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