For reference:

BOOLEAN FUNCTIONS

A Boolean function is function* whose domain is, for some natural number k, the set of k-tuples over {0 1} and whose range is a subset of {0 1}.

* With the exception of '1' mentioned below as a 0-place function even though it is not an actual function.

There are two 0-place Boolean functions:

1
the value 'truth, sometimes represented by the constant 't'

0
the value 'falsehood', sometimes represented by the constant 'f'

There are four 1-place Boolean functions:

{<1 1>
<0 1>}
the constant function that maps any value to 'truth'

{<1 0>
<0 0>}
the constant function that maps any value to 'falsehood'

{<1 1>
<0 0>}
the identity function

{<1 0>
<0 1>}
negation

There are sixteen 2-place Boolean functions:

{<<1 1> 1>
<<1 0> 1>
<<0 1> 1>
<<0 0> 1>}

reduces to t

{<<1 1> 1>
<<1 0> 1>
<<0 1> 1>
<<0 0> 0>}

disjunction

{<<1 1> 1>
<<1 0> 1>
<<0 1> 0>
<<0 0> 1>}

converse of material implication

{<<1 1> 1>
<<1 0> 1>
<<0 1> 0>
<<0 0> 0>}

identity on the first coordinate

{<<1 1> 1>
<<1 0> 0>
<<0 1> 1>
<<0 0> 1>}

material implication

{<<1 1> 1>
<<1 0> 0>
<<0 1> 1>
<<0 0> 0>}

identity on the second coordinate

{<<1 1> 1>
<<1 0> 0>
<<0 1> 0>
<<0 0> 1>}

material equivalence

{<<1 1> 1>
<<1 0> 0>
<<0 1> 0>
<<0 0> 0>}

conjunction

{<<1 1> 0>
<<1 0> 1>
<<0 1> 1>
<<0 0> 1>}

negation of conjunction

{<<1 1> 0>
<<1 0> 1>
<<0 1> 1>
<<0 0> 0>}

negation of material equivalence

{<<1 1> 0>
<<1 0> 1>
<<0 1> 0>
<<0 0> 1>}

negation of the second coordinate

{<<1 1> 0>
<<1 0> 1>
<<0 1> 0>
<<0 0> 0>}

negation of material implication

{<<1 1> 0>
<<1 0> 0>
<<0 1> 1>
<<0 0> 1>}

negation of the first coordinate

{<<1 1> 0>
<<1 0> 0>
<<0 1> 1>
<<0 0> 0>}

negation of the converse of material implication

{<<1 1> 0>
<<1 0> 0>
<<0 1> 0>
<<0 0> 1>}

negation of disjunction

{<<1 1> 0>
<<1 0> 0>
<<0 1> 0>
<<0 0> 0>}

reduces to f


For k>2, a k-place Boolean function can be expressed as a combination of 2-place Boolean functions.

The connectives are interpreted as Boolean functions:

~ is interpreted as negation. You can see how the truth table for ~ is another way of representing this Boolean function.

v is interpreted as disjunction. You can see how the truth table for v is another way of representing this Boolean function.

& is interpreted as conjunction. You can see how the truth table for & is another way of representing this Boolean function.

-> is interpreted as (material) implication. You can see how the truth table for -> is another way of representing this Boolean function.

<-> is interpreted as (material) equivalence. You can see how the truth table for <-> is another way of representing this Boolean function.

There we mentioned only one 1-place connective and only four 2-place connectives. That's okay, because this is an adequate set to represent any other Boolean function by a combination of these connectives.


There are combinations of Boolean functions that are adequate too:

negation of conjunction ("not both")

negation of disjunction ("neither nor")

negation with disjunction

negation with converse of implication

negation with implication

negation with equivalence

negation with conjunction

negation with negation of equivalence

negation with negation of implication

negation with negation of converse of implication


In other words, these are adequate:

negation of disjunction

negation of conjunction

negation with any one of these: disjunction, conjunction, implication, negation of implication, equivalence, negation of equivalence, converse of implication, negation of converse of implication

For reference:

LANGUAGE

symbols:

sentence letters P Q R ...
connectives ~ v & -> <->
left and right parentheses ( )

formulas:

a sentence letter alone is a formula
if X and Y are formulas, then so are ~(X) (XvY) (X&Y) (X->Y) (X<->Y) [we omit parentheses if no confusion results*]
nothing else is a formula

* And if we used Polish notation, then we wouldn't need parentheses at all.

You quote the first line of a post and you ignore the rest. — Metaphysician Undercover

The rest doesn't mitigate your contradiction. I've been through this before with you where you shift your position making coherent discussion impossible.

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

Philosophy of mathematics requires knowing the mathematics being philosophized about.

You've been addressed:

A few posts ago:

I cannot agree that abstractions are objects, unless we restrict "object" to refer only to abstractions.
— Metaphysician Undercover

Now:

I readily conceive of abstract objects
— Metaphysician Undercover — TonesInDeepFreeze

But you also put words in my mouth:

do not seem to be ready to accept the dualism required for a true understanding of "abstract objects"
— Metaphysician Undercover

I haven't said anything about duality. This is yet another instance of you putting words in my mouth (except weaseling with "do not seem"). — TonesInDeepFreeze

A few posts ago:

I cannot agree that abstractions are objects, unless we restrict "object" to refer only to abstractions. — Metaphysician Undercover

Now:

I readily conceive of abstract objects — Metaphysician Undercover

We should stop right there.

But you also put words in my mouth:

do not seem to be ready to accept the dualism required for a true understanding of "abstract objects" — Metaphysician Undercover

I haven't said anything about duality. This is yet another instance of you putting words in my mouth (except weaseling with "do not seem").

I've given you adequate opportunity to explain the principles that you adhere to — Metaphysician Undercover

I don't advocate any particular philosophy. But I have explained to you crucial notes about the mathematics itself, and I have touched on certain aspects of frameworks in which mathematics is discussed. You have made it a point to either ignore, evade, misconstrue or strawman all of it.

you now insist that I ought to just drop them, and take up some "different perspectives" — Metaphysician Undercover

I don't insist that you do or don't do anything (other than that you don't put words in my mouth or lie about me).

And I did not even hint that you have to "drop" your philosophy. I only said that you are not capable of also giving fair consideration, let alone study, to other points of view in the philosophy of mathematics, not to even the basics of mathematics on which you have such vacuously dogmatic opinions. It's characteristic of childish mentality to think that you can't look at things from other people's points of view without giving up your own.

And putting "different perspectives" is scare quotes is also jejune. Even if you disdain other perspectives, it should not be at issue that the notion of a different perspective is common and doesn't need scare quotes. That's not even a big point, but it is an emblematic detail.

Sure, you can't conceive of an empty set. But lots of people do.

But the problem is more fundamental with you. You can't conceive of abstract objects.

Here's a difference between you and me: You're a dogmatist. I am not.

In these kinds of matters, you cannot be bothered to give fair consideration to frameworks other than your own. Not only do you know nothing of the mathematics involved, you know hardly anything (if anything) of the many views of modern philosophers of mathematics. And you willfully avoid knowing anything about them. So you are brutally stunted in your capability to view from different perspectives, to intelligently compare different frameworks, to conceive. No matter that there is a rich, intensive, and intellectually competitive cornucopia of thinking in and about mathematics, you insist that all of those very smart and dedicated people must be wrong all the way to the core, while you alone stand above.

On the other hand, I can see the attractions of various points of view - from among even physicalist, materialist, idealist, phenomenologist, platonistic, nominalist, structuralist, finitist, constructivist, intuitionist, etc. Unlike you, I don't demand that my own personal framework for understanding mathematics is the only reasonable framework.

A dialogue between Georg and Ernst:

G: How's your rock collection these days?

E: I sold all my rocks. Now my collection is empty.

revolutionize science! — Agent Smith

Come the revolution everyone will like strawberries and cream.

I'm sorry you feel this way! — Agent Smith

It's not a matter of feelings. They're facts.

For many months you have continued to post disinformation, even repeating items on which you were already corrected or refuted. That is a pattern not merely of a beginner, but of a crank.

And you recently lied about me personally.

I don't know what that emoticon you keep posting means.

/

I found out more about the -(1/12) thing. It requires taking the infinite sum in a different sense from the usual sense. It doesn't imply that there is a contradiction in mathematics.

So those who're learning are guilty of intellectual dishonesty? — Agent Smith

I said nothing that can be construed as "learning is intellectual dishonesty".

TIDF split the scene — Metaphysician Undercover

Wrong again.

to say that there is a collection of objects with no objects is contradictory — Metaphysician Undercover

You skipped my previous remarks about that.

you are now trying to reduce many to one, by saying that a set is an object. — Metaphysician Undercover

If I recall, we discussed this at length many months ago. Again, I don't mean a physical object. I mean a set is an abstract object.

If you don't accept the notion of abstract objects, then I admit that there's not much for us to discuss. I do not feign to be able to explicate the notions of 'abstract' and 'object' beyond ordinary understanding of such basic rubrics of thought as acquired ostensively or by whatever means people ordinarily understand them.

If you do accept the notion of abstract objects, then I point out that a set theoretical intuition may begin with the notion of a thing being a member of another thing: The notion of membership. First, that obviates the need even to use the word or notion 'set', as instead we merely discuss the membership relation. Second, even though the word or notion 'set' is dispensable, we can still go on to define it. Also, in the formal theory itself, the word or notion 'object' does not occur, though when we informally talk about informal theories it would be cumbersome to eschew the word and notion 'object'. In that context, mathematicians readily understand that a set is an abstract object. There is nothing in the definition of the word 'object' that precludes an abstract object nor that precludes that certain abstract objects, viz. sets, are in relation to others, viz. membership.

students are taught very specific principles, and their minds are funneled down a very narrow path — Metaphysician Undercover

That offers at least these prongs of refutation:

(1) I am mostly (but not exclusively) self-taught from textbooks; and textbooks in mathematics don't indoctrinate. Rather they put forth the way the mathematics works in a context such as presented in the book. A framework is presented and then developed. There is no exhortation for one to believe that the framework is the only one acceptable.

(2) Indeed, mathematics, especially mathematical logic, offers a vast array of alternative frameworks, not just the classical framework, including constructivism, intuitionism, finitism, paraconsistency, relevance logic, intensional logic .... And mathematics itself does not assert any particular philosophy about itself, as one is free to study mathematics with whatever philosophy or lack of philosophy one wants to bring to it.

(3) It is actually cranks who are narrowminded and dogmatic. The crank insists that only his philosophy and notions about mathematics are correct and that all the mathematicians meanwhile are incorrect. The crank doesn't even know anything about the mathematics yet the crank is full of sweeping denunciations of it. The crank makes wildly false claims about mathematics, and then doesn't understand that when he is corrected about those claims, the corrections are not an insistence that the crank agree with the mathematics but rather that the corrections merely point out and explain why what the cranks says about mathematics is untrue. It's as if the crank says, "classical music is all wrong because classical music never has regular meter" and then when it is pointed out that most of classical music does have regular meter, the crank takes that as narrow minded demand that he like classical music. And the crank is not even aware that mathematics, especially mathematical logic, offers a vast array of alternatives. Meanwhile, the crank's usual modus operandi is to either skip, misconstrue, or strawman the refutations and explanations given to him, thus an unending loop with the crank clinging to ignorance, confusion, and sophistry.

The students are discouraged from looking outside the program — Metaphysician Undercover

Does Metaphysician Undercover have actual incidents to cite? Is there a particular incident to which he is witness, or widespread reports of them that would justify such a sweeping generalization or even a more modestly limited claim?

Please edify me then. How does intuition work in math? How is it related to so-called mathematical/logical rigor? Talking to you is like conversing with a computer. DOES NOT COMPUTE! DOES NOT COMPUTE! From start to finish, that's all you say! I should call tech support! — Agent Smith

There's lot to unpack there.

(1) (a minor point) You were responding to a quote of mine about intuitionism. That's the wrong quote, since the subjects of intuition and intuitionism are related but very different subjects. Instead the pertinent quote is "I quite understand that human thinking, including about mathematics, involves intuition. Indeed I'm interested in the relation between formal theories and intuitions."

(2) "How does intuition work in math?" The subject of intuition and mathematics is a big one. I could not even summarize my thoughts about it in an ad hoc post. And I do not have conclusive things to say about it. I said that I think about the subject a lot; but I do not claim to have arrived at firm conclusions about it.

(3) The subject of the relation between intuition and rigor is narrower than just mentioned, but still a big one. It's not clear to me where the best place to start would be. But perhaps one area to begin is the notion that formalization in its most basic intuitive sense can, in principle, be reduced to series of discrete observations about discrete objects conceptualized as being indivisible, such as, witnessed in physical form, tally marks on paper or 0s and 1s on paper. My own personal imagination doesn't provide that there could be any form of participation in mathematical reasoning more basic.

(4) Even if it seems to you that my postings read as mechanical or computer-like, that is not a refutation of anything I've posted. Moreover, it would be a non sequitur to infer that I don't think about the subject of intuition in mathematics from your premise that my writing is computer-like. Moreover, as to writing style, one should take into account that my purpose is not to entertain you, nor to present to you as loosely gabbing. When you post plainly incorrect claims about mathematics, then usually my main purpose is to clearly point out that you are incorrect and often to explain why; and often that is best achieved by explanations that use uniform terminology and parallel forms.

(5) You are lying. It is not even remotely true that I only say that you are incorrect. Over many months, I have also given you quite generous and detailed explanations why you are incorrect. It is remarkably dishonest and boorish of you to say otherwise.

Is mathematics inconsistent? — Agent Smith

No derivation of a contradiction has been shown in ZFC.

And you write (I'm using plain text):

Sum[n = 1 to inf] = inf
Sum[n = 1 to inf] = -(1/12)

As far as I can tell, those are not even well formed.

Sum[n=0 to inf] requires a term on its right*, otherwise it's just a dangling variable binding operator.

* E.g., Sum[n=0 to inf] 1/(2^n) is well formed and meaningful.

x + x = x → 2x = x → 2 = 1 — Agent Smith

For finite cardinals, if ~x = 0, then ~x+x = x.

For infinite cardinals, x+x = x and 2x = x.

But you cannot infer 2 =1 from 2x = x when x is an infinite cardinal. Cardinal addition where one of the multipliers is an infinite cardinal does not have the cancellation property.

Once again, your ignorance and intellectual dishonesty have enabled you to post a false claim.

I have a minor use-mention quibble with the penultimate sentence, but otherwise it seems to me that he or she gave a reasonable explanation of the identity of indiscernibles. Why do you ask?

I made a bad typo. I just now corrected it.

x=x ... reflexivity
with
(x=y & Px) -> Py ... indiscernibility of identicals (aka substitutivity)

is a complete axiomatization of identity theory and they imply:

x=y -> y=x ... symmetry
and
(x=y & y=z) -> x=z ... transitivity

The converse of the indiscernibility of identicals is the identity of indiscernibles. Interestingly, if the language has infinitely many predicates, then the identity of indiscernibles is not expressible.

Another complete axiomatization (from Wang) is:

Ex(x=y & Px) <-> Py

That proves

x=x
and
(x=y & Px) -> Py

His invention of the predicate calculus is great intellectual wisdom.

When people say the three laws of thought are

A=A ... identity
A v ~A .... excluded middle
~(A & ~A) ... non-contradiction

they are using 'A' for two different things.

For identity, 'A' ranges over objects.

For excluded middle and non-contradiction, 'A' ranges over propositions.

Indeed, that is not a good presentation. 'A' should not be mixed up that way.

A better statement is:

For all individuals x, we have x=x.

For all propositions P, we have P v ~P and we have ~(P & ~P).

But the "three laws of thought" paradigm does not express the full scope of reasoning about identity or reasoning about propositions. There are other principles that are also needed for reasoning about identity and for propositional logic. The paradigm has been surpassed by those of symbolic logic that are more comprehensive.

And your emoticon doubly seals it! Who could ever defeat an emoticon?

Henri Poincaré! — Agent Smith

Well that truly settles the question!

Empty generalization and bluster.

I quite understand that human thinking, including about mathematics, involves intuition. Indeed I'm interested in the relation between formal theories and intuitions. And I know vastly more about the school of intuitionism compared with your lack of knowledge about it.

Mathematics is allowed its own special definition of "contradiction", — Metaphysician Undercover

It is a formal definition. But it still captures the ordinary sense of "To claim a contradiction is to claim a statement and its negation." For example, in ordinary conversation we may say, "'Mike is a car mechanic and Mike is not a car mechanic' is a contradiction."

explain to me how a set which has no objects also has a collection of objects — Metaphysician Undercover

You switched form "is a collection of objects" to "has a collection of objects".

I said nothing about "has a collection of objects". Rather, I said

"a set is a collection, possibly empty, of objects" — TonesInDeepFreeze

What if you had one object? — Metaphysician Undercover

An object that has no members is either the empty set or an urelement. And of course, an object that has in it only one object is a non-empty set.

So my explanation stands and all the rest of my remarks demolishing your ignorant and self-misleading remarks stand.

Whether an expression is well formed or not is irrelevant to whether it is self-contradictory, because to determine contradiction we must analyze the meaning, and this is the content, not the form. — Metaphysician Undercover

You are entirely ignorant of what contradiction is in mathematics.

Moreover, even if contradiction were, in some sense, couched semantically, then no contradiction, even in some sense of a semantic evaluation, has been shown from ZFC.

Moreover, if an expression is not grammatical, then it does not admit of semantic evaluation.

The empty set for instance, involves contradiction. — Metaphysician Undercover

"Involves contradiction" has not been given meaning by you. Either the theorem that there exists an empty set implies a contradiction or it does not. No contradiction has been shown to be derived from the theorem that there exists an empty set.

If you mean that the notion of 'set' is not compatible with a set being empty, then that just entails that your conception differs from a different conception in which there is an empty set.

Moreover, the phrase "the empty set" is not in the formal theory. Rather, there is a theorem:

E!xAy ~yex

and definition:

x=0 <-> Ay ~yex.

Moreover, even if you persisted to object to mathematicians using the informal locution 'the empty set', then mathematicians could say, "Okay, we won't say 'empty set' anymore. Instead we talk about sets and one particular object, whether it is a set or not, such that that object is the only urelement (an object that has no members), and then all these things - the sets and the urelement - are called 'zets'. So there is an empty zet." That would not alter the mathematics of set theory one bit, especially formally, and even informally except that the mouth pronounces a 'z' instead of an 's' for that one word.

So you are terribly ignorant and self-misguided in every aspect of this matter.

you have a very odd notion of what constitutes contradiction — Metaphysician Undercover

No, I have the standard logical and mathematical notion.

The law of noncontradiction states that the same object cannot both have and not have, the same property, at the same time — Metaphysician Undercover

That is one informal formulation. It is equivalent though to the standard formulation. That is:

~Ex(Fx & ~Fx)

is equivalent to

~(P & ~P).

let's say that a set is a collection of objects — Metaphysician Undercover

First, 'set' is not a primitive of set theory. An actual definition can be:

x is a class <-> (x=0 or Ey yex)

x is a proper class <-> (x is a class & ~Ey xey)

x is a urelement <-> (~x=0 & ~Ey yex)

x is a set <-> (x is a class & ~x is a proper class)

Second, even informally, you mention a certain definition of 'set'. Mathematicians are not then obliged to refrain from having an understanding in which "collection of objects" does not preclude that it is an empty collection of objects, notwithstanding that that seems odd to people who have not studied mathematics, and so more explicitly we say, "a set is a collection, possibly empty, of objects". You are merely arrogating by fiat that your own notion and definition must the only one used by anyone else lest people with other notions and definitions are wrong. That is an intellectual error: not recognizing that definitions are provisional upon agreement of the discussants and that one is allowed to use different definitions in different contexts among different discussants. It's like someone saying "a baseball is only one such that is used in major league baseball" and not granting that someone in a different context may say, "By 'baseball' I include also balls such as used in softball". It is intellectually obnoxious not to allow that. And it is one in the deck of calling cards of cranks.

The statement that there exists sets that are infinite is not a logical impossibility.

The idea of allowing 'there exist potentially infinite sets' but not 'there exits infinite sets' is fine as a motivation for an alternative mathematics. But, if one cares about mathematics being formal (in the sense that it is utterly objective by algorithmic checking whether a sequence of formulas is indeed a proof from axioms), then the notion of 'potential infinity' requires formal definition and axioms to generate the desired theorems about it.

Saying, "I don't like the notion of infinity so I'll use potential infinity instead" but without even hinting at how that alternative would be formulated is no better than saying "I don't like that human life requires breathing oxygen so we should breathe hydrogen instead" but not giving a hint as to what technology would allow hydrogen to do the job of oxygen.

when TDIF refuses to divulge the secrets of the mathemagician's smoke and mirrors. — Metaphysician Undercover

There is no magic. Very much to the contrary. At a bare minimum, it is algorithmically verifiable whether a given formal expression is well formed and then whether a given sequence of formulas is a formal proof. That is a courtesy given by formal logic that is not hinted at in various handwavings and posturings by cranks as often found in a forum such as this. And I have given extensive explanation of many of the formulations I have mentioned.

Every time infinity is employed in the application of mathematics, it's like employing a contradiction. — Metaphysician Undercover

Yet no one who says things like the above has ever demonstrated that Zermelo set theoretic infinitistic mathematics implies a contradiction.

This becomes very clear in an analysis of the common mathematician's claim to have resolved Zeno's paradoxes. — Metaphysician Undercover

Zeno's paradox is not a formal mathematical problem. Saying that calculus provides a problem solving tool in which paradox does not occur does not imply a contradiction.

Yet, again, we remind that a contradiction is a statement of the form "P and not-P". No such statement has ever been derived from Zermelo set theoretic infinitistic mathematics, no matter that, perpetually, cranks groundlessly and ignorantly claim otherwise.

.

Look at how the subject of identity is handled in formal logic, philosophical logic, mathematical logic, set theory, and mathematics. Those clear up a lot of questions (though there are still some philosophical questions that arise).

Zofran — Agent Smith

From your personal stash. Side effects include confusion and disorientation. Explains a lot.

You regularly and willfully spread ignorant disinformation about mathematics throughout many threads, over the course of at least several months, even repeating yourself after you've been given detailed explanations why you are incorrect. It's disgusting.

You should not be disappointed. You should be encouraged. I gave you an outstanding example: when one does not have sufficient knowledge then one should defer from making wild claims.

Is there a finite number (Nmax) such that no calculations ever in physics will exceed that number? — Agent Smith

I know so little about physics or cosmology that I can't answer that.

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.

This is another example of you running your mouth off on this technical subject of which you know nothing because you would rather just make stuff up about it rather than reading a textbook to properly understand it.

describe this universe — Agent Smith

'Describe the universe' is a notion not defined by you.

Clearly we don't know how to do math with infinity — Agent Smith

Clearly that is false. Infinite sets are basic for calculus.

No calculation ever would exceed that number — Agent Smith

Let that number be M. Then M+1. Poof.

Physicists tend to throw their hands up in the air with disgust mixed with utter frustration when they see ∞∞ when number crunching. — Agent Smith

Example please.

pi to its final decimal place — Metaphysician Undercover

There is no last decimal place of pi.

Maybe this is a place to start
Infinitary logic:

https://en.wikipedia.org/wiki/Infinitary_logic — Paul S

That's not a good place to start. Infinitary logic is a special topic in mathematical logic that is not so much directed to the basic notions of infinity in mathematics. To understand the basic notions of infinity in mathematics, one would turn to axiomatic set theory, in which definitions of 'finite' and 'infinite' are formulated and in which theorems about infinite sets are proven.

But infinity is a purely abstract concept. In fact, Mathematics that delves deeply into it seems to be filled with paradoxes. — Paul S

The mathematics that in particular delves into infinity is set theory. There are no known contradictions in axiomatic set theory.

There is a fixed point lemma in the proof of incompleteness:

G <-> prov(#G)

But I don't know what fixed point is involved in the sequence of theories under consideration.

Paradoxes are an existential threat to epistemology (truth) & logic. When these two are assaulted (successfully), our world comes crashing down around our ears! — Agent Smith

There have been controversial puzzles in epistemology for centuries. I don't see any crashing down of the world related to this. What do you think will happen, the media will announce that philosophers still haven't reached agreement on solutions to the logical and linguistic paradoxes and then the financial markets will all collapse, followed by all the populations lapsing into chaos and war?