### What is the difference (if any) between "not true" and "false"?

• A fairly simple question I hope someone can help me with.

Looking besides the crude restrictions of classical logic: Just think of "neither", "both", and especially "we do not know" = "indeterminable". All these can be labelled "not true", but "false" wouldn't be appropriate.

"This claim is false" is certainly not true. But I don't think you can claim it's false either, unless you define "claim" in an unusual way.

it seems simple but in fact it's one of the most complex questions in modern logic.

Questions like this differentiate logicists, who care about meanings, and formalists, who don't care about meanings.

there are many logics on offer today. some are 2-valued (e.g. true or false) but there are also many-valued logics. there are still others that do not involve truth values. so this is a complex question. in classic 2-valued logic not true is the same as false. but in e.g. intuitionistic logic you cannot go directly from "not true" to "false". Google "law of excluded middle" to see why.

I think aphorism 17 on page 55 has some relevancy here.

Beyond many given answers, I find very strange that nobody pointed out the concept of undecidability. Even in formal math this is a very well-known concept, regarding a set of those sentences whose truth cannot be established. A classical example is the "Liar paradox" (see https://en.wikipedia.org/wiki/Liar_paradox ): in this case a simple sentence is not true, but it's not either false.

Or take any metaphysical dichotomy or antinomy. In Western thought the extremes are either true or false. In Eastern thought (big generalisation) they would both be be not-true but neither would be strictly true or false in an Aristotelian sense. Indeed, this would be the entire solution for metaphysics, since it allows us to dissolve all metaphysical dilemmas as logical misunderstandings. I would recommend Whittaker's book on Aristotle's 'De Interpretatione'. . .

• "p is false" implies "p is not true", but not vice verse because p can also be nonsense.

"2 + 2 = 5" is both false and not true.

"2 + 2 > red" is neither true nor false because it is nonsense. If it were false, its negation "2 + 2 ≤ red" would be true, which is not the case.

I think you can equate false and nonsense. In this case the negation of a false statement is false as well.

While your distinction between falsehood and non-sense is overall correct, the second example you gave is dubious, since one might use logical atomism to show that is actually false: you can break down "2 + 2 > red" into (a) ^ (b) ^ (c) into (a) "2+2 is a number", (b) "red is a number" and (c) "The number from (a) > the number (b)". Since (b) is false, (a)^(b)^(c) is false as well.

@AlexanderSKing - Thought provoking. Actually we can drill down even further according to the theory of types: a number is at least a class of classes of things; red is a universal defined by the class of all red things. In other words, red and a number do not even belong to the same type. To say "red is a number" commits the same folly as "Huston Rocket is a Basketball League," and according to Theory of Types, expressions like this is meaningless. But I think the Theory of Type has a lot of room for expansion; obviously we understand expressions like this.

whether or not your first sentence holds depends on which logic you are using. under classic truth-conditional logic it is false.

note that "not" is a logical constant in classical logic but "false" is not.

@John Am: under classical truth-conditional logic "false" is definitely not synonymous with "nonsense". The former is a truth-value; the latter is not. there is no concept of "nonsense" in t-c logic; every proposition is either true or false. a nonsensical sentence is not a proposition, by definition.

I feel like you have to define "2+2>red" in a meaningful manner before you can legitimately talk about "negating" the statement. Though I certainly agree both the statement and its "negation" are undefined (under normal math) and therefore "not true" while also being "not false".

@mobileink: The question didn't mention propositions.

You don't even have to go so far as to get into nonsense. "How are you doing?" is neither true nor false for the same reason.

@hvd - Thought provoking. English speakers obviously understand sentences like this. I think it is a propositional function with an implicit variable waiting to be substantiated, but comments like this one lead to deeper thoughts. Thanks.

"2 + 2 > red" is neither true nor false because it is nonsense. If it were false, its negation "2 + 2 ≤ red" would be true, which is not the case." I don't see how this is implied by the law of excluded middle. The law of excluded middle implies here that "'2+2 > red' is false" is true, not a change in the operation, unless it comes from the axioms of the system being considered, which as a mathematical fictionalist I'm going to push back against. If something doesn't satisfy a criterion, then it has no place on the cardinal scale of things that satisfy a criterion so the ordinalrelation fails

@LothropStoddard - thought provoking. Your comment lead me to many different meanings of the words "true" and "false". But, be warned, every page in this part of the book deserves weeks or perhaps months of slow meditation.

@LothropStoddard, you are not understanding the concept put forth. Your misunderstanding is most likely you study math and think math is logic. Clearly this is a misteaching & confusing for literal readers. Even in math you must have like terms before you can do any mathematical operations. Clearly you know red is not a number. How can you multiply or add red the concept? You know better. In this way "nonsense" is typically used in philosophy, not math, that something in the wrong category is being compared or some impossible attribute is being applied to some x that can not maintain it.

@AlexanderSKing isbit possible, with atomism, to arrive at a true sentence by negation? Or does atomism work in a logic with more than two truth values?

• In the classical logic something is neither true nor false if it is grammatically malformed to have a truth value, so 2+5 or "x is blue" are not "true", but not "false" either, they are not truth-apt. The classical assumption was that all truth-apt expressions can be distinguished by syntax alone, i.e. there is a clear way to tell from how they are formed whether it is truth-apt or not, without inquiring into what they mean. However, it is easy to come up with expressions that are grammatically well-formed but problematic semantically, sometimes crudely called gibberish, e.g. category errors like "electrons are blue". Those are also neither true nor false, at least intuitively. Wittgenstein even suggested that in natural languages there is no clear distinction between syntax and semantics, and there is no way to clearly prescribe what is well-formed, all rules are "grammar".

There are non-gibberish expressions that have problematic relation to the truth for other reasons, e.g. "such and such will win the election tomorrow". Is it already true (or false) today? Aristotle and modern intuitionists say "no". What about undecidable mathematical statements, like the continuum hypothesis? Same idea. There is also another dimension to the difference between true and false. The classical logic assumes for simplicity that that those are the only truth values that truth-apt expressions might take, this is called bivalence, often confused with the law of excluded middle. Multivalued logics remove this assumption. In particular, popular in applications fuzzy logic allows certain claims (usually "vague" ones) take any truth value between 0 and 1, with 0 being false and 1 true. So something like "15 degrees centigrade is cold" will be neither true nor false but have the truth value of say 0.6.

All of these phenomena led to the idea of logics with "truth value gaps", where we either interpret some expressions as having no truth value at all, or one different from "true" and "false". Sometimes we are forced to do this by the classical logic itself, e.g. the Liar sentence "I am false" leads to a contradiction if we assume that it has one of the two classical truth values. There is a whole field of semantic paradoxes like tha Liar, to resolve which Kripke specifically developed a whole semantic theory with truth value gaps. Paradoxes of vagueness, like the paradox of the heap (one grain is not a heap, adding a single grain won't make not a heap a heap, therefore no amount of grains makes a heap) can also be resolved using truth value gaps.

• In classical logic these are the same by definition.

But in very tentative logics like Constructivism or Intuitionism, things are only said to be true or false if they meet quite stringent conditions. People using criteria like this require a truth to be proved in a given way, or captured by a certain kind of generalization, and a falsehood to proceed from a clear counterexample that meets the standard for truth. (The idea is that truth is ultimately negotiable, as our intuition improves, or that we should avoid claiming truths we cannot back up with computations.) That means that just not being false is not enough to make them true. There is a vast middle ground of things that remain inaccessible to truth or falsehood.

I don't know what you mean by "very tentative", many serious mathematicians have studied both constructivism and intuitionistic logic.

I don't think this is intended to be pejorative so much to say that the logic systems by their nature accept a reduced set of deductive axioms, and thus are more "tentative" about accepting conclusions

truth-conditional (i.e. "classical") logic uses axioms. constructive/intuitionistic logic does not use axioms. there is nothing "tentative" about either.

I think you meant to say that constructive/intuitionistic logic also uses axioms. Well, that they're both formal deduction systems based on well-defined axioms and rules of inference.

'Tentative' in the sense of withholding judgement. Intuitionism does not aggressively assign truth values when they are not needed.

• There is some ambiguity in what a person means precisely by the phrases.

For example, sometimes people use "P is true" (respectively "P is false") to mean that P can actually be proven (resp. disproven) in whatever logical system you're using.

With such a meaning, if P were an undecidable statement — one that can be neither proven nor disproven — then one would assert "P is not true" but not assert "P is false".

Similarly, if we assign truth values to propositions in a multi-valued logic, natural language doesn't do a good job distinguish between

• We did not assign the value "true" to the proposition P
• We assigned the value "true" to the proposition "not P"
• We assigned the value "not true" (i.e. "false") to the proposition P

so again it's somewhat ambiguous exactly what a person means if they say "P is not true" or "P is false"

• Let me try to clarify the difference. Lets start by assigning a value of -1 to false, and +1 to true, and 0 to something "in between".
When someone says something is false, it has only a value of -1.
When someone says something is not true, it can have a value not only of -1, but also of 0. Therefore, not true (0, -1) is not the same as false (-1).

• The three classical laws of thought that form the basis of propositional logic are the law of identity, the law of non-contradiction, and the "law of the excluded middle". The latter holds that every proposition is either true or false; there is nothing in-between. In his book on Metaphysics, Aristotle notes the law of non-contradiction and then explains the law of the excluded middle as follows:

But on the other hand there cannot be an intermediate between contradictories, but of one subject we must either affirm or deny any one predicate. This is clear, in the first place, if we define what the true and the false are. To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true; so that he who says of anything that it is, or that it is not, will say either what is true or what is false. (Metaphysics, Book IV, Part 7, translated by Ross)

The law of the excluded middle implies that any proposition that is not true must be false, and any proposition that is not false must be true. The other answers on this question are right to point out that these are the same in logic, but the above principle tells you the axiom from which this comes.

The LEM does not say that every proposition is true or false. The rule is that for the LEM to apply to a legitimate contradictory pair one of the pair must be true and the other false. Whether this is the case for some proposition is an extra-logical question. This might be the most regularly-muddled issue in the whole of philosophy and it causes havoc. It leads to a large collection of intractable metaphysical dilemmas and the stagnation of philosophy.

• I skimmed through other answers and think nobody mentioned this. Well, if we consider classic logic, then their "meaning" is the same. We call this as equivalence. So both "not true" and "false" are equivalent. But they are not identical at least syntactically they are different and consist of different "symbols" if we can say so.

To understand the difference you can think of two citizens John and Drake, for instance. They are equal under the law, i.e. equivalent. But it doesn't mean they are the same thing/person/entities. It is like two 1€ money one made of paper and another is coin. They are equivalent as they carry the same value. But not identical, at least because there exist vending machines that accept only coins and vice versa. So, having one euro in not suitable form will not allow to buy your favorite cold beverage although you possess the exact amount of money to be able to buy it.

P.S. I think identity and equivalence are context dependent notions.

• Please make a distinction between something not "being true" and something being "not true". In the second case "not true" means exactly "false", which is the strict negation of it being true.

"The car is not white"

1) If "The car isn't white", then it means it is any other color.

2) If "The car is not-white", then it assumes the "opposite" of white exists (perhaps black), and the car is specifically of that color.

In the sentence "x is not true", is "not" the negation of the verb? If so, then x might be either false or nonsense. If "not" is part of the direct object (x is y, where y = "not true" or "untrue"), then x is false, because "false" is the negation of "true".

This answer is mistaken. In the domain of discourse of colors, the proposition not(white) is exactly equivalent to all other colors besides white, the "opposite" of white ***is*** the union of all other colors.

When speaking of colors, physics makes it necessary for one to distinguish between "white" light vs. "white" paint.

• In some cases not true could be either false or nil, but mostly not true just means false.

Truth is a condition of statements (utterances, propositions, sentences, and such - see chapter 9 of John R. Searle's "The Construction of Social Reality"). This condition is satisfied when utterance matches (fits, corresponds to...) what is (the case, the world, states of affairs, et cetera. The adjective "true" describes the satisfaction of this condition.

"Not true" and the synonymous adjective "false" describe a state of affairs where this condition of utterance is unsatisfied (met, obtained...) "Not true" is also used in a sense which "false" is not commonly accepted to indicate a nil (or null) status regarding truth value. For example, the statement is neither true or false that "Ruebens is a better painter than Pollock" - simply a matter of agreement, or what is commonly described as "true to you" (or me, or us, or true to them). Such are matter of sentiment, opinion, poetic use of language and such. Note that the objects of this sentence (Ruenbens, Pollock, their paintings, and the utterers opinion of them) are non-fictional. I mention this to distinguish a sentence with only perspectival (or situational) truth value (not empirical or axiomatic truth value) from sentences which has no rationally assessable truth value, such as "colorless green ideas sleep furiously" or "god did it" or "Yay!" or "go away".

This is to say that such sentences are not therefore falsehood, only that evaluating the truth condition is inadequate to the occaision of their utterance. Consider, even a sentence which is rationally assessable as true or false may in fact be uttered without concern for such an evaluation. For example, if I am in my shower while practicing French and reciting the sentence, "il pleut" it is not a comment upon microclimate conditions. If I enter a French cafe completely soaked from a sudden Summer rainstorm to which the patrons were unaware and upon receiving odd looks I motion upwards and utter, "il pleut!" then it is an explanation of my circumstance including a comment upon the weather which can be assessed a truth value (all the patrons need do is look outside and see that either it is still raining or that everything which was dry when they entered the cafe is now wet to empiricaly verify the state of affairs and render an accurate truth value to my utterance).

In a specialized discourse, "truthy" and "falsey" have broader use of "not true" but only in the technical arena and not for empirical statements of the case.