A non-circular definition of "not"

The notion of "not" is used throughout all languages as far as I know. For instance, consider the sentence,

"Trees are not blue."

There are various ways of expressing this notion without using the word "not", but they still mean the same. For instance,

"No tree is blue."

"All parts of trees have colors that are different than blue."

"The negation of the statement, 'some trees are blue', is true."

I'm curious if there is a definition of "not" which is not circular, that does not use any synonyms (as were used above) to define it.

• Would a "set theory" explanation work? Inside vs outside a set, with "not" being outside? "Me and not me. Sometimes you all go away, but I'm always right here." Mar 2 '20 at 20:14
• A popular way to "define" the meaning of all logical connectives is to give introduction and elimination rules for them, as in natural deduction. This is called Gentzen semantics, see logical harmony. The introduction rule adds ¬ to A when any B can be derived from A, and the elimination rule removes it when the same happens with ¬A in place of A. Mar 2 '20 at 20:48
• @Conifold I didn't see your comment when I wrote my answer! To the OP, Conifold's comment (besides introducing very useful terms) is very similar to my answer but not quite identical: the former introduces the further level of explicit syntax and concrete manipulations of strings. The underlying idea is the same, though: we understand logical operations by their interactions with deduction, and can try to look for characterizations in terms of what does (as opposed to does not) occur in the relevant context. Mar 2 '20 at 21:21
• Further elaborating on Conifold's comment, it's worth noting that there are serious subtleties with the Gentzen approach to logical connectives. Consider an imagined logical connective @ ("tonk") defined by the introduction rule "From a infer a@b" and the elimination rule "From a@b infer b." Obviously as soon as our syntax admits this connective everything collapses; so what criteria do we use to determine "meaningfulness" of proposed intro/elim definitions? This is exactly what logical harmony is meant to address. Mar 2 '20 at 21:29
• Maybe useful Jaakko Hintikka, Negation in Logic and in Natural Language (2002) as well as Dov Gabbay & Heinrich Wansing (editors), What is Negation? (Springer, 1999) Mar 3 '20 at 7:21

There's a "purely algebraic" approach, albeit one of dubious success: namely, we take the view that negation is characterized by explosion (avoiding non-contradiction since the latter is negative).

We work in the context of abstract deductive systems. There are many different notions of such floating around in the literature; what I'll mean in this answer is a rather simple one:

An a.s.d. is a pair (X, >) where X is some set of things called "sentences" and > is a relation between sets of sentences and individual sentences called "entailment."

(If this site supported mathjax, I'd replace ">" with the symbol with LaTeX code "$\vdash$" - this is standard usage in algebraic logic.)

A priori a.s.d.s could be extremely boring or pathological; generally we consider them in the presence of additional assumptions, e.g. compactness (if A>p then there is some finite subset B of A such that B>p), recursive enumerability (elements of X are canonically coded by natural numbers and > is compact and when restricted to finite sets of hypotheses is r.e.), presence of conjunction (for all p, q in X there is some r in X such that for all s in X we have {p,q}>s iff {r}>s), or so forth.

The key point I care about here is having negation:

An a.s.d. (X,>) has negations if for each p in X there is some q in X such that the following hold:

• For all r in X we have {p,q}>r.

• If q' is in X and for all r in X we have {p,q}>r, then we have {q'}>q.

Such a q is said to be a negation of p. (Note that negations need not be unique, merely "unique up to equivalence"!)

Note that there is no negative information here. In particular, ">" isn't assumed to be asymmetric at all (and it isn't in most natural examples, e.g. when (X,>) is just classical propositional logic we have {p&p}>p and {p}> p&p).

To repeat the remark at the beginning, the above reflects the following perspective on negation:

Negation is characterized by explosion.

This is the "positive" flipside to non-contradiction, which is the other main way to think about negation.

Of course, this is extremely objectionable to some people: e.g. paraconsistent logic does not accept the principle of explosion (note that constructivism/intuitionism does - it's the excluded middle they reject, which is different). Since paraconsistent logic - whether I hold with it or not (I don't) - is something which makes sense and is interesting, it's clear to me that the approach above is not good. However, it's also not nothing, and it introduces algebraic logic as a topic of interest.

The other big potential dissatisfaction with the approach above is that it steps away from natural language. But that doesn't mean it's irrelevant: thinking of formal/algebraic logic as a "laboratory" for analyzing natural language, we can still talk about informal deductions and more-or-less follow the same theme as above. So I'd argue that that's ultimately not as serious an issue.

• It says in the answer, "If q' is any other element of X with the above property". What above property are you referring to? Mar 2 '20 at 22:06
• @CraigFeinstein The property of the previous bulletpoint - I've edited to clarify (and remove a possible worry about negativity in the word "other"). Mar 2 '20 at 23:42
• I’ve seen this in natural deduction interpretations a bit, and it makes sense in that context. Interestingly, for non-classical logics, there is a variation using Sequent Calculus frameworks with multiple conclusions; you can “left introduce” a negation by removing a conclusion from an entailment sequence. This does something pragmatically similar, and explosion can be seen as a specific case of a more general phenomenon! Mar 3 '20 at 0:36
• I'm not 100% sure this answer is correct, but I am 100% sure that if there is a correct answer, this is it. Mar 3 '20 at 14:47

According to the research of the Natural semantic metalanguage project, the problem of circular definitions is solved through the identification of semantic primes, the basic blocks of meaning which are shared by all languages and cannot be meaningfully subdivided. Any attempt to define a prime will end up circular or more convoluted than the word itself.

NOT is one of these primes. Proponents of NSM say that empirical research from languages all around the world show that NOT is a fundamental and irreducible concept in all languages. Being a prime also means that NOT doesn't really need to be defined - it's a base level concept which everyone will already understand. Any attempt to give a definition will be more complex than the word itself. The meaning of "not" is "not"!

• This doesnt answer the question. Jul 10 at 13:21

I'm curious if there is a definition of "not" which is not circular, that does not use any synonyms…to define it.

I am not certain that this problem has a solution. The central dilemma is that at some point the definition must exclude some category of things, and describing exclusion requires some form of the concept “not”. As curiousdannii discussed, perhaps there are some notions which are semantically prime, and are simply a part of what languages are.

The best that I can offer is this: Can the dilemma be avoided by crafting a definition whose content is both necessary and sufficient to completely define a set? Only affirmative statements would be used to create the definition. But anything whose qualities were not identical to those in the definition would, by implication, not be the same thing.

So, for example, assume that when Bertrand Russell proved that 1+1=2, he did so affirmatively. He began from a set of assumptions and followed them until he reached 1+1=2. When he did so, he did not need to add that 1+1 does not equal 5 or 12. That conclusion follows, but it does not need to be spoken to conclude that 1+1=2. The erroneous conclusions simply do not follow from the premises.

The precision of the premises enables the implication that anything outside the premises is superfluous or contradictory.

Negation is a (unary) logical operator, and can be considered either syntactically or semantically.

(1) From a syntactical point of view , negation is a function which associates any propositional input "phi" to the output " ~ phi". No circularity here, since the symbol" ~ " ( at the purely syntactic level) has no meaning.

All this amounts to saying that, among our formation rules, we have the following one :

If "phi" is a well-formed-formula , then "~ phi" is also a wff.

(2) From a semantical point of views, no circularity either.

Semantically , negation is a function from the set {0,1} to the set [0,1} , let's call it function N.

This function is defined in the following way :

N(1)= 0

N(0) =1.

Saying that Q is the negation of P means that

the image of the value of Q by function N is always identical to the value of proposition P, and reciprocally.

"Always" meaning " for all possible value assignements, namely 0 or 1 for each atomic sentence of the language".

The words " no" or " not " nowhere appear in this defnition.

Now, we are free to say that "1" means "true", and that "0" means "false". We are also free to say that "false" means "not true", and that "true" means " not false". There is indeed some circularity here, but it does not affect the definition of the function. It only affects the interpretation you give to this function, and the use you make of it.

• What about non-binary things? "Not tall" doesn't equal "short". Apr 23 '20 at 3:01

We can define "not" in terms of the numbers one and two:

Consider the sentence "A tree is a tree." There is one noun in the sentence, "tree".

Now consider the sentence "A tree is not grass." There are two nouns in the sentence, "tree" and "grass".

The word "is not" is used when there are two. The word "is" is used when there is one.

Is this a circular definition?

• How do you determine inequality of nouns? This is just a linguistic repackaging of defining "not" in terms of "not equal to," which seems circular to me. (This is why an "explosion" rule is useful: it's a purely positive characterization.) May 13 '20 at 15:34
• It is circular in the sense that one can argue that it is linguistic repackaging. However, if one regards numbers as more fundamental than logic, it seems like a better definition than other circular definitions. May 13 '20 at 17:02
• I regard numbers as more fundamental than logic for the following reason: Binary logical operations like "and", "or", "if then" always need two statements, and two is a number. May 13 '20 at 17:55

Here is a three-line algorithm for determining whether X = Y or X != Y.

1) Define the variable XisNotY=true.

2) If X = Y then let XisNotY=false.

3) End

At the end of the algorithm, XisNotY will be true if and only if X != Y. Note that the algorithm never uses the word "not" or anything like it, so it is a noncircular definition of "not".

• A criticism of this answer that I just realized is "how does one define an "if then" statement?" Can one define it without using the notion of "not"? May 15 '20 at 15:24

You can define negation by paraphrasing the law of non-contradiction:

not can be defined as a function such that given a subject A and a predicate B, A can never be both B and not(B).

• But what does "never" mean? Mar 2 '20 at 20:57