An analysis of truth

This article by Dan Polansky investigates the concept of truth and some closely related topics. One could think the concept of truth is so simple and direct that it needs no analysis at all. However, since various thinkers have analyzed the concept and cast it into doubt, it seems worth a little while to analyze it anyway. Moreover, one finds that assignment of truth values to sentences in natural language is not entirely unproblematic: some sentences have no truth value, some are ambiguous (and therefore, the truth value is not given before the ambiguity is resolved) and some would arguably benefit from truth value being a real or fractional number ranging from 0 to 1 rather than being one of true and untrue.

If one does not really care about truth (and thus, truth about truth), one may sweep various concerns and quandaries concerning the concept of truth under the carpet, and happily claim that all is fine with truth and that any putative problems are likely to be pseudo-problems. This shows how important truth is as a regulative principle of intellectual endeavors.

Keeping truth undefined

One approach to the definition of truth is this. Truth is something very simple and basic, a concept that children master at the age of, say, three. The child learns how to respond to certain sentences with "that is not true" (or the like) productively. Later, a school child can learn logic without ever bothering with a definition of truth; one can take university courses on various logics without ever bothering with the definition. There is no serious problem concerning truth from which one could learn something important. It is fine to leave the concept of truth undefined.

This anti-philosophical approach has its merits, but we will look at truth anyway.

First impression

Truth is a property of certain kinds of objects. Candidate kinds of objects are as follows:

• sentences
• interpreted sentences
• sentence meanings abstracting away from the sentence and its language
• propositions
• propositional beliefs (sentential beliefs)

Some kinds of representation do not have a truth value:

• drawings
• paintings
• photographs
• maps
• non-propositional beliefs

In the following, we will work with the hypothesis that it is sentences that have truth value, albeit via their meaning.

Not all sentences have truth value. Thus, question sentences and imperatives have no truth value. Furthermore, one can debate about whether certain kinds of ought-sentences have truth value or rather are something like imperatives in disguise.

One can get an idea by means of simple everyday examples:

• The sentence "two plus two equals four" is true.
• The sentence "two plus two equals five" is false/untrue.
• The sentence "there is no maximum prime number" is true.
• The sentence "there is a maximum prime number" is false/untrue.
• The sentence "potable water is poisonous to humans" is false/untrue.
• The sentence "there is a dog barking on the street" is false/untrue if there is in fact no dog barking.

But then, what is an abstract characterization or criteria that distinguish true sentences from untrue sentences? One characterization is this: the word "true" is X such that stating "It is X that Y" means the same as "Y" for relevant kinds of Y. More on this is in the next section.

Tarski-inspired sketch of a theory of truth

By considering Tarski's theory of truth, mentioned by Popper, we may obtain a simple and clear idea that can serve as a practical definition of truth. The idea is captured in the following axiom schema, where X is a sentence:

• Sentence "X" is true if and only if X.

The specific axioms are obtained from the scheme by replacing X with specific sentences. Thus:

• Sentence "It rains in Oslo" is true if and only if it rains in Oslo.

Above, X is not a variable of quantification but of textual substitution. We can wish to obtain a definition that characterizes the truth of sentence s by linking s to other objects or characteristics of s, but this is avoided. In predicate logic, when s is a variable of quantification, "s" is not a formula, whereas "isTrue(s)" can be a formula, as well as e.g. "correspondsTo(s, reality)". The right-hand part of the axiom schema does not say anything about X; it just textually places X after "if".

Thus, if the learner of English did not know the meaning of the word "true", now they should have a decent first idea. However, possible problems with truth values of sentences such as ambiguity, vagueness, being merely approximate, etc. are not addressed by this treatment.

The above is merely inspired by Tarski's theory of truth rather than being identical to it: Tarski's theory is a technical one, depending on the contrast between object language and metalanguage.

Wikipedia's section about folk theory of truth states: "The folk theory of truth is useful in everyday life but, upon deep analysis, turns out to be technically self-contradictory." This is unconvincing since: 1) folk theory is not technical and therefore cannot be technically self-contradictory, 2) it is not obvious that liar paradox (see dedicated section below) cannot be resolved by the claim that some descriptive sentences have no truth value or by other means and 3) even if there is a locus of contradiction in the folk theory, a member of the folk would not blindly run formal inference from the contradiction to produce all well formed sentences as allegedly true but would rather isolate the locus of contradiction and avoid making any inferences from it. Thus, arguably, the folk theory has more merits than the critics admit.

Links:

• 2. Tarski’s theory of truth in Truth, SEP
• Tarski’s Truth Definitions, SEP
• 4. Tarski’s Semantic Theory in Truth, IEP
• Truth#Folk beliefs, wikipedia.org

Ambiguity

Since in general, words have multiple meanings/semantics, sentences also have multiple meanings/semantics. This limits out ability to assign unique truth value to sentences. It is less problematic when there is enough context for disambiguation. However, whether the disambiguation is ever perfect is perhaps debatable.

Expressions in imperative programming languages have radically unambiguous execution semantics. This suggests a great success can be reached in certain fields of unambiguous codification. One can expect to be able to reach similar unambiguity when one codifies mathematical axioms using the language of first-order logic.

Software requirements, written in natural language, are often required to be unambiguous.

Approximate truth

There many everyday examples of perfectly true sentences. However, many sentences are approximately true, and we often do not bother to explicitly say so. Thus, we may say:

• "The orbit of Mars around the Sun is elliptical."

This cannot be exactly true since Mars is gravitationally impacted not only by the Sun, whereas the elliptical orbit would be perfectly so only if Mars and Sun were the only mass objects impacting the trajectory.

We may choose to be more accurate:

• "The orbit of Mars around the Sun is very closely elliptical."

By a similar token, Newtonian gravitational theory is a very good approximation for small speeds, but not for high speeds. Instead of saying that Newtonian theory is true, we may use the Popperian term "verisimilitude": the theory has a good but imperfect verisimilitude, or is similar to truth but not exactly true.

Another approximation is the statement that "humans have four nasal cavities"; some individual humans do not have all four of them. Similarly, "humans have one head", but that is not true for all humans. Also, "humans have either XX or XY chromozomes", but that is not exactly accurate either.

Accuracy

The word "accuracy" can sometimes be used instead of "truth". One speaks of higher or smaller accuracy. Accuracy stands in contrast to precision, following Russell's prescription. Per this usage, precision captures something like the resolution or detail with which one says something. Loose or non-native speakers of English may say "precision" or "imprecise" and mean "accuracy" or "inaccurate".

Theoretical entities

Mendelian genetic theory posits Mendelian genes. But no one has ever observed these directly, and as far as I know, there is no mapping from Mendelian genes to DNA letter sequences. Nonetheless, the theory seems to be empirically adequate: it makes testable predictions about entities that can be directly observed. Thus, on one hand, one may deny the truth of statements about genes, since genes do not really exist, and yet, one may be very ready to make statements about genes, since their predictions are empirically adequate and as long as one speaks within Mendelian theory, genes are posited. At the same time, as long as the communication partners know they are talking within the Mendelian theory and know that the genes are theoretical rather than observational entities, they do not deceive each other by talking about genes.

Artificial slicing into entities

Sentences imply a certain slicing of reality into entities or entity types which is not always objectively given. Nonetheless, as long as the slicing is well defined, one can argue that sentences are true even if the slicing is arbitrary. However, the arbitrariness of the slicing may limit the degree to which the sentences reflect reality.

As an example, we may slice the phases of English into Old English and Middle English, positing no other entity in between. Then we may assign words as belonging to one or both of these phases. But there is something conventional about this slicing. The slicing and the sentences using it posit an implied sharp boundary between the two entities that may not exist in reality. The conventionality of a possible slicing stands in contrast to the efforts of biologists to as-if cut the reality at its joints when identifying taxa such as species and genera. Thus, the taxa are aimed to correspond to something real and observable, not merely something conventional and practical.

The arbitrariness of certain slicings can be perhaps likened to artificial borders between countries. Some borders are less artificial, such as those following the course of a river or following the structures of mountains. Upon first impression, one could think a Martian could not detect the borders upon a quick inspection. But it is not so simple; once different regulatory regimes are adopted on different sides of the border, the border can very well be physically recognizable: for instance, one side of the border can have a very loose policy for cutting tree, unlike the other side.

This is to some extent relating to the subject of fuzzy logic below, but seems distinct.

Timelessness of truth

Truth value of an interpreted sentence utterance or occurrence is timeless, not changing in time. However, this is not so for the truth value of an uninterpreted context-free sentence understood as a sequence of words with no additional context information such as the speaker/writer, time of utterance, etc.

One concern is of changes in definition: the sentence "Pluto (heavenly body) is a planet", taken as a sequence of words with no time or context of utterance, has no definite truth value since the definition of the concept of planet has changed in such a way that Pluto was a planet before but not after the change. One could imagine expanding the sentence with a link to a defining dictionary, e.g. "Pluto (heavenly body) is a planet [term definitions from Merriam-Webster, 2020-01-01]", but this is not customary, and Merriam-Webster does not present entry revision histories for this to actually work. Moreover, Merriam-Webster's definition is not technical enough to exclude Pluto; one would have to load the definition from International Astronomical Union (IAU).

The above problem is reduced when one considers not sentences as context-free sequences of words but as sentence utterances or occurrences, having an auditory or textual context, the speaker/writer, utterance time, etc. One can then figure out from context the time of utterance and the applicable definition of "planet" at the time of the utterance.

Another case of failed timelessness is with sentences like "it is raining [here] today". Again, the context-free sentence has no truth value although it does have a context-free meaning/semantics, but the sentence-utterance/occurrence does have a truth value since it provides specific referents for "today" and "here".

Kuhnian paradigm

One may wonder about the impact of Kuhnian paradigms on the concept of truth and slicing into entities. For example, prior to chemical revolution, there was no way to say that X is a physical mixture while Y is a chemically pure substance since the conceptual distinction was not there. Similarly, Priestly is alleged to have thought to hold in his hands dephlogisticated air, having no concept of oxygen. Thus, the general state of knowledge limits the kind of sentences (from the conceptual rather than lexical perspective) one can form, and which truths can be expressed. Whether this detracts from the concept of truth is unclear; even before the chemical revolution, one could accurately say "water is a substance" as opposed to the inaccurate/untrue "water is a solid object".

Scientific truth

The problems of scientific truth seem to be related to multiple subjects picked above, including approximate truth and theoretical entities. One interesting thing about scientific truth is that it seems more liable to these problems than everyday statements such as "there is a dog on the street". Paradoxically, in some ways, ordinary people dealing with ordinary affairs may seem to more often trade in perfect truth than scientists relying on models that are to some extent tentative or approximate. However, this is rather speculative and would require a deeper analysis and perhaps sourcing; it concerns an empirical hypothesis, to be tested empirically to some extent.

A field of inquiry adept in eliminating ambiguity and arriving in near-certain truth that is nowhere close to approximate is mathematics. However, according to modern classification terminology, mathematics is not science; "science" refers to empirical sciences such as the prime science physics. Moreover, some theories of existence of mathematical objects (numbers, geometric shapes, etc.) deny genuine existence to them.

Fuzzy logic

Many words such as "hill" intuitively do not apply either fully or not at all; they may apply somewhat. Thus, one may think of something as a 0.25-hill, 0.5-hill or 0.75-hill. Indeed, one would think of the transition from a hill to a mountain to be continuous. Thus, if one says "This is a hill" but really thinks "This is a 0.75-hill", this makes the original sentence not entirely true, but far from untrue. This is a limitation from the point of view of the kind of logic that classifies sentences into true and untrue ones. Nonetheless, one can say "This is something of a hill" or "This is something between a hill and a mountain", and the problem disappears.

To understand this simple concept, one does not need to study the technicalities of fuzzy logic, which is a mathematical structure with engineering applications.

Self-reference

This very article aims at presenting true sentences to the reader, even if it may fail here and there. If the article had no such intention, it could very well define truth as "any black cat that broke a vase" and be done with it.

Self-reference is also of concern in the criticism of the concept of truth. Since, if the critic does not aim to raise true or at least valid objections or reservations against the concept of truth, why should we care? However, the critic may object that one can reveal problems with the concept of truth even by means of sentences that are not perfectly true, e.g. are merely approximately true, are metaphorical, etc. That response seems to be true or valid enough: things said by a sentence and things revealed by a sentence are two distinct things.

Self-reference plays a role in section Liar paradox.

Theories of truth

As of now, this article treat of most theories of truth in a limited way. This shortcoming is planned to be addressed by a later expansion. In the meantime, one can learn about various theories of truth in the linked further reading.

Truth as correspondence to fact

The content of this section is largely delegated to further reading. One observation: this definition shifts the definition burden to the concepts of fact and correspondence. Nonetheless, this definition or characterization is superficially plausible enough.

Further reading:

• The correspondence theory in Truth, britannica.com
• The correspondence theory in Truth, Stanford Encyclopedia of Philosophy
• Correspondence Theory of Truth, SEP

Truth as a correspondence to reality

Some sources use the language of correspondence to reality. Superficially, that seems plausible enough. However, it raises some questions.

One can argue that our perceptions and experiences are created by Descartes' evil demon, by Matrix, or to put it in lay terms, that world is but a dream. If that is the case, a sentence like "there is a dog on the street" based on one's visual perception would be untrue as long as one interprets it in this anti-realist fashion. But then, one would hardly obtain any true statements concerning the empirical world, and one would have little hope of obtaining any in future. That seems unsatisfactory.

To address the above, one can interpret sentences in a non-realist fashion. Thus, the dog on the street is not really an extra-mental object but rather a disposition to perceptions of the sole existing perceiver. But then, this is no longer correspondence to reality, unless one uses the word reality to contrast dream experiences of the sole perceiver from non-dream experiences. But even if we disregard dreams, there are other perceptual phenomena at odds with reality, including optical illusions and mirage. On the other hand, the idea is clear: if, before conversion to non-realism, we were able to distinguish what is real from what is unreal using our perceptions and experience, we should be able to use the same method under a new non-realist (in the sense of no world outside of the mind) interpretation.

Moreover, one can truthfully state, "I had a dream, and in that dream, there was a dog barking on a street". That seems true enough, provided I did in fact have such a dream, and yet, it does not point to an extra-mental reality. In any case, the dog in the dream does not need to correspond to any extra-mental dog.

A further complication is that if in math, one is not a Platonist, one may believe that mathematical objects are not real in some sense, and that they are in the minds only. And yet, one naturally hesitates to deny truth to such statements as "two plus two equals four".

Further reading:

• Evil demon, wikipedia.org
• Evil Genius Doubt in Descartes' Epistemology, Stanford Encyclopedia of Philosophy
• Platonism in the Philosophy of Mathematics, Stanford Encyclopedia of Philosophy

Truth as coherence of a system

The content of this section is largely delegated to further reading. One objection, possibly invalid, is that what we require in practice is not only correspondence between statements/propositions but also between certain statements/propositions and perceptions. Thus, there can be a coherence between the statement "There is a cat on the street", and perceptions of those who happen to be on that street.

Further reading:

• Coherence and pragmatist theories in Truth, britannica.com
• The coherence theory in Truth, Stanford Encyclopedia of Philosophy

Pragmatic theory of truth

American pragmatism is characterized by the slogan "truth is what works" by William James. This makes it possible to assign truth value not only to descriptive sentences but also to duty-imputing or normative sentences. However, Stanford Encyclopedia of Philosophy provides a different characterization. Let us stop here and delegate the rest to further reading.

Further reading:

• Pragmatist theories in Truth, Stanford Encyclopedia of Philosophy
• The Pragmatic Theory of Truth, Stanford Encyclopedia of Philosophy
• Pragmatic theory of truth, wikipedia.org

Deflationism

The content of this section is largely delegated to further reading. Deflationism as regards truth seem to take a position similar to that we have taken in section Tarski-inspired sketch of a theory of truth.

Further reading:

• Deflationism in Truth, britannica.com
• Deflationism in Truth, Stanford Encyclopedia of Philosophy

The duty to avoid stating untruth

In various contexts, there is arguably a moral duty to avoid stating untruth. One case of failure to heed the duty is intentional stating of untruth, another one is failing to take reasonable precautionary steps to prevent ending in error and as a consequence stating untruth, even if not expressly intentionally.

Kant argued that lying is always morally problematic. However, this is not plausible. When German Nazis knock on the door of a villager and ask whether he is hiding Jews or resistance fighters, the villager has no duty to respond truthfully. A similar point was made by Russell when he opined he had no duty to speak truth in response to an inquiry from game hunters about where the game is located or hiding. One may argue that Hitler would have no chance if avoidance of lying were hard-coded into the physical laws of the universe; however, since it is not so hard-coded, one has to make moral decisions in an imperfect world, and be forced to respond to people to whom one does not want to talk at all.

Further reading:

• Justice: What's The Right Thing To Do? Episode 07: "A LESSON IN LYING", youtube.com

Mandatory canonical untruth

Expanding on the above, while societies nominally require their members to say true things, in fact, they often require their members to say canonical untrue things. Thus, in a strongly Christian society, one may be required to say the right/canonical things about God and Jesus. In a Marxist-Leninist society, one may be required to say the right/canonical things about bourgeoisie, proletariat and dialectical materialism. In the times of Galileo, one may be required to speak the accepted dogma about the relationship of the Sun and the Earth.

The requirement to speak untruth may in part be driven by sincere belief of those requiring that these things are true. But not necessarily; it seems far more likely that the priestly classes of various religions know all too well that they are inculcating untruths. And where the priestly classes are gone, analogues of priestly classes are likely to appear, especially if their political ideology is based in part on the idea that truth does not exist.

The value of misrepresentation

As something of a speculation, truth and accuracy are probably more cared about in engineering than in management. In engineering, untruth can have grave consequences; in management, untruth is all too often required to overcome the limitations of poorly designed systems of rules; there is the phrase "work to rule" referring to a form of worker protest consisting in sticking to rules exactly as they are. These often not-fully-functional systems of rules often do not get corrected since as long as they are being constantly violated, the consequences of their defects are tolerable. This principle cannot be used to develop computer software: the compiler, linker and CPU do not break the rules to correct defects created by the programmer.

As an aside: As a result, when a company has a poorly designed systems of rules and business processes, and these rules and processes get implemented in business software and strictly enforced in machine fashion, this can cause grave problems to the business. In this case, it is not necessarily the business software that is at fault.

Expression of uncertainty

One who is aiming at only saying true things is well advised to consider stating things with uncertainty or expressly indicating that something is a hypothesis or a speculation. Thus, one can use the following forms:

• "X probably holds true."
• "It seems X is an Y."
• "It is not completely clear whether X is true."
• "X seems very likely but no proof has been delivered."

The great advantage is that one can communicate not only things that are certainly true (there are limits to certainty anyway) but also things that one is in the process of figuring out or is in another way uncertain about.

There are limits to what uncertainty one can reasonably wish to explicitly express. Thus, during the acceptance of Newtonian physics, it would be artificially precisionist to state things like this: "According the tentative not-yet-falsified Newtonian theory, the gravitational force has the size of so-and-so". (Let alone that this Popperian language was not available in this form.) Therefore, on some level, one often runs some risk of reporting inaccurately.

Figures of speech including metaphor

Sentences using metaphor seem to be tricky as for truth, and many other figures of speech present similar problems. One may claim that even if they are not literally true, they are metaphorically true. However, if one accepts "metaphorically true" as a species of "true", this will dramatically reduce the refutability/falsifiability of sentences since the defender will be able to claim that the intended meaning was metaphorical, hyperbolic, or non-literal in another sense. A better plan seems to be to treat "metaphorically" in "metaphorically true" as alienans (somewhat similar to "fake" in "fake inspector"), and therefore, if something is metaphorically true, it is not really true.

This may present complications for statements like "Species originated by natural selection". Technically speaking, one would be forced to say "Species originated by natural analogue of artificial selection". One may want to continue the former less loquacious practice without being accused of untruth.

On the other hand, a conceptual metaphor can bring the reader's mind to notice something about the world that he would not notice otherwise. Thus, it can have reality-bearing cognitive value. Indeed, "Species originated by natural selection" is a statement that can achieve as much to someone not acquainted with Darwinian theory; they only need to ask: "what, if anything, could be natural selection?"; "to what context does 'selection'" refer to and how can I bring it over to the context of origin of species"?

Fictional entities

Novels and short stories in fiction report about fictional entities (characters, places, etc.) as if they were not fictional. One is not ready to admit that novels report truthfully, but one is not complaining about their reporting untruthfully. Interestingly, one is ready to draw inferences and perform consistence checking within the fictional world. The sentence "in the Tolkien's fictional world, Bilbo is a hobbit" seems true enough. (There we go: "true enough" as opposed to "true"?) The case seems to be interesting.

Another case is of a novel or short story that is hinting at truth about real-world entities. Thus, in the most trivial case, one would report truthfully about real-word entities and events but change proper names; however, one could be all too easily accused of libel since change of proper names is too thin a masking. One can perform deeper masking by modifying various details to be fictional and thus remove the description from reality to various degrees. But even so, there is something truthful or reality-bearing about the resulting description. The case is interesting as well.

Whose truth

There are some arguably bizarre theories of truth asking such questions as "whose truth: your truth or my truth" and similar. I am not properly acquainted with them. To my mind, they are hard to take seriously. However, a deeper analysis could hypothetically reveal that I am wrong and that they have something significant to say or reveal.

This kind of idea is mentioned in Jesus Christ Superstar, in the words of Pilate: "But what is truth? Is truth that changing law? We both have truths. Are mine the same as yours"?

Links:

• John 18:38, wikipedia.org -- on Pilate's truth-relating utterance

Subjectivity and relativity

Some sentences can be claimed to be remarkably subjective or relative, limiting an unambiguous determination of truth value. Examples:

• "X is good": good in reference to what objective or purpose?
• "X is beautiful": beautiful in whose eyes?
• "X tasted great": to whom?
• "X is an excellent film": according to what critics?

One could think these statements have no truth value and are entirely subject-dependent or depending on unstated reference object. But that does not seem entirely accurate. A good kitchen knife can be good for slicing food such as onion, a default purpose of the knife (as opposed to as a weigh). A beautiful woman can be widely recognized as beautiful. The food in a particular restaurant can be reported by many customers as tasting great. A film can have received good raking by film critics. That said, this is for a more extensive analysis and debate.

The above ideas suggest how to rephrase the above examples for greater explicitness:

• "X is good for its default purpose or use case" or "X is ethically good"
• "X is widely considered to be beautiful" or "person P found X beautiful"
• "X is reported to taste great by the many people who tried it" or "person P said X tasted great"
• "Film X has excellent ratings on website Y" or "person P likes film X very much"

Revelations by untrue sentences

True things can be revealed by untrue sentences. Examples:

• A speaker who said fluently things in English reveals their English speaking competence, if the things said are untrue. Thus, they reveal the truth of the statement "Speaker so-and-so is a fluent English speaker."
• Someone who says "The password to the box is not 35475829232302" has revealed the password if it is in fact the correct password.
• An untruthful speaker reveals which entities they are able to recall as existing or worth mentioning.
• As a more general example, two communicating parties sharing sufficient context may perform certain distortions of things being said as long as the receiver is able to perform unambiguous decoding of the distortion.

This topic can be developed further, to point at various ways and levels of hinting, distortion/deformation, and non-literal interpretation (e.g. as a metaphor, metonymy, hyperbole, etc.), resulting in a rather challenging topic of its own.

Limits of knowledge

Some seem to point out to limits of knowledge as limits of truth. However, limits of knowledge do not in fact present any problem for the concept of truth, which is independent of/orthogonal to the concept of knowledge.

As for knowledge and truth, the following combinations are possible:

1. X is true and the truth value of X is known
2. X is false and the truth value of X is known
3. X is true and the truth value of X is unknown
4. X is false and the truth value of X is unknown

Since the cases 3 and 4 cannot be told apart by the party not knowing the truth value of X, one can think of it as 3 cases in a combined truth/knowledge domain: X is known to be true, X is known to be false, X is unknown.

It is trivial to come up with empirical hypotheses that do in fact have truth value, or empirical questions that have true answers, but the truth value or the true answer is unknown by most people or not at all. Examples of statements not known to be true or false provided the parameters are plausible and non-refuted:

• District/region X has the number N of trees.
• There is N number of words in English.
• The number of households in district/region X that have an electronic tablet is N.
• At time T, Socrates the Greek philosopher was at location given by latitude and longitude LAT, LON.
• At time T, person P rolled a die and the result was R.

On a related note, discovery does not create facts; it discovers them. For example, the discovery of Saturn's rings did not create the fact that planet Saturn has rings. If no one discovered that Saturn had rings, for failure of development of telescopes, it would still be true that planet Saturn has rings, even if no one knew that to be the case.

Limits of knowledge apply to mathematics, a non-empirical field of inquiry, as well. We will never know all true statements about positive integers, starting with singular statements of the form "N is a prime number"; there is an infinite number of such true statements, and they cannot be represented in a compact manner, unlike e.g. trivially known statements of the form "N equals M + 1". We do have an algorithm for deciding whether a number is a prime number, but it hits practical computational limits concerning computation time and memory. Moreover, there are algorithmically undecidable problems such as the Turing machine halting problem, which further highlights the contrast between something being true (e.g. machine X halts when run on input Y rather than running indefinitely) and something being known to be true.

The above considerations give a clear picture of the contrast between X being true and X being known to be true.

There is another related subject: upon ultimate analysis, we cannot have absolute certainty. If this is so, we only somewhat tentatively accept statements as true, using the concept of truth as a regulative principle, which leads some of us to correct statements in the light of new evidence, discovery, realization, etc. If truth was just a name for whatever statements command strong conviction in most people, that would be a whole different concept of truth.

Truth as a limit case

One can suspect that, in some areas, perfect truth is unattainable. At least for some applications, truth can perhaps be likened to a mathematical circle or a sphere as a limit case for what can be achieved in our empirical world. We cannot manufacture a mathematically perfect spherical ball, but we can achieve remarkable precision in manufacturing, and we are satisfied with having a name for that limit object at which we aim, here "sphere". To what extent the case of truth bears a good analogy to the manufacturing example, perhaps in some areas, is not quite clear.

Applications

Whatever the quandaries about the concept of truth, the concept has practical applications.

One application is in those libel laws that require statement to be untrue in order to be libel. A defendant accused of libel who states that he is a truth-nihilist and that there is no such thing as truth and that therefore he could not possibly have said untruth is unlikely to get very far with this line of defense.

The workings of contracts depend on one's ability to tell whether a condition in the contract was violated.

Approximate truth or mere-model quasi-truth has such practical applications as the Ptolemaic astronomy in medieval sea navigation. For Global Positioning System (GPS) calculations, Ptolemy and Newton are not sufficient, and one needs Einstein; and thus, being closer to truth makes a difference.

In software making, Boolean variables usually take the values of true and false. Software applications make use of such variables, as well as logical operators acting on them.

Tarski's theory of truth

The content of this section is delegated to further reading.

Further reading:

• Tarski and truth conditions in Truth, britannica.com
• Tarski’s Truth Definitions, Stanford Encyclopedia of Philosophy
• Semantic Theory of Truth, Internet Encyclopedia of Philosophy
• Semantic theory of truth#Tarski's theory of truth, wikipedia.org
• Tarski, A. (1983). "The Concept of Truth in Formalized Languages", thatmarcusfamily.org

Tarski's undefinability theorem

Tarski's undefinability theorem states, informally, that in the context of a formal logic the predicate is-true applying to positive integers interpreted as codes for formulas of Peano arithmetic cannot be defined in Peano arithmetic itself. The rest of this section is delegated to further reading.

Further reading:

• Tarski’s Theorem on the Undefinability of Truth in Gödel’s Incompleteness Theorems, Stanford Encyclopedia of Philosophy
• Tarski's undefinability theorem, wikipedia.org

Desiderata other than truth

In sentence production, e.g. in journalism, there are desiderata (things desired) additional to truth, and these should not be confused with truth. One may require text to be non-misleading, neutral, balanced, objective or complete (with respect to a defined scope). One can create a wrong impression about an issue by presenting only one side of the argument or by highlighting only some facts and not other relevant facts.

For instance, BBC editorial standards single out the desiderata of truth, fairness, accuracy and impartiality: "In our journalism in particular, we seek to establish the truth and use the highest reporting standards to provide coverage that is fair and accurate. Our specialist expertise provides professional judgement and clear analysis. We are impartial, seeking to reflect the views and experiences of our audiences – so that our output as a whole includes a breadth and diversity of opinion and no significant strand of thought is under‑represented or omitted." (Boldface from the original set in italics.)

Another set of desiderata applies to persuasive rhetoric. One set is given by Robert Pirsig: "He singled out aspects of Quality such as unity, vividness, authority, economy, sensitivity, clarity, emphasis, flow, suspense, brilliance, precision, proportion, depth and so on; kept each of these as poorly defined as Quality itself, but demonstrated them by the same class reading techniques." In the quotation, Pirsig's "Quality" is predominantly rhetorical quality and includes neither accuracy (although it includes "precision") nor completeness.

Another set applies to technical requirements specification. One example is from NASA and includes clarity, completeness, compliance, consistency, traceability, correctness, functionality, performance, maintainability, reliability, and verifiability/testability.

Further reading:

• Journalistic objectivity, wikipedia.org

Liar paradox

Liar paradox is one problem with assigning truth value to sentences. The problematic sentence is "this sentence is untrue", which can be alternatively rendered as two sentences: "1) the next sentence is true" and "2) the previous sentence is untrue", nominally avoiding self-reference on a single-sentence level.

Attempts were made in mathematics to ensure the paradox cannot occur. Nonetheless, since we admitted that some sentences (especially questions and imperative sentences) do not have truth value, it seems relatively painless to accept that the sentence embodying the paradox has no truth value either. Other solutions are given in the Wikipedia article.

Further reading:

• Liar paradox, wikipedia.org
• Self-reference and Meaning in Ordinary Language in Conjectures and Refutations by Karl Popper

Etymological interpretation

As an exercise, one can have a look at what the names of the concept of truth in various languages would mean if the meaning followed the etymology.

Let us have a look at a Slavic example, Czech pravda (Russian is similar). That sounds as if it was related to pravit, say or tell. Thus, one could etymologically interpret pravda as, that which is being said or perhaps said by many. That does not match the semantics (or at least the core semantics that is subject of this article): even if all language users say that the Earth is flat, that does not make the statement true. But this kind of etymological quasi-meaning is interesting to note since all too many language users (including pseudo-philosophers) are liable to use the word in this way, leading to terminological confusion.

For English, Merriam-Webster traced the etymology to Old English trēowe, faithful. But in which sense of faithful? Is it like a spouse being faithful to another spouse? Or a picture being a faithful depiction of reality? Or is it faith + -ful, to be interpreted as leading to faith or worthy of faith? Worthy of faith, while not too far away, does not match the semantics: given we can never or hardly ever attain perfectly certain knowledge (not even in mathematics if we analyze things to their ultimate ends), we are bound to have faith in (and believe and trust) things that may turn out to be untrue. Leading to faith also does not match the semantics: one only needs to recall the Ancient Greek sophists giving the advice that, in order to convince, one should better state plausible yet untrue things than implausible/hard to believe yet true things.

For German, Duden traces the etymology of wahr to wār, of which it says "eigentlich = vertrauenswert", worthy of trusting or trustworthy (but Duden does not have an entry for the adjective vertrauenswert).

Truth, untruth, error and lie

Let us determine the relation of the concept of truth (and untruth) to the concepts of error and lie.

One might naively think that truth is the opposite concept of error. But it is not so. Truth is the opposite of untruth/falsity, both truth values. Error is not a truth value. Rather, error contains an epistemic element, unlike truth. To believe an untruth is to be in error; sure enough. However, a statement can be untrue without anyone being in error about it; no one can be considering the statement as a candidate for believing.

Similarly, lie is not an (perfect) opposite concept of truth; the concept of lie contains an element of intent, unlike truth and untruth/falsity.

See also

• Truth theory

Further reading

• Truth, wikipedia.org
• Truth, britannica.com
• Truth, Stanford Encyclopedia of Philosophy (see section Related Entries for other truth-related articles)
• Truth, Internet Encyclopedia of Philosophy
• Truth, New World Encyclopedia