# Can a definition be true/false?

Can a definition be false or, for that matter, true?

Dog (noun): A tamed lupus canis.

Unicorn (noun): A horse with a horn growing out of its forehead; may be of any color, but are usually pink or white.

To clarify (to the extent my ignorance permits) ...

1. x = 2
2. x + 4 = 6 means x = 2

What's the difference, if any, between 1 and 2?

• Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Philosophy Meta, or in Philosophy Chat. Comments continuing discussion may be removed. Commented Oct 31, 2023 at 5:10
• x=2 is a mathematical equation, i.e. a formula that can be either true or false. Commented Oct 31, 2023 at 7:39
• The second one is an inference: assuming the first formula, the second one follows. Neither are definitions. Commented Oct 31, 2023 at 7:39
• See The logic of definitions for some basic concepts. Commented Oct 31, 2023 at 7:41
• @MauroALLEGRANZA, gracias. Perhaps we should look at more types of definitions. Commented Oct 31, 2023 at 8:10

1. x = 2
2. x + 4 = 6 means x = 2

What's the difference, if any, between 1 and 2?

The expression "x + 4 = 6" does not mean x = 2. It asserts the equation x + 4 = 6, so it means that it is true that x + 4 = 6.

So, (1) and (2) don't mean the same thing, although x = 2 and x + 4 = 6 have the same solution.

And (1) may be used to mean the solution to (2).

. . .

Definitions are typically just equivalent to subject-predicate sentences, and as such are either true or false.

Definition:

lexicographer n. One who produces a dictionary.

This is equivalent to the subject-predicate sentence:

A lexicographer is someone who produces a dictionary.

Most lexicographers would agree that this sentence is true, so the definition has to be true as well because it says the same thing.

So, definitions are true or false unless there are somehow meaningless.

. . .

You point at the sun and you say this is the sun. In doing this, you do two things. First, you specify for the benefice of someone else which word is used in your language to denote the sun; second, you define the word "sun" in a way which makes the definition true. The definition allows you to use the word "sun" to refer to the sun when speaking to English-speaking people. Not bad.

A dictionary might say:

sun

1. (Astronomy) the star at the centre of our solar system. It is a gaseous body having a highly compressed core, in which energy is generated by thermonuclear reactions (at about 15 million kelvins), surrounded by less dense radiative and convective zones serving to transport the energy to the surface (the photosphere). The atmospheric layers (the chromosphere and corona) are normally invisible except during a total eclipse. Mass and diameter: 333 000 and 109 times that of earth respectively; mean distance from earth: 149.6 million km (1 astronomical unit).

This is not what very nearly all speakers of English use the word "sun" to refer to most of the time. Most of the time, we use the word "sun" to refer to the one thing which we can see in the sky by a clear day and is brighter than anything else we normally see.

Even if you understand every word of the dictionary definition, it is possible to fail to understand that the thing referred to is the one thing which we can see in the sky by a clear day and is brighter than anything else we normally see. Instead, you may think that the definition applies to one particular star we can see... at night!

Some true definitions are easy to make. Others can be very hard.

• Arigato gozaimus for your answer. Indeed, veritas that a word like "sun" denotes <insert your definiens>. You should read the other replies plus comments. Commented Oct 31, 2023 at 8:05
• What would you say is the difference between my mathematical examples in the OP? Commented Nov 1, 2023 at 3:42
• @AgentSmith See my edit. Commented Nov 2, 2023 at 11:16
• It should be noted that in traditional notions of definition starting with Frege, definitions were not allowed to be true or false. There are a number of people on this site who still endorse the view, though I'm not one of them. This answer has no references.
– J D
Commented Nov 3, 2023 at 18:39

The Stanford Encyclopedia lists several different kinds of definition, so it worth distinguishing some of them.

A dictionary definition aims to give the meaning of a word to speakers of the language. The definition usually takes the form of some synonyms. This kind of definition can be incorrect. One might also say that a statement of it can be false.

A descriptive definition aims to specify the necessary and sufficient conditions for the way a term is actually used. Or at least to list the features that provide the family resemblance of its extension. If you ask a philosopher for a definition of 'knowledge' prepare for a lengthy essay. Again, this kind of definition may be incorrect.

Sometimes, a definition may be offered as a way of capturing the essential intention behind a word, while acknowledging that it does not agree with ordinary usage. Gilbert Ryle referred to this as a 'persuasive redefinition' in contradistinction to an analysis. Such things are sometimes useful, but they can lead to equivocation if the redefinition is not highlighted. They may be rejected as false by those who do not agree with the proposed redefinition.

A stipulative definition introduces a fresh term to a language and specifies its meaning, or sometimes applies a fresh meaning to an existing term. This kind of definition mostly occurs in the context of technical disciplines where precise terminology is required. Sometimes it has limited scope, i.e. the author may state that for the purposes of this book or paper I shall be using this word to mean that. Applying a fresh meaning to an existing term can easily lead to confusion. We are accustomed to physicists assigning precise meanings to words like force, energy, power, momentum, etc., and we know that these may differ from their ordinary usage. But when botanists apply a fresh meaning to 'nut' it is awkward and confusing. Stipulatively defined words make up a small minority or words in a natural language. People who work mainly in technical disciplines have a highly unfortunate tendency to think they are entitled to impose any meaning on any word they choose and say they have stipulated it to be so. Stipulative definitions strictly speaking cannot be false, but they can definitely be inappropriate and misleading when applied to an existing word.

• Good that you brought up the issue of devilish details in re definiyions. How is, hazarding loose terminology, a definition born? Commented Oct 30, 2023 at 10:09
• You are not so in danger @AgentSmith for there is all sorts of talk of the meaning of names being by metaphorical/causal "baptism" in this connection. Commented Oct 30, 2023 at 11:18
• Words and their definitions can come into existence in different ways. Sometimes by ostension (pointing). Sometimes an author invents a new word (Shakespeare invented over a dozen). Some are combinations of existing words. Some are borrowed from foreign languages. With technical terms there is often a standards body with official responsibility for setting definitions. But most words don't have any stipulated definition, they just come into being naturally and dictionaries document their use once they are sufficiently common. Commented Oct 30, 2023 at 12:20
• @AgentSmith Language is a shared hallucination. Terms are "born" when one person proposes them and others all agree to use that term to mean what they all agree it means. (This is an imperfect process. Look to "false friends" for blatant examples of different groups of people using the "same word" to mean different things.) For this reason, people sometimes clarify the particular definition of the word they'll be using for a single book/paper/legal document, and hope the reader agrees to use that definition when reading that particular document.)
– R.M.
Commented Oct 30, 2023 at 14:50
• @Bumble, a paragraph with no obvious errors, gracias. Am I getting mixed up between language and philosophy? Commented Oct 30, 2023 at 18:21

First, we can reformulate definition statements as truth-apt assertions about how a word is used: "I stipulate that unicorn refers to usually-pink-or-white goat/horse-like animals with one horn growing out from the center of any unicorn's head," which will be true just in case I am honestly trying to refer in general to such beings; or, "Most people use unicorn to refer..." with the attendant satisfaction conditions.

Second, we might be inclined to say that definitions, not so reformulated, are accurate or inaccurate; this is similar to, but perhaps subtly distinct from, saying that definitions are true or false.

Third, so-called "real" definitions, of objects or concepts instead of words, would seem to be such as are correct or incorrect, or at least trivial or nontrivial. Of anything, we might trivially define it as "something-or-other," which might seem true but almost pointless to say. But so of some specific thing, a given dog say, we might then really-define it as "a member of a species with genetic tendencies and compatibilities x, y, z," and this would be less trivial/pointless.

• A definition is correct/incorrect but they're not true/false. A tamed canis lupus is, however, true, no? A horse with a horn on its forehead, though, is false. Commented Oct 30, 2023 at 10:06
• @AgentSmith I think there is room to talk of true and false definitions; if it sounds odd to speak so, it is perhaps because, 'I stipulate that X refers to Y," is the form of a speech act, like, "I now pronounce you man and wife," i.e. it makes true what is being said, in the act of saying it. As for whether a definition is false when no real thing conforms to it, that's an interesting question: but take a Triangle in General, not obtuse nor isosceles nor... nor whatever in particular; no real triangle is like that; so is the general definition of "triangle" false? Commented Oct 30, 2023 at 11:05
• @AgentSmith In what sense is "a horse with a horn on its forehead" per se false? If I have a painting of a horse with a horn on its forehead, is that painting not a painting? Is there not something depicted in that painting? Is it a "false painting" ... at least anymore so than a painting of a horse with a white stripe on its forehead which doesn't match any known horse? -- Saying "this thing/concept does not exist in the real/physical world" is a bit different than a bald "this thing/concept is false".
– R.M.
Commented Oct 30, 2023 at 14:37
• @KristianBerry, well, this just in, I recall reading about good and bad definitions, but nowhere have I come across the statement "this/that definition is false" Commented Oct 30, 2023 at 18:17
• @R.M. yes I believe this "obvious" misapprehension of a such a "simple" notion as a definition is a linga (mark/sign) of noobs like me. What exactly is a definition? We should start there, no? Commented Oct 30, 2023 at 18:20

A definition means basically giving a name to the thing described in it. A definition is neither true nor false, it is more like an opinion: I would like to call that thing by this name.

The thing described in the definition can be a real phenomenon, an abstract idea, an imaginary thing or even an illogical concept. The description just has to make it clear, what kind of thing is in question. Otherwise the definition is incomplete and nobody knows what is being discussed.

• A definition, as you can see I'm quite confused, is a basically the meaning of a string of symbols, following some kinda rule. Commented Oct 30, 2023 at 10:08
• A definition is an assertion that can either be accepted or not; it is not, in and of itself, either true or false. Commented Oct 31, 2023 at 13:48

I will try to put the content in the excellent answers above in a concise form. A definition of a symbol can be one of two things:

1. A list of meanings that the symbol is commonly used to represent.
2. A declaration that the the symbol is to be used for an express purpose.

Statements in the form of 1) can be true or false. If I state that one of the common meanings of the word 'dog' is 'computer motherboard', my statement is patently false. Whereas if I say 'unicorn' is commonly used to refer to the idea of a horse with a horn, my statement is true.

• On the mark, "excellent answers". Since there are so many good answers to choose from, can you help me pick one that's the best? Commented Nov 1, 2023 at 3:41
• That's tricky. Perhaps Bumbles' is the most comprehensive. Kristian's provides an interesting alternative 'accurate/inaccurate' take. Dheeraj evades the direct point of you question, but raises interesting distractions. But let's go for Bumble. Commented Nov 1, 2023 at 6:47

Strictly speaking, a definition is an axiom involving a new term that makes a conservative extension of the underlying formalism. What that means is that any statement that can be proven in the expanded formalism using the definition either (1) makes explicit reference to the new term introduced by the definition or (2) can be proven in the original formalism without using the definition. What makes the definition useful is that the proofs, in case (2), may be much more complex than proofs obtained using the definition.

A definition may (and usually does) involve preconditions that have to be satisfied for the newly-introduced term to even exist. If the preconditions are false, then so is the definition.

It can also be false in the sense of being overly-restrictive - false-framing. Example:

Unicorn (noun): A horse with a horn growing out of its forehead; may be of any color, but are usually pink or white.

where a more appropriate definition (that speaks directly to the etymology of the term, itself) would be:

Unicorn (noun): An ungulate with a single horn growing out of its forehead; such as those in myths that are usually pink or white; or those in real life, that the myths may ultimately be based on, like the African rhinoceros, or the (now extinct) Siberian rhinoceros (elasmotherium); or depictions thereof, such as the Thai and Khmer representations of the Hindu god, Agni (as a rhinoceros), or the ancient art in the Chauvet Cave in France, dating from 10000 to 30000 years ago (of one-horned ungulates).

Another example:

0!! = 1, 1!! = 0, (n+2)!! = (n+2) n!!, for n = 0, 1, 2, ⋯

0!! = 1, 1!! = 0, (n+2)!! = (n+2) n!!, for n = ⋯, -5, -3, -1, 0, 1, 2, ⋯

(or a further generalization thereof), which would allow one to state:

1/√(1 - x) = (-1)!!x⁰/0!! + 1!!x¹/2!! + 3!!x²/4!! + 5!!x³/6!! + ⋯

• So as an axiom, definitions can be true/false? Commented Nov 1, 2023 at 3:03
• Yeah, if it has preconditions that fail to hold. Axioms can have preconditions. Commented Nov 1, 2023 at 21:11
• What may these preconditions be? Commented Nov 1, 2023 at 21:13
• Here's an example: if ⋁(AB) = ⋁A·⋁B, for all A,B ∈ 𝔄M, where AB ≡ { ab ∈ M | a ∈ A, b ∈ B } ... requires ⋁A and ⋁B before it can be stated; maybe granted by an earlier axiom, e.g. for all A ∈ 𝔄M, ⋁A ∈ M exists. You need the latter axiom merely to state the former one. Dependency chains between axioms and definitions can occur. In systems, like AutoMath or Martin-Lőf or Coq, "context" is usually put on the left side as stipulations, e.g. "Γ ⊢ A". Commented Nov 2, 2023 at 20:07

Definitions cannot be true or false. The meaning of a word is its use in language, and definitions will reflect that shared understanding to greater or lesser degrees.

However, in some domains, there is an authorative, prescriptive source for the definitions of words. The most extreme example of this is in law. In such contexts, propositions about what a definition is can be true or false. E.g. the proposition

in the Criminal Code, day means the period between six o’clock in the forenoon and nine o’clock in the afternoon of the same day

is true.

There are many kinds of definition. Verbal definition e.g. as found in laughing or crying , textual definition e.g. Hindi or English , visual definition e.g. symbolic language for deaf , odour definition e.g. smell of burgers, abstract definition e.g. democracy , contact definition e.g. hot or cold , feelings definition e.g. pain or pleasure etc.

Definitions are built upon other definitions in its domain. For example - textual definitions, as defined using English ,use simpler definitions to create more complex definitions.

The purpose of definition is to convey its meaning. Definition by definition are true to the meaning they are trying to convey.

Definition may employ several types of dictionaries to convey the meaning.

The purpose of such an approach is to clarify the meaning in the best possible way.

I think the core of this kind of question is in what's meant by True or False, and the nature of definitions. I wanted to come at this in a more concrete way and in laymans terms, though some of the existing answers (@Bumble, @KristianBerry, @Lowri) touch on it.

Important notes to start:

"All models are wrong; some are useful."

You cannot boil a complete valid system down to a pure set of axioms. Every definition depends on other definitions, and there's no escape or bottom, though we can wrap it in higher- and higher-level formalism. In other words:

"It's turtles all the way down".

I'd argue that definitions are inherently arbitrary, and cannot be True or False in any Real sense. They are each a symbol arbitrarily linked to a concept; nothing more or less.

We often speak of them as such though, or as correct or incorrect, right or wrong; this is merely a shorthand for one or more of the underlying true traits of a definition: whether it's widely-shared and to a lesser degree useful or intuitive / consistent.

This is complicated by the common conflation of statements of fact and definitions.

Note that this argument is itself a definition, of 'definition'. I don't know if it's the most widespread one, but I'll argue it's a better model (that is, more useful) than the alternative I've seen argued - that every word has some Real fixed Aristolean meaning, decipherable with enough study or divine enlightenment*.

* This isn't an idle quip.

At the simplest level: If I tell someone "My dog is a fligabert - a poodle crossed with a wolf", they're liable to respond "That's not a real word". Of course, what differentiates 'Fligabert' from 'Labrador' but widespread use? If in 20 years this wolf-poodle breed becomes extremely popular, and widely known as a Fligabert, 'Fligabert' would naturally become the 'correct' word. For now it's 'incorrect', but only because there's only one person holding to the definition. If however I tell them it's a 'woodle', they're much more likely to accept it as a word (albeit perhaps with some rolled eyes) because despite not having heard it before, it matches the pattern of how cross-breeds are named.

If I designate a new mathematical operation φφ like f(x) = xφφ + y - z, and define it to mean "Cube every other positive term in this equation, skipping the one bearing the φφ", I can certainly do that.. But it'd probably not be a useful definition. In addition to not being shared by anyone else, it's also inconsistent with the way mathematical operations are usually styled.

All of those meanings vary in usefullness, but what meaning could there possibly be to calling any of them True or False on any deeper level?

Sufficiently-intuitive words can communicate their meaning perfectly well even if not previously widely-shared. Similarly, widely-shared words can become useless if their meaning becomes sufficiently counter-intuitive. See "Inflammable".

An important distiction here needs to be made between a definition and a statement of fact.

On their own, meant as definitions, there is no difference between `x = 2` and `4 = x + 2`; one's simply cleaner. I think some of the other answers have mis-dissected it, calling it a statement of fact. As an equation, it could be read as "I define `X` as the value that satisfies the equation `4 = x + 2`. Such a value exists given the common definitions of all the symbols in that equation, but to call the equation True or False without a pre-determined `x` is meaningless. The equation can only be True or False if all the values it depends on come in with prior constraints, and is therefore presented as a statement of fact.

Simiarly, if someone says "A unicorn is a species of maize that grows in isolated stalks", it's a False statement only if the speaker knows what the rest of us see as the meaning of "unicorn", and is claiming an equivalence between the underlying concepts - a horned horse and a stalk of corn. If the speaker honestly means it as a definition though, then we can still call it 'wrong', but only in the sense that it differs from the definition the rest of the english-speaking world agrees on for that word. The 'wrongness' goes no deeper. Humanity made up the word unicorn, and so the only 'true' meaning it can possibly have is whatever the group as a community believes it to mean.

Most of this doesn't matter much in everyday life; words are 'correct' when used the way everyone else uses them, or perhaps if their meaning is intuitive even if not well-known, and 'incorrect' otherwise. The model described above becomes incredibly important though when a group lacks a mutually-shared set of definitions.

Some concrete examples:

• Subgroups disagree on the meaning of a particular word or symbol

• The statement "The X historical flag flying above my house means Y" actually boils down to some combination of "The X flag means Y to me/my group" and "When this X flag was historically flown, it meant Y to the people flying it". The first statement is a definition, and cannot possibly be true or false - though it may certainly be useful or useless at communicating the intended meaning. The second is a provable statement of fact. Using terms like "true", "real", or "incorrect" as a shorthand here without breaking it down into pieces, substantially adds to the perceived disconnect between groups.

• A common trend is for an older group to call the slang of the younger generation "made-up / incorrect" or "confusing"; This dispute is clarified by translating those complaints to 'not widespread among their community' or 'unintuitive'.

• A document a group believes or defines to be true (like a Holy Book or Constitution, respectively) contains words or phrases which no longer have a commonly-understood meaning, and so one has to be defensibly determined.

• This need not overlap with the above, though it often will.
• A fundamentalist with a Holy Book has the fall-back of believing that every word there does in fact have some Real True meaning - whatever the Writer intended, which can in fact theoretically be discovered through sufficient study or divine enlightenment. Even here though, the True thing is the meaning itself, not the link between that word and that meaning - i.e. you may believe that the phrase 'dragon' has a very specific meaning in a particular chapter of a particular Holy Book, but it'd be fallacious to believe that every use of 'dragon' in secular media actually refers to that same specific meaning.
• With legal documents, we're in much more of a bind. Humanity's worked hard to shape an objective framework out of subjective words, but we can't escape the core nature of language; it means whatever everyone agrees it means. The best we can do, is make sure everyone shares the same definitions, and that all those definitions are consistent. If someone comes up and says "This phrase of this law means Z, not Y", the only place the court can turn is "What has that phrase meant in other parts of the law, the judicial record, or common parlance" (i.e., is any particular definition shared among its peers), and "Of any viable interpretations which one will yield a more reasonable result, both in this case and in future cases?" (i.e., is this or that definition useful, consistent, and intuitive). Even if a phrase does have a firmly established definition which clearly applies to a case, that doesn't save us from having to make arbitrary judgements; the definition itself is made of words which require definitions, and it's turtles all the way down. At no point in that chain will you hit an inarguable definition, merely (hopefully) eventual limited common agreement.

Logic is the formal expression of the rules of thinking. Logic is circular; that is, all rules that validate Logic are logical rules, then, Logic validates itself by using logic. When we say Logic is formal, it means it is a formal system, of concepts and axioms. Considering that, truth is the set of rules that are logically consistent with such axiomatic set. In simple words, truth is consistent with Logic.

The circularity of Logic does not mean that it is not valid. It is valid, if you live in a universe which seems to fit perfectly such circularity. In other words, even if it is metaphysically (rationally) circular, it is validated physically (empirically). Test a logical implication empirically, you will find it normally works (I say normally because empirically nothing is sure, there are no 100% probabilities).

When you state something, you are creating a logically consistent link between your statement and Logic (more precisely, with Propositional Logic, which is the core of Logic). So, you are proposing a truth.

From a academically philosophical perspective, a statement positing truth is precisely what Positivism is about. See my answer here.

• Are you trying to link logic and language? Do you see circularity in the issue I raised in re truth-aptness of definitions? Commented Oct 31, 2023 at 8:08
• Logic is the formal expression of the rules of thinking. A formal expression (see formal system) already implies a language. Moreover, formal was classically understood as written (formalizing an agreement typically involves writing it). Thinking / reason do not depend necessarily on a language, but Logic does. Commented Oct 31, 2023 at 9:48
• Well, definitional implications of logic, is that what you're trying to elaborate on? Circularity is a big issue with the credibility of logic; definitions too can be circular, and word has it that they are. How very uplifting all this is, oui? Commented Oct 31, 2023 at 10:24
• @RodolfoAP "Logic is the formal expression of the rules of thinking" That cannot be. People thinking logically don't apply any rule. You are confusing logic with formal logic. Commented Oct 31, 2023 at 17:39
• @Speakpigeon there is no "informal logic". "Thinking logically" means thinking according to the formal rules. Do not believe everything you hear. Commented Nov 1, 2023 at 5:32

Can a definition be true/false?

A definition of something in the real world can be accurate or inaccurate. I suppose that a definition can be so far off the mark that it is fair to call it false. This is especially true when the inaccuracy leads to a false conclusion; the poor quality definition is part of the falsehood.

What about the example of the unicorn? Here, when using this term, the user is suspending disbelief in order to enter a fictional world. Within the context of that alternate world, the defined object must still act consistently within itself and other things in that world. It’s definition must allow that to happen. A definition which did not do so could fairly be called false.

As I write, this is the eleventh response to this question. Congratulations.

• Gracias for the 11th response. An interesting take. Dogs are domesticated wolves is an ok definition? Commented Nov 1, 2023 at 3:09

Yes.

To the extent that a definition is a statement of fact about a thing, and that such a statement may be either true or false, it stands to reason that a definition may be true or false.

Example of definitions that may be considered to be true:

Dog (noun): A tamed lupus canis.

Unicorn (noun): An imaginary creature resembling a horse.

IF these definitions are true, THEN reversing them would make them false. Examples of definitions that may be considered to be false:

Dog (noun): An imaginary creature resembling a horse.

Unicorn (noun): A tamed lupus canis.

• This answer is completely unaware of the grue-bleen problem; the reason why your second pair of definitions is "false" is because you (or, more broadly, some English-speaking societal memeplex) have prior expectations which have evolved through time. For example, it is not clear whether CTVT (disgusting tumors, not linking) count as dogs; they are dogs according to genetics but not cultural expectations. Commented Oct 30, 2023 at 18:09
• @Corbin, certain shades of blue and green are far more similar than dogs and unicorns. In any case, if there is confusion over body shape due to both being quadrupeds, the horn is a clear differentiator. (as well as the feet, and teeth...) Commented Oct 30, 2023 at 18:16
• @MichaelHall The Unicorn is not a horse. English dictionaries define the Unicorn as a "an imaginary creature *resembling a horse*". Commented Oct 31, 2023 at 17:33
• @Speakpigeon, the point of my answer is not to precisely define what a unicorn is, or is not. I'm simply using the example provided by the OP. Please make your comment on the question, not here. Commented Nov 3, 2023 at 1:16
• @MichaelHall "IF these definitions are true, THEN reversing them would make them false." This is not true since distinct definitions may mean the same thing. Commented Nov 4, 2023 at 10:57