And the way for you to do that is to read a book on the subject.

I will not impose upon myself a restriction from commenting on your posts.

Aren't odd numbers a part of natural numbers? Is it not true that the cardinality of the former equals that of the latter? — Agent Smith

Yes. That has never been in question here. Indeed I reiterated just what you said in my post that you are replying to now! What is in question here is your your ignorant misunderstanding that equality of cardinalities in set theory implies that set theory says that a (proper) part can be equal to the whole. Please pay attention to exactly what you have said and what I have said.

Again, in set theory it is the case that infinite sets are such that they have proper subsets of the same cardinality as the set. But that is not at all to say that there are sets S and T such S is a proper subset of T and S is equal to T.

If T is infinite, then there exists a proper subset S of T such that there is a bijection between S and T (thus card(S) = card(T)). But there are no sets S and T such that S is proper subset of T and S = T.

By the way. In set theory we have these definitions:

T is infinite <-> T is not finite

T is Dedekind infinite <-> there is a proper subset S of T such that there is a bijection between S and T

In set theory, even without choice, we prove:

there is a proper subset S of T such that there is a bijection between S and T -> T is infinite

In set theory, with choice, we prove:

T is infinite -> there is a proper subset S of T such that there is a bijection between S and T

So, in set theory with choice, the definitions of 'infinite' and 'Dedekind infinite' are equivalent anyway:

T is infinite <-> T is Dedekind infinite.

Also, even without choice, we can prove that there do exist Dedekind infinite sets, such as your example of the odds and the naturals.

So I take it that restricting 'object' to refer only to abstractions is not acceptable to you. Thus, indeed you do not agree that abstractions are objects. Thus, indeed you contradict yourself when you also said: — TonesInDeepFreeze

Why do you conclude that? — Metaphysician Undercover

I take it that you don't take it that the only objects are abstractions, because you went on to say why you don't take it that the only objects are abstractions. You wrote:

"I cannot agree that abstractions are objects, unless we restrict "object" to refer only to abstractions. But then we could not use "object" to refer to anything else, or we'd have equivocation. And we would have to create a special form of the law of identity, such that when 'the same' abstraction exists in the minds of different people, we can still refer to it as "the same" abstraction, despite accidental differences between one person and another, due to different interpretations. The current law of identity requires that accidental differences would constitute distinct 'objects' which are therefore not the same, so we'd need a different law of identity." [Bold added]

The bold part is your argument why you don't restrict 'object' to refer only to abstractions.

Or, are you now saying that the only objects are abstractions?

Yes, I knew you would reply by shifting back around again from your own words but pretending that you haven't. This will go on indefinitely with you, as you play a silly game that is the forum equivalent to a child's peek a boo.

The basic set theoretic structure of the reals underlies almost everything I have done, but I haven't used infinity as a "point" — jgill

Makes sense.

Classical means the tools of analysis like limits, differentiation and integration and all those entail. Nitty gritty. Actual specific results vs broad generalities. — jgill

That differs from how I find 'classical' is used. I find that 'classical' mathematics means all and only those results that can be formalized as theorems of ZFC with classical logic. And classical logic means the first order predicate calculus including the law of excluded middle.

Clarification:

When you said "I have never used infinity as anything more than unboundedness", perhaps I misunderstood you. I thought you meant 'infinity' in the sense of infinite sets. That is, I thought you meant that you recognize that certain sets are infinite, but you don't use them.

But maybe you didn't mean that you don't use those sets. But that you do use them, but you don't use the extended real line with its points of infinity? As instead you simply deploy the fact that the reals are unbounded?

Extra brownie points. — Kuro

Your defensive sarcasm is misplaced.

I reiterated the point that I was correct to support the additional point, which you did not mention, that I was also not arrogant about it.

I'm not sure how your post relates to my quote. — jgill

You said, "I have never used infinity as anything more than unboundedness."

I don't opine as what 'used' means there. I only pointed out that the calculus uses certain infinite sets, even if not explicitly. Just the real line alone is based on having the infinite set of real numbers.

It's just that conversations involving cardinalities beyond ℵ1 don't usually occur in classical or even much of modern analysis. — jgill

(1) I only said that infinite sets are used. I didn't say infinite sets with cardinality greater than aleph_1 are used.

(2) Just to be clear, we don't know what aleph is the cardinality of the set of real numbers.

(3) What is the difference you have in mind between classical and modern? Ordinary contemporary analysis is classical analysis.

/

As to the Wikipedia quote, of course I agree, and I mentioned the distinction between the notion of infinite size and the notion of points of infinity.

What I quoted and my reply:

a part is equal to the whole
— Agent Smith

No, set theory does not say that there is a proper subset of a set such that the proper subset is the set. Set theory does say that there are sets such that there is a 1-1 correspondence between a proper subset of the set and the set. — TonesInDeepFreeze

I made clear that I was responding regarding set theory. Granted, I didn't include in his quote the part - that makes even more explicit that the context is set theory - where he specifically mentioned 'infinity' and 'Cantor'. Also, I didn't belabor that the previous context was infinity especially as Agent Smith faults the notions of infinity and set theory (with your comments about mereology running alongside but separate from the particular exchanges between Agent Smith and me). In all that context, if one paid attention to the conversation, rather than just knee-jerking to one quote in it, it was clear that set theory was what was being discussed at that juncture.

"part is equal to whole" at face-value, just is a mereological truism — Kuro

(1) It's my very point that Agent Smith seems to have conflated the set theoretic notion that an infinite set T has proper subsets equinumerous with T with the mereological notion of part/whole.

(2) Agent Smith was rejecting the notion that a part is equal to a whole. Obviously, that is not about a non-proper part being equal to a whole, but rather a proper part being equal to a whole. So that he's not adducing the truism you mentioned.


/

I would have made it easier for you if I had quoted him more fully. But even then, I did not distort him by quoting more narrowly. I was correct in my reply to Agent Smith. It was not arrogant of me. You were wrong to claim I was not correct or that I was arrogant about it.

Is infinity a contradiction? It does lead to some rather odd conclusions: a part is equal to the whole and all that. No wonder many mathematicians (recall Kronecker's vitriol against Cantor) were dead against it. — Agent Smith

No, set theory does not say that there is a proper subset of a set such that the proper subset is the set. Set theory does say that there are sets such that there is a 1-1 correspondence between a proper subset of the set and the set. — TonesInDeepFreeze

Part and whole have nothing to do with set and subset — Kuro

I didn't say that subset and set align with the mereological notions of part and whole.

Agent Smith said that said set theory allows that a part can be equal to a whole. I correctly pointed out that that is not true. (For that matter, 'part and whole' are not even terms of set theory). And I correctly pointed out that what set theory does say is that in some cases a proper subset is equinumerous with its superset.

And Agent Smith made no reference to 'part' and 'whole' as technical terms of mereology. And it doesn't matter anyway. Whatever mereology has to say, set theory does not say that a part can be equal to a whole.

does not excuse your hostility — Kuro

I am hostile to him only in a broad sense that includes that I decry his continual (over many posts and many months) ignorant and willful falsehoods, misrepresentations and confusions of the subject.

So AgentSmith was correct, and your "correction" of him is a result of conflation of mereology with set theory on your part — Kuro

You are either confused about the context of the posts or you are willfully fabricating about it.

What I responded to:

Is infinity a contradiction? It does lead to some rather odd conclusions: a part is equal to the whole and all that. No wonder many mathematicians (recall Kronecker's vitriol against Cantor) were dead against it. — Agent Smith

He's not talking about infinity per mereology. He's talking about the Cantorian notion. The mathematicians who were against Cantor's notion of infinity were not taking Cantor to have presented a mereology but rather indeed a mathematical notion of sets.

Agent Smith is claiming that the notion of infinity, as in set theory (for example, as he mentioned, set theory engendered by Cantor), leads to the conclusion that a part is equal to the whole. And I correctly replied that that is not true, though it is true in set theory that in some cases a proper subset S or T is equinumerous with T.

/

Bringing a mereological perspective to the subject is fine. But it is a red herring to put my exchange with Agent Smith in context of mereology when the context was clearly set theory.

this is a philosophy forum, not a math forum. — Metaphysician Undercover

If part of one's philosophizing about mathematics includes criticisms of certain actual mathematics, then one should know enough about that actual mathematics that one doesn't misconstrue and misrepresent it. And if one proposes a certain alternative philosophy of mathematics, then it is natural to ask "To what actual alternative mathematics does your alternative philosophy pertain?"

"I cannot agree to abstractions as objects, without specific restrictions", does not contradict with "I can readily conceive of abstract objects". — Metaphysician Undercover

What you wrote:

I cannot agree that abstractions are objects, unless we restrict "object" to refer only to abstractions. But then we could not use "object" to refer to anything else, or we'd have equivocation. And we would have to create a special form of the law of identity, such that when 'the same' abstraction exists in the minds of different people, we can still refer to it as "the same" abstraction, despite accidental differences between one person and another, due to different interpretations. The current law of identity requires that accidental differences would constitute distinct 'objects' which are therefore not the same, so we'd need a different law of identity. — Metaphysician Undercover

So I take it that restricting 'object' to refer only to abstractions is not acceptable to you. Thus, indeed you do not agree that abstractions are objects. Thus, indeed you contradict yourself when you also said:

I readily conceive of abstract objects — Metaphysician Undercover

To put it starkly:

"I cannot agree that abstractions are objects" is tantamount to "abstractions are not objects".

"We restrict 'object' to refer only to abstractions" is tantamount to "only abstractions are objects".

So what you said is tantamount to: Abstractions are not objects unless only abstractions are objects. But you also deny that only abstractions are objects. Thus you affirm that abstractions are not objects.

Or:

Let 'P' stand for 'abstractions are not objects'.

Let 'Q' stand for 'only abstractions are objects'.

You say 'P unless Q'. But you deny Q. So you affirm P.

You host a continually silly shell game. I shouldn't indulge in responding indefinitely.

maybe we need someone else, an arbiter — Agent Smith

Actually, you could start by just refraining from making claims for which you have no basis.

But if you do want to know more about Cantor's life then there is the Dauben biography.

you're too technical for my taste. — Agent Smith

That's such a cop out. When a person such as you posts a bunch of wildly intellectually irresponsible, incorrect and confused bull, it's not being "too technical" for me to flag it and sometimes even, gratis, provide explanation regarding it.

Anyway, I am truly curious why you thought of extrapolating from the fact that Cantor had breakdowns and depressions and was in sanitoriums to claiming that he "lost his mind" and was in a "lunatic asylum". Or you just like to bolster your point of view about mathematics by making stuff up and post it as if it's fact?

you would never try to provide an infinite list of points to completely describe a line (Cantor) — keystone

Cantor doesn't do that. In fact, Cantor proved that that CAN'T be done. It's his MOST famous result.

You have it completely backwards.

What articles have you read about Cantor that have led you to your terrible misunderstandings?

EDIT: I struck out my message here. I misunderstood the poster. He did not say that Cantor said the continuum can be listed; rather he said that Cantor said that that cannot be done, which is correct. My message is fully retracted.

I have nothing more to contribute. — Agent Smith

The word 'more' there is excess.

logicism, the ideology that there is a single correct logical definition of a mathematical object — sime

That is not what logicism is.

infinite object, something beyond our comprehension — keystone

I comprehend the notion of an infinite set.

I'll leave you to discuss with the other experts. Good day. — Agent Smith

Yeah, you often ditch an exchange with that arch "Good day" sign off, while not ingesting a single bit of the information and explanation given to you.

Cantor lost his mind (theia mania) and spent his later years in a lunatic asylum for instance. These concepts & paradoxes of which there are many seem to have a deletorious effect on the brain/mind - constantly mulling over them may lead to a nervous breakdown. — Agent Smith

(1) By what source do you assert that Cantor "lost his mind"?

Cantor had collapses and severe depression. I don't know of any source that says he "lost his mind" in the sense of insanity such as schizophrenia, delusions or hallucinations. One does not ordinarily say of people who are depressed that they "lost their mind".

(2) By what source do you assert that Cantor was in a "lunatic asylum" (thus suggesting that he was himself a lunatic)?

Cantor went to sanitoriums for his collapses and depression. I don't know of any source that says he was instutionalized as a lunatic.

(3) It is not a safe inference that Cantor's mathematical work itself caused his collapses. Famously there were other stressors at work.

/

Now, let's deal with the heart of this. Whatever mental problems Cantor had, they do not refute the insights of his work. Otherwise, would be an ad hominem argument. And I mentioned also that the core of his work is easily detachable from his religious beliefs.

It's fine to be interested in Cantor's biography, but it's beneath the dignity of even as common a forum as this to argue or even insinuate that his mental difficulties enter into a fair evaluation of his work.

No. You're just being glib and not even thinking about what I wrote; just pouncing in an ill considered way about it.

First, 'finitist' has many different senses in the philosophy of mathematics.

Second, recognizing that set theory doesn't have 'infinity' as a noun but rather 'is infinite' as an adjective, does not at all entail that one shouldn't also assert that there exist sets that are infinite.

∞ isn't and object — Agent Smith

That's right.

In previous threads I've pointed out that in set theory there is not a noun 'infinity'.* Rather there is the adjective 'is infinite'.

This is a crucial point for understanding how the subject of infinite sets is approached in set theory. To continue to ignore this point is to commit to continual confusion about the very foundational notions.

* Notations such as 'as n goes to infinity' are abbreviations for formal writing that dispenses with 'infinity' as a noun. And points of infinity and negative infinity are something different too.

it is challenging for me to envision the existence of a set of all natural numbers. Without assuming its existence, accepted set theory doesn't get far off the ground. — keystone

The existence of the set of natural numbers is derived axiomatically. Granted, the key axiom is that there exists a successor inductive set, which is an infinitistic assumption.

On the other hand, the notion of "potential infinity" demands alternative axioms.

Take just the non-infinitistic axioms of set theory. What axioms does the "potential infinity" proponent add to get real analysis?

I think I have a grasp of how real numbers play into accepted set theory — keystone

You don't.

A bit of magic is needed to make the leap from a finite collection of points forming nothing to an infinite collection of points forming a continuum. — keystone

As I mentioned, that is not how it is done. You would do yourself a favor by reading a good textbook on the subject so that you would have a basis to critique the actual mathematics rather than what you only imagine is the actual mathematics.

Why can't we just say that pi is not a number? Instead, it is an algorithm — keystone

Fine. But it's not easy to axiomatize real analysis that way.

One can philosophize all day about how one thinks mathematics should be. But other folks will ask "What are your axioms?" They ask because they expect that an alternative mathematics should have the objectivity of set theory, which is utter objectivity in the sense that, by purely algorithmic means, we can definitively determine whether a purported proof is actually a proof.

I've heard people say that the paradoxes entwined with actual infinities are beautifully mysterious...I just think they demonstrate the flaws of the concept of actual infinity. — keystone

What specific paradoxes do you refer to?

Keep in mind that no contradiction has been found in ZFC.

[Revised post:]

Consider the sentence:

'Snow is white' is true if and only if snow is white.

That sentence is written in a metalanguage.

The left side of the biconditional is:

'Snow is white' is true.

There 'Snow is white' is a quotation of the sentence:

Snow is white.

which is a sentence of the object language.

The whole sentence is of the form:

'P' is true if and only if P.

/

But we can make this even more precise, using Tarski's method of models:

Consider a formal sentence such as:

0+2 = 2

'0+2 = 2' is true if and only if the denotation of '0+2' is the same as the denotation of '2'.

If, with the interpretation of the language we are using, the denotation of '0' is the number zero, and the denotation of '2' is the number two, and the denotation of '+' is the addition operation, and the denotation of '=' is the identity relation, then:

'0+2 = 2' is true in this interpretation if and only if zero plus two is identical with two.

/

With the original example:

If, with the interpretation of the language we are using, the denotation of 'snow' is:

precipitation in the form of small white ice crystals formed directly from the water vapor of the air at a temperature of less than 32°F (0°C)

and the denotation of 'white' is:

has the achromatic object color of greatest lightness characteristically perceived to belong to objects that reflect diffusely nearly all incident energy throughout the visible spectrum

then:

'Snow is white' is true in this interpretation if and only if precipitation in the form of small white ice crystals formed directly from the water vapor of the air at a temperature of less than 32°F (0°C) has the achromatic object color of greatest lightness characteristically perceived to belong to objects that reflect diffusely nearly all incident energy throughout the visible spectrum.

I find it hard to imagine that something (an n-dimensional continuum) can be constructed from nothing (0-dimensional points). — keystone

First, there are two different notions of 'the continuum'. One is that the continuum is the set of real numbers R. The other is more specifically that the continuum is R along with the standard ordering on R, or formally the ordered pair <R L> where L is the standard 'less than' ordering on R.

Second, where can one read of a notion of the real continuum as an "n-dimensional continuum"? What does it mean?

Third, the set of real numbers is not constructed from nothing. The set of real numbers is constructed as the set of Dedekind cuts of rational numbers (alternatively, as equivalence classes of Cauchy sequences of rational numbers), and the rational numbers are constructed as equivalence classes of integers, and the integers are constructed as equivalence classes of natural numbers, and the set of natural numbers is derived axiomatically from the set theory axioms. That is not nothing.

/

Suggestion: Since you are interested in formulating an alternative to infinitistic mathematics, then you would do yourself a favor by first reading how infinitistic mathematics is actually formulated, as opposed to how you only think it's formulated, and also you could read about non-infinitistic alternative formulations that have already been given by mathematicians.

/

I do feel that there is a little bit of Cantor's nonsense implied in any view that supports actual infinities. — keystone

That's what you feel. But you've not supported it. If by "Cantor's nonsense" you mean his religious beliefs, then it is plain, flat out false that axiomatic infinitistic mathematics implies Cantor's religious beliefs.

This is important to recognize:

(1) Cantor's work is not axiomatic. His work was from before the modern axiomatic method reached a satisfactory stage. There are mathematical difficulties with Cantor's work due to the fact that it's not axiomatic.

(2) No matter what one thinks of Cantor's religious beliefs and how he related them to mathematics, we can separate the wheat from the chaff by recognizing the conceptual advantages of Cantor's set theoretic work without also including his religious views about it.

(3) The inconsistencies that he tried to explain by religious notions do not (as far as we know) occur in actual axiomatizations by later mathematicians.

(4) Given (2), other than for historical appreciation and for informal motivation, it is not needed for mathematicians to refer to Cantor at all. Once we had Zermelo's work, and then as it itself is rendered by the methods of symbolic logic, for purposes of formal mathematics, we could forget that Cantor even existed.

The OP mentions Aristotle's distinction of actual vs. potential infinities. The Wikipedia page on the subject doesn't explain the difference between the two all that well. — Agent Smith

The adjective 'is infinite' is mathematically defined in the formal theory, set theory. I have seen no formal theory in which the adjective 'is potentially infinite' is mathematically defined.

A mathematical definition of an adjective 'P' is of the form:

Px <-> Q,

where 'Q' is a formula that has no free variables other than possibly 'x' and no symbols other than the primitives or previously defined symbols.

It's something to keep in mind that people who use 'is infinite' are at least backed up by a mathematical definition, while those who rely on a notion of 'potentially infinite' are not.

I have never used infinity as anything more than unboundedness. — jgill

In Calculus 1 classes, there is not a concern that the subject be axiomatized. But if we are concerned with having the subject axiomatized, then the ordinary mathematical context is one in which there are infinite sets. Just take an infinite sequence of real numbers. A sequence is a kind of function, and every function has a domain, and the domain of an infinite sequence is the infinite set of natural numbers.

the domain of the metalanguage — Banno

What do "the domain of the metalanguage" and "the world of that metalanguage" refer to?

You're talking about Tarski. It was Tarski who invented the now usual method of interpretations of languages. A language itself doesn't have a domain nor a world. Rather, an interpretation of a language has a domain of discourse. Ignoring that very basic and crucial distinction leads to deep confusions.

Relations don't exist. — Banno

Relations on the domain of discourse exist.

EDIT NOTE: I see that Tarski does talk about languages being inconsistent. However, that is not in accord with the basic mathematical logic regarding languages, models and theories that Tarski spearheaded. I don't know what to make of that situation.

this thread is not intended to be so formal but to get on with outlining what is going on. — Banno

Confusing fundamental concepts is not mere informality. And any outline based on such fundamental confusions cannot be other than itself more confusion.

rain on the parade — Banno

I'm not doing any raining. I'm giving you information and explanation.

Some simplified detail might be fun. — Banno

Are you suggesting that I provide more detail, or are you suggesting that you might be providing more detail?

'trivial' in mathematics, in the context of this discussion, is not a formal notion. What mathematicians mean is that a trivial theorem is one that we can see the proof of it without much effort. If theorem has such an easy, simple, short proof that one would grasp it in an instant, then we say the theorem (or sometimes, the proof of the theorem) is trivial.

Also, for example, we might say an inconsistent theory is trivial, because every sentence is a theorem, so, given that a theory is inconsistent, there is no work involved in determining which sentences are theorems.

Another example, the empty set is a function, since the empty set vacuously satisfies the definition of 'is a function'. We could say it is the most "trivial" function. In a case such as the empty set being a function, we might also say "it's vacuously the case that the empty set is a function".

'Triviality' is not a deep notion in this context. It's just a way for mathematicians to point out that they recognize that certain claims are correct but quite obviously so, or that certain objects have certain properties but in a not very informative way.

proof of consistency — Banno

Consistency follows from soundness. Proving soundness is not deep. We ordinarily just do induction on the length of derivations.

inconsistent language - or theory, if you prefer — Banno

'inconsistent language' doesn't make any sense.

A language is not something that can be consistent or inconsistent.

It's important to understand why that is the case:

A language doesn't make assertions, perforce, it doesn't assert contradictions. Sentences made in a language do make assertions, and a theory is a set of sentences closed under derivability, so theories are consistent or inconsistent.

Informally, think of it this way, for example with a natural language:

English doesn't make assertions. English is used to make assertions.

[with an inconsistent theory] every theorem can be deduced; on in which everything is true. — Banno

That is wrong in two ways:

(1) It is not informative. It should be, "With an inconsistent theory, every sentence can be deduced". By definition, a theorem is a deducible sentence, so with every theory (consistent or inconsistent), of course every theorem can be deduced, because being deducible is what it means for a sentence to be a theorem.

(2) It is not the case that every sentence is true in an inconsistent theory. Sentences are not even true or false in theories. Rather, sentences are true or false (and never both) per any given model. An inconsistent theory asserts contradictions, but that doesn't make the contradictions true. Indeed contradictions are false in all models.

if a contradiction is true in our system, then anything is derivable. — Banno

Same thing. Sentences (including contradictions) are not true or false in theories. Rather they are true or false in any given model. Meanwhile, a model of a theory is a model in which every theorem of the theory is true. But a contradiction is not true in any model, so an inconsistent theory has no models. There are models for the language of an inconsistent theory, but those are not models of the theory.

What we do say is, "From a contradiction, we may derive any sentence." But there is no such thing as a "contradiction that is true in a theory". Again: sentences are not true or false in a theory (sentences are true or false in models), and (in ordinary propositional logic) there is no such thing as a "true contradiction".

a language strong enough to talk about its own sentences, because directly it will be able to generate a sentence of the form

This sentence is false — Banno

We need to be careful not to conflate 'language' with 'theory', or with 'a theory and an axiomatization' or 'a logic' or 'logistic system'. These are related but different notions.

A theory can have sentences that "talk about" sentences in the language of the theory, without contradiction. However, a consistent theory adequate for "a certain amount" of arithmetic, cannot have a defined truth predicate in the theory.

In general, I see in this thread uses of 'language' that should be 'theory' or other specific notions.

A language is just a set of symbols and a signature that assigns kind (predicate symbol or operation symbol) and arity to the predicate and operation symbols. We also add formation rules for terms and formulas.

A theory is a set of sentences closed under deduction. (Some authors say a theory is just a set of sentences, but I prefer when authors add "closed under deduction".) Every theory has a language, which is the language used to form the sentences of the theory. In this sense, if L is the language and T is the theory, we may write <L T> for the language and theory.

A theory and an axiomatization is a pair <S T> where T is the theory and S is a set of sentences in the language for the theory such that every member of T is deducible from S.

A logic is an entailment relation.

A logistic system (deductive system) is comprised of the logical axioms and rules.

EDIT NOTE: I see that Tarski does talk about languages being inconsistent. However, that is not in accord with the basic mathematical logic regarding languages, models and theories that Tarski spearheaded. I don't know what to make of that situation.

∃(x)Fx≡ Fm v Fn v Fo... and (x)Fx≡Fm&Fn&Fo...? — Banno

If the domain is finite, then an existential statement is equivalent to a finite disjunction, and a universal statement is equivalent to a finite conjunction. But, in ordinary logic, there are no infinite disjunctions nor infinite conjunctions, but, for infinite domains, we can think of the quantifiers as "in a sense" representing "infinite disjunctions" and "infinite conjunctions".

Notice that:

ExS <-> ~Ax~S compares with (P v Q) <-> ~(~P & ~Q)

AxS <-> ~Ex~S compares with (P & Q) <-> ~(~P v ~Q)

conjunction and disjunction are analogous with:

multiplication and addition in Boolean algebra

intersection and union in set theory

universal quantifier and existential quantifier in predicate logic

Each pair is a pair of duals.

Any sentential formula has an equivalent using a combination of 'not' and 'or'. That is only one of the eight cases I mentioned where 'not' with another connective is adequate. Moreover, 'not both' (Sheffer stroke) is adequate by itself, and 'neither nor' (Nicod's dagger aka 'Pierce arrow') is adequate by itself'.

For reference:

A NATURAL DEDUCTION SYSTEM:

Comment: Unlike quantifier logic with predicate symbols of arity greater than 1, sentential logic doesn't really need a calculus, because checking for sentential validity is mechanical (for example, using truth tables). But we like to give a calculus anyway:

Notation:

P, Q and R are any formulas

G, H and J are any sets of formulas

u for union

|- for implies


RULES:

Assumption

Enter P on a line and charge that line to itself.

{P} |- P
______

Deduction

If Q is inferred from P along with possibly other lines, then infer P->Q and charge it with all lines charged to Q except the line for P.

If Gu{P} |- Q, then G |- P->Q
______

Modus Ponens

From P and P->Q, infer Q and charge it with all lines charged to P and to P->Q.

If G |- P and H |- P->Q, then GuH |- Q
______

Intuitionistic Contradiction

If a contradiction is inferred from P, along with possibly other lines, then infer ~P and charge it with all lines charged to the contradiction except the line for P.

If Gu{P} |- Q and Hu{P} |- ~Q, then GuH |- ~P
______

Classical Contradiction

If a contradiction is inferred from ~P, along with possibly other lines, then infer P and charge it with all lines charged to the contradiction except the line for ~P.

If Gu{~P} |- Q and Hu{~P} |- ~Q, then GuH |- P
______

Conjunction Introduction

From P and Q, infer P&Q and charge it with all lines charged to P and to Q.

If G |- P and H |- Q, the GuH |- P&Q
______

Conjunction Elimination

From P&Q, infer P and charge it with all lines charged to P&Q.
From P&Q, infer Q and charge it with all lines charged to P&Q.

If G |- P&Q, then G |- P
If G |- P&Q, then G |- Q
______

Disjunction Introduction

From P, infer PvQ and charge it with all lines charged to P.
From Q, infer PvQ and charge it will all lines charged to Q.

If G |- P, then G |- PvQ
If G |- Q, then G |- PvQ
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Disjunction Elimination

From PvQ, P->R and Q->R, infer R and charge it with all lines charged to PvQ and to P->R and to Q->R.

If G |- PvQ and H |- P->R and J |- Q->R, then GuHuJ |- R


DEFINITION:

P <-> Q stands for (P->Q)&(Q->P)