truth theory

Semantic concept formalized by the Polish-American mathematician and logician Alfred Tarski (1902-1983), although other thinkers had previously discussed the idea.

Truth theory concerns the truth-values of sentence structures in various formal logical languages. Tarski suggested a table by which these values could be determined (although he was less sure about whether the same rules could be applied to natural languages).

Also see: correspondence theory of truth

Source:
A Tarski, ‘The Concept of Truth in Formalized Languages’, Logic, Semantics and Metamathe-matics (1956)

Truth in perspective

Notions of truth are notoriously difficult to disentangle from many of our most basic concepts — meaning, reality, and values in general, to mention just a few.

The subjects of meaning and truth are commonly treated together, the idea being that a thing must be meaningful before it can be true or false. This association is found in ancient times, and has become standard in modern times under the heading of semantics, especially formal semantics and model theory. Another association of longstanding interest is the relation between truth and logical validity, “because the fundamental notion of logic is validity and this is definable in terms of truth and falsehood” (Kneale and Kneale, 16). Though not the main subjects of this article, meaning and validity are truth’s neighbors, and incidental inquiries of them can serve to cast light on truth’s character.

Beyond this minor note of accord, hardly universal, suggesting that meaning is necessary to truth, reflectors on the idea of truth just as quickly disperse into schools of thought that barely comprehend each other’s thinking. A few of the more notable points of departure are these:

1. One of the first partings of the ways occurs at the watershed between literal and symbolic meanings, leading to a corresponding division in truths. People often speak of truth in art, truth in drama, truth in fiction, human truth, moral, religious, and spiritual truth, along with the difference between truth in principle and truth in practice. These topics demand a perspective on meaning, reality, and truth that looks beyond the bounds of literal truth and the branches of philosophy that are limited to it.
2. Merely resolving that meaning precedes truth, logically speaking, only brings up a host of new questions, since the meaning of the word meaning is notoriously hard to pin down. There are just to start at least two different dimensions of meaning that are commonly recognized, namely, connotative meaning and denotative meaning.

In one classical formulation, truth is defined as the good of logic, where logic is treated as a normative science, that is, an inquiry into a good or a value that seeks knowledge of it and the means to achieve it. In this scheme of ideas, truth is the positive quality of a sign that indicates the right course of action for reaching a value that we value for its own sake. As such, truth takes its place among justice and beauty, whose normative sciences are ethics and aesthetics, respectively. Viewed in this light, it is pointless to discuss truth in isolation from a frame of reference that encompasses the topics of inquiry, knowledge, logic, meaning, practice, and value, all very broadly conceived.

Historical overview

In an ancient fragment of text called the Dissoi Logoi, a writer is evidently trying to prove the impossibility of speaking consistently about truth and falsehood. One of the conundrums put forward to confound the reader cites the case of the verbal form, “I am an initiate”, which is true when A says it but false when B says it. Escape from befuddlement seems easy enough if one observes that it is not the verbal expression, the sentence, to which the predicates of truth and falsity apply but what the sentence expresses, the proposition that it states. (Cf. Kneale and Kneale, 16). This same tension between strings of characters and their meanings remains with us to this day.

In his early work Περι Ερμηνειαs (Peri Hermeneias or On Interpretation) Aristotle strikes a chord that not only sets the key for a number of philosophical movements down through the ages but supplies the initial motif for many themes in the logic of meaning and truth that are still undergoing active development in our time.

Words spoken (phoné) are symbols or signs (symbola) of affections or impressions (pathemata) of the soul (psyche); written words (graphomena) are the signs of words spoken. As writing, so also is speech not the same for all races of men. But the mental affections themselves, of which these words are primarily signs (semeia), are the same for the whole of mankind, as are also the objects (pragmata) of which those affections are representations or likenesses, images, copies (homoiomata). (Aristotle, On Interpretation, 1.16^a4).

Some of the points to be noted in this passage are these:

1. Aristotle employs a distinction in Greek that is drawn between natural or physical signs (semeia) and artificial or cultural signs (symbola).
2. The passage mentions three principal domains of elements, namely, the objects (pragmata), the signs (semeia, symbola), and the psychological elements (pathemata). The last domain extends over the full range of a human being’s affective and cognitive experiences, for brevity summed up as ideas and impressions, where these words are taken in their broadest conceivable senses.
3. This means that the phenomena under investigation have to do with the types of three-place relations that conceivably exist among three domains of this sort. As a general rule, three-place relations can be very complex, and a commonly-tried strategy for approaching their complexity is to consider the two-place relations that are left when the presence of a selected domain is simply ignored.
4. There are two types of two-place relation on the face of the overall three-place relation that Aristotle takes the trouble to mention, namely these:Sign {\displaystyle \longrightarrow } Idea. Words spoken are signs or symbols of pathemata.

   Idea {\displaystyle \longrightarrow } Object. Pathemata are icons (homoiomata) of pragmata.

5. More incidentally, but still bearing heavily on many later discussions, Aristotle holds that the relation between writing and speech is analogous to the relation between speech and the realm of experiences, feelings, and thoughts.Writing {\displaystyle \longrightarrow } Speech. Written words are symbols of spoken words.

   Speech {\displaystyle \longrightarrow } Ideation. Spoken words are symbols of impressions.

Elements of theory

It is customary in philosophy to refer to a distinctive treatment of a particular subject matter, frequently summed up in a succinctly stated thesis, as a theory, whether or not it qualifies as a theory by strict empirical or logical standards. When there is any risk of confusion, an informal thesis of this kind may be referred to as an account, a perspective, a treatment, or so on, reserving the term theory for the type of formal system that serves in logic and science.

Theories of truth can be classified according to the following features:

• Primary subjects. What kinds of things are potentially meaningful enough to be asserted or not, believed or not, or considered true or false?
• Relevant objects. What kinds of things, in addition to primary subjects, are pertinent to deciding whether to assert them or not, believe them or not, or consider them true or false?
• Value predicates. What kinds of things are legitimate to say about primary subjects, either in themselves, or in relation to relevant objects?

In some discussions of meaning and truth that consider forms of expression well beyond the limits of literally-interpreted linguistic forms, potentially meaningful elements are called representations, or signs for short, taking these words in the broadest conceivable senses.

Most treatments of truth draw an important distinction at this point, though the language in which they draw it may vary. On the one hand there is a type of incomplete sign that is nevertheless said to be true or false of various objects. For example, in logic there are terms such as “man” or “woman” that are true of some things and false of others, and there are predicates such as “__is a man” or “__is a woman” that are true or false in the same way. On the other hand there is a type of complete sign that expresses what grammarians traditionally call a complete thought. Here one speaks of sentences and propositions. Some considerations of truth admit both types of signs, terms and sentences, while others admit only the bearers of complete thoughts into the arena of judgment. In a number of recent discussions that focus on linguistic analysis, the vehicles of complete thoughts are described as truthbearers, with no intention of prejudging whether they bear truth or falsehood. The things that can be said about any of these representations, signs, or truthbearers are expressed in what most truth theorists describe as truth predicates.

Most inquiries into the character of truth begin with a notion of an informative, meaningful, or significant element, the truth of whose information, meaning, or significance may be put into question and needs to be evaluated. Depending on the context, this element might be called an artefact, expression, image, impression, lyric, mark, performance, picture, sentence, sign, string, symbol, text, thought, token, utterance, word, work, and so on. For the sake of brevity, it is convenient to use the term sign for any one of these elements. Whatever the case, one has the task of judging whether the bearers of information, meaning, or significance are indeed truth-bearers. This judgment is typically expressed in the form of a specific truth predicate, whose positive application to a sign asserts that the sign is true.

Considered within the broadest horizon, there is little reason to imagine that the process of judging a work, that leads to a predication of false or true, is necessarily amenable to formalization, and it may always remain what is commonly called a judgment call. But there are indeed many well-circumscribed domains where it is useful to consider disciplined forms of evaluation, and the observation of these limits allows for the institution of what is called a method of judging truth and falsity.

One of the first questions that can be asked in this setting is about the relationship between the significant performance and its reflective critique. If one expresses oneself in a particular fashion, and someone says “that’s true”, is there anything useful at all that can be said in general terms about the relationship between these two acts? For instance, does the critique add value to the expression criticized, does it say something significant in its own right, or is it just an insubstantial echo of the original sign?

Theories of truth may be described according to several dimensions of description that affect the character of the predicate “true”. The truth predicates that are used in different theories may be classified by the number of things that have to be mentioned in order to assess the truth of a sign, counting the sign itself as the first thing. In formal logic, this number is called the arity of the predicate. The kinds of truth predicates may then be subdivided according to any number of more specific characters that various theorists recognize as important.

1. A monadic truth predicate is one that applies to its main subject ? typically a concrete representation or its abstract content ? independently of reference to anything else. In this case one can say that a truth bearer is true in and of itself.
2. A dyadic truth predicate is one that applies to its main subject only in reference to something else, a second subject. Most commonly, the auxiliary subject is either an object, an interpreter, or a language to which the representation bears some relation.
3. A triadic truth predicate is one that applies to its main subject only in reference to a second and a third subject. For example, in a pragmatic theory of truth, one has to specify both the object of the sign, and either its interpreter or another sign called the interpretant before one can say that the sign is true of its object to its interpreting agent or sign.

Several qualifications must be kept in mind with respect to any such radically simple scheme of classification, as real practice seldom presents any pure types, and there are settings in which it is useful to speak of a theory of truth that is “almost” k-adic, or that “would be” k-adic if certain details can be abstracted away and neglected in a particular context of discussion. That said, given the generic division of truth predicates according to their arity, further species can be differentiated within each genus according to a number of more refined features.

The truth predicate of interest in a typical correspondence theory of truth tells of a relation between representations and objective states of affairs, and is therefore expressed, for the most part, by a dyadic predicate. In general terms, one says that a representation is true of an objective situation, more briefly, that a sign is true of an object. The nature of the correspondence may vary from theory to theory in this family. The correspondence can be fairly arbitrary or it can take on the character of an analogy, an icon, or a morphism, whereby a representation is rendered true of its object by the existence of corresponding elements and a similar structure.

Signs

In some branches of philosophy and fields of science the domain of potentially meaningful entities may include almost any kind of informative or significant element. The generic terms sign or representation suffice for these, with the qualification that the terms are used equivocally up and down a full spectrum from the more abstract types to the more concrete tokens that are associated with each other. More specifically, the linguistic turn in analytic philosophy begins with a focus on the syntactic character of the sentence, from which is abstracted its meaningful content, referred to as the corresponding proposition. A proposition is the content expressed by a sentence, held in a belief, or affirmed in an assertion or judgment.

Truthbearer is used by a number of writers to refer to any entity that can be judged true or false. The term truthbearer may be applied to propositions, sentences, statements, ideas, beliefs, and judgments. Some writers exclude one or more of these categories, or argue that some of them are true (or false) only in a derivative sense. Other writers may add additional entities to the list.

Truthbearers typically have two possible values, true or false. Fictional forms of expression are usually regarded as false if interpreted literally, but may be said to bear a species of truth if interpreted suitably. Still other truthbearers may be judged true or false to a greater or lesser degree.

Higher order signs

As predicate terms, most discussions of truth allow for a number of phrases that are used to say in what ways signs or sentences or their abstract senses are regarded as true, either by themselves or in relation to other things. Theorists who admit the term call these phrases truth predicates. A truth predicate that is used to ascribe truth to something, in and of itself, in effect treating truth as an intrinsic property of the thing, is called a one-place or monadic truth predicate. Other forms of truth predicates may be used to say that something is true in relation to specified numbers and types of other things. These are called many-place or polyadic truth predicates.

In ordinary parlance, the things that one says about a subject are expressed in predicates. If one says that a sentence is true, then one is predicating truth of that sentence. Is this the same thing as asserting the sentence? This question serves as useful touchstone for sorting out some of the theories of truth.

Propositional attitudes

What sort of name shall we give to verbs like ‘believe’ and ‘wish’ and so forth? I should be inclined to call them ‘propositional verbs’. This is merely a suggested name for convenience, because they are verbs which have the form of relating an object to a proposition. As I have been explaining, that is not what they really do, but it is convenient to call them propositional verbs. Of course you might call them ‘attitudes’, but I should not like that because it is a psychological term, and although all the instances in our experience are psychological, there is no reason to suppose that all the verbs I am talking of are psychological. There is never any reason to suppose that sort of thing. (Russell 1918, 227).

What a proposition is, is one thing. How we feel about it, or how we regard it, is another. We can accept it, assert it, believe it, command it, contest it, declare it, deny it, doubt it, enjoin it, exclaim it, expect it, imagine it, intend it, know it, observe it, prove it, question it, suggest it, or wish it were so. Different attitudes toward propositions are called propositional attitudes, and they are also discussed under the headings of intentionality and linguistic modality. The formal properties of verbs like assert, believe, command, consider, deny, doubt, hunt, imagine, judge, know, want, wish, and a host of others, are studied under these headings by linguists and logicians alike.

Many problematic situations in real life arise from the circumstance that many different propositions in many different modalities are in the air at once. In order to compare propositions of different colors and flavors, as it were, we have no basis for comparison but to examine the underlying propositions themselves. Thus we are brought back to matters of language and logic. Despite the name, propositional attitudes are not regarded as psychological attitudes proper, since the formal disciplines of linguistics and logic are concerned with nothing more concrete than what can be said in general about their formal properties and their patterns of interaction.

The variety of attitudes that a proposer can bear toward a single proposition is a critical factor in evaluating its truth. One topic of central concern is the relation between the modalities of assertion and belief, especially when viewed in the light of the proposer’s intentions. For example, we frequently find ourselves faced with the question of whether a person’s assertions conform to his or her beliefs. Discrepancies here can occur for many reasons, but when the departure of assertion from belief is intentional, we usually call that a lie.

Other comparisons of multiple modalities that frequently arise are the relationships between belief and knowledge and the discrepancies that occur among observations, expectations, and intentions. Deviations of observations from expectations are commonly perceived as surprises, phenomena that call for explanations to reduce the shock of amazement. Deviations of observations from intentions are commonly experienced as problems, situations that call for plans of action to reduce the drive of dissatisfaction. Either type of discrepancy forms an impulse to inquiry (Awbrey and Awbrey 1995).

Reflection and quotation

The study of propositional attitudes is no sooner begun than it leads to the all-important philosophical distinction between (1) using a meaning-bearer to bear its meaning in an active manner and (2) mentioning a meaning-bearer in a form that keeps its meaning in a more inert or inhibited state. The reasons for doing the latter are various, but involve the need to reflect on a potential meaning, to compare and contrast it with others, to criticize and evaluate both its logical implications and its practical consequences, all before deciding whether to put its meaning into action or not.

The word “quote” derives from the Latin verb quotare, which refers to the practice of numbering references and referring to pieces of text by marking their numbers. There is a certain aesthetic distance involved in this practice, and it leads, if only for moments at a time, to viewing each piece of text as a string of characters that bears its own litter of meanings, but meanings to be reflected on and critically compared with others, both in and out of their litter. It is hardly an accident, then, that matters of Gödel numbers, quotation, and reflection are bound up with each other in mathematical logic and computation theory.