelian syllogisms?". — TonesInDeepFreeze

Please do not elide that again.

This is by all appearances an Aristotelian logic game. You appear to admit as much: — tim wood

I said no such thing.

The question was "which conclusion can be drawn?" The question was not "which conclusion can be drawn by the method of Aristotelian syllogisms?".

You also more-or-less plainly imply that the law of undistributed middle does not apply. — tim wood

Undistributed middle is a fallacy. I never said otherwise. But that in no way vitiates that from

"All Seas are Bees"

we may validly conclude

"No Ayes that are not Bees are Seas".

Period.

I don't understand why you don't understand that, except that it seems you have stuck in your head that valid inferences regarding "Some, All, and No" must be within the scope of the method of Aristotelian syllogisms.

Now it is for you to demonstrate how it does not apply in any of your standard logics - without adducing premises or information not already provided to make it seem as if it does not. — tim wood

What? It's basic everyday logic. And if that doesn't satisfy you, then one could formalize it in basic symbolic logic.

Indeed, the two salient principles used are Modus Tollens and Monotonicity of Entailment. It's pretty much that simple.

Or in short, how can you say anything categorical about something that has not already been categorically defined - without somehow adding the missing qualifications? — tim wood

I don't know what you think you mean by "categorically defined".

Meanwhile, it is a plain fact that "All Seas are Bees" entails "No Ayes that are not Bees are Seas".

And there are no "qualifications" needed. It is as clear as day in everyday reasoning, and it is as clear as a day on the sun with symbolic logic: Put another way:

Any circumstance in which All Seas are Bees is a circumstance in which No Ayes that are not Bees are Seas.

That is logical entailment.

From the premise "All Seas are Bees" we most certainly can validly draw the conclusion "No Ayes that are not Bees are Seas."

Period.

I already walked you through an English language demonstration of that. Or, I could do it formally in symbolic logic if anyone was captious enough to demand it.

So, in Boolean or first-order or whatever order logic are undistributed middles no longer fallacious? — tim wood

Of course it is a fallacy. But that is not relevant because I am not using that fallacy. I am not arguing in an Aristotelian syllogism. I am not in any way committing undistributed middle, because I'm not even inferring syllogistically.

You really still don't understand this?

The problem, noted above, is called undistributed middle. — tim wood

If by 'above' you mean the sentence you wrote before that one, then, yes, undistributed middle.

But if by 'above' you mean the discussion about Ayes, Bees, and Seas and your thought that (D) is not entailed by the premises, then you still have a severe misconception. An argument may be valid even if its validity is not within the Aristotelian syllogistic forms.

Some animals are cats: True — Alkis Piskas

It is true that some animals are cats. But it is not entailed by your premises.

Some cats are mammals: False, — Alkis Piskas

Wrong. In basic logic such as this, 'Some' means one or more. 'Some' does not mean 'Some but not All'.

No cats are animals: "No cats" is ambiguous — Alkis Piskas

Wrong. 'No' means 'none of' and is unambiguous.

there are two true statements, (A) and (D). — Alkis Piskas

(A) is true, but it is not entailed by your premises.

what, if any, house rules may be in effect. — tim wood

The house rules are everyday common reasoning.

Aristotelian logic, on the other hand, may be not so useful in some modern applications, but it is not wrong. — tim wood

It's not wrong. It's just that it is nowhere close to covering much of everyday deductive reasoning.

only in outdated, Aristotelian categories it's "E" — DavidJohnson

Because Aristotelian syllogisms do not exhaust even very basic reasoning.

You get your D, but only on expanding the terms of the problem. — tim wood

False. I expanded nothing. I made no assumption other than the premise "All Seas are Beas".

"All Seas are Beas" implies "All non-Beas are non-Seas" implies, perforce, "All things that are both non-Ayes and non-Beas are non-Seas".

And that argument holds whether or not there are Ayes or Beas or Seas.

There is no existence premise nor existence conclusion involved.

/

It is alarming that someone would fail to understand the correctness of the inference, irrespective of training in logic, but rather just as a matter of commonly acquired reasoning.

Else we willy-nilly prove the existence of God, Zeus, unicorns, and the two-horned rhinoceros sleeping in my bed. — tim wood

This argument does not prove the existence of anything.

it asks for a conclusion, not an inference — tim wood

The problem was not what conclusion is true, but rather what conclusion can be drawn from the premises. That is inference.

If, for example, there were As that at the same time are not Bs, then D follows. But that hypothetical is not given. — tim wood

(D) follows from the premises (actually, only one of the premises is needed) no matter what does or does not exists.

I spelled out the logic of the argument explicitly. There is no existence assumption in the argument. Moreover, A's are irrelevant to the argument.

Don't need a logic professor.

If x is a C, then x is a B.
Therefore, if x is not a B, then x is not a C.
So, perforce, if x is both an A and not a B, then x is not a C.

That argument involves no assumptions about whether or not there is a something that is an A, B, or C.

the middle term is not distributed, which means no valid conclusion can be drawn. — tim wood

Whether the conclusion is or not inferred according to syllogistic forms, it is inferred validly.

This follows iff there is an A that is not a B. — tim wood

All Seas are Bees

entails

No Ayes that are not Bees are Seas

whether or not there is an Aye that is not a Bee. Ayes don't even have anything to do with it. And there is no need for an existence assumption.

You've got something stuck in your head that is not the case.

Run it through your mind, or write it in symbols [where 'U' and 'E' are the universal and existential quantifiers]:

Ux(Sx -> Bx)

entails

Ux((Ax & ~Bx) -> ~Sx)

or couched equivalently:

Ux(Sx -> Bx)

entails

~Ex(Ax & ~Bx & Sx)

It's clear as day:

If there were a thing that is both not a Bee and is a Sea, then that would contradict that all Seas are Bees. And that is the case whether there is or is not anything that is an Aye or a Bee or Sea or any combination.

Some Ayes are Bees
All Seas are Bees

.

1. Some Ayes are Seas (true, but a premise)
2. Some Ayes are not Seas (undecidable)
3. All Ayes are Seas (undecidable)
4. No Ayes are Seas (false) — Agent Smith

1. Is not a premise and it is not entailed by the premises.

4. Is not made false by the premises.

If the proposition, "No Ayes are Seas" were added, would D still be valid? — DavidJohnson

The question is not the validity of (D) but the validity of inferring (D) from the premises.

(D) is validly inferred from the premises. Adding an additional premise cannot vitiate an otherwise valid argument. This is the monotonic property of basic logic.

D looks tempting, but depends on the implicit added proposition that if some As are Bs, then some As are not Bs — tim wood

That is not needed to see that (D) is the correct answer.

It is quite simple, without even need for Venn diagrams or symbolic logic.

(D) can be couched in two equivalent ways:

"If a thing is an Aye and not a Bee, then it is not a Sea".

"There is no thing that is an Aye and not a Bee and a Sea".

And either way you couch (D) it is entailed by the premise:

All Seas are Bees, or couched equivalently:

"If a thing is a Sea then it is a Bee."

Theorem: There is no S such that S is in the domain of S.

Proof: Use axiom of regularity.

Outside of mathematics, I believe that some people do experience infinity in mediation, other spiritual practices, and with psychedelics. Also, one can experience the law of non-contradiction transcended, so that all dualities are one.

It cause for misunderstanding to say that in mathematics infinity is an object. Granted, there are objects sometimes called 'infinity', such as points on the extended real line. But the more general set theoretic notion of infinity is not of the noun 'infinity' but of the adjective 'is infinite'. Overlooking that distinction often leads to serious misconceptions about how set theory and mathematics treat the subject.

'is infinite', as a set theoretic notion, is a 1-place predicate defined:

S is finite iff there is a bijection between S and a natural number

S is infinite iff S is not finite

S is Dedekind-infinite iff there is a bijection between S and a proper subset of S

Salient about infinite sets is that every Dedekind-infinite set is infinite (this is the other side of the coin of the "pigeonhole principle"), and, with an appropriate choice axiom, every infinite set is Dedekind-infinite.

And mathematical infinity is not just magnitude or number. Yes, there are infinite ordinals and infinite cardinals, which are called "numbers" (there is not, as far as I know, a mathematical definition of 'is a number') but there are infinite sets that are not ordinals or cardinals.

Regarding a remark in a previous post, yes I am happy to suggest study materials that would provide understanding of these points:

(1) A bijection between a proper subset, or even the entire set, of terminating decimal expansions (which reduces to the set of finite sequences of natural numbers) and the set of natural numbers is not a bijection between [0 1] (which is represented by the set of denumerable (not terminating) expansions) and the set of natural numbers.

(2) Among the members of [0 1] not represented by terminating expansions are both some of the rational numbers in [0 1] and all of the irrational numbers in [0 1]. Those numbers are not represented in the domain of the bijection.

(3) To answer "What are examples of members of [0 1] that are left out of the domain of the bijection?", we could mention for example: .333.. (i.e. 1/3) and .14159265358979323846264338327950288419716939937510... (i.e. the decimal portion of pi).

Also, the domain of the bijection is only a proper subset of the set of terminating expansions, as for example, not even .01 is a member of the domain. However with a more careful construction, of course, we can show a bijection between the entire set of terminating decimal expansions and the set of natural numbers.

(4) To answer "What natural number is not in the range?", we say that no natural number is not in the range. But so what? Uncountably many members of [0 1] are not in the domain, so it is not a bijection between [0 1] and the set of natural numbers. Of course, any mathematician knows that there is a bijection between the set of terminating expansions and the set of natural numbers. Indeed it is a basic tool of formal languages and computing that the set of finite sequences on a countable set is countable. And, indeed, Cantor proved that there is a bijection between the set of rational numbers and the set of natural numbers, thus, a fortiori, there is a bijection from the set of terminating decimal expansions into the set of natural numbers. But Cantor also proved that there is no bijection between the set of denumerable (non-terminating) expansions and the set of natural numbers. Versions of the proof may be found in any good textbook on set theory.

Suggested study materials (studied in this order).

Logic: Techniques Of Formal Reasoning - Kalish, Montague and Mar. I consider this to be the best textbook for learning symbolic logic, which is extremely helpful (for me, crucial) for understanding set theory.

Introduction To Logic - Suppes. This has the best explanation (it is superb) of formal definitions that I have found.

Elements Of Set Theory - Enderton. My favorite textbook on set theory. It is beautifully written.

Axiomatic Set Theory - Suppes. Good as a backup to Enderton.

Philosophy Of Set Theory - Tiles. An overview of the intuitions and philosophy behind various views on set theory and mathematics.

So maybe what you mean by "0.1-1" is [0.1 1]?

If you're going to ask me to explain why you have no proof, then it would help for you to use recognizable notation, and would be more courteous too.

Now you say

Between 0.1 and 0.99999.... you use all numbers of N — AgentTangarine

What does "use the numbers mean"?

Do you mean they are all in the range of a function from N to some set? WHAT function? You have not adduced any function. For any given k in N, what is f(k)?

1,2,3,....9999999.... — AgentTangarine

So maybe you just mean that all the natural numbers are in the domain? Well, duh, yeah. So what? The domain is not at issue. It is the range that is at issue. You need to prove that there is a function whose domain is N and whose range is R (i.e. every real number is in the range).

So I ask:

Are you clear that your task is to prove that there is a function whose domain is N and such that every real number is in the range of the function?

I asked you first:

Cannot N be mapped onto 0.1-1?
— AgentTangarine

Do you mean to suggest that there is a 1-1 function from N onto 0? — TonesInDeepFreeze

You say it doesn't apply, yet you mentioned it directly in connection.

I might think you are trolling me, but even worst trolls don't usually have your endurance.