Truth

Common dictionary definitions of truth mention some form of accord with fact or reality. There is, however, no single definition of truth about which scholars agree. Numerous theories of truth continue to be widely debated. There are many other issues about which scholars disagree. What sorts of things can properly be called true or false? What tests can establish a claim as being true? How do we know something to be true? Which truths, if any, are subjective, relative, objective, or absolute? Does truth, as a concept, have a rigorous definition, or is it unavoidably imprecise?

Philosophy of truth
Since only meaningful things can be true or false, the association of the subjects of meaning and truth is standard. Both are studied as part of semantics, especially formal semantics. Truth is also related to logical validity, because the latter concept is defined in terms of truth and falsehood. For these reasons, meaning and validity are touched upon frequently both in this article and in published discussions of truth. It is conventional to refer to a philosophical treatment of a particular subject matter as a theory, whether or not it qualifies as a theory by strictly empirical or logical criteria. Most of the discussion below follows this convention.

Signs, sentences, and propositions
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, described by the generic terms sign or representation. Such entities may include words, pictoral representations, logical or mathematical symbols, etc., and also may include a wide variety of meaningful combinations or clusters of signs. Modern analytic philosophy, by contrast, begins with a focus on the words and syntax of a 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.

Sentences with relative pronouns, such as "I", "it", "now", "here", and so forth, can be true when uttered by one person but false when uttered by another, or even by the same person in a different place and time. For example, "I am a football fan", is true for some persons in some contexts and false for others. This suggests that it is not the sentence to which truth and falsity apply but what the sentence expresses, the proposition that it states.

Difficulties in human communication often arise from the fact that persons are capable of taking up different attitudes, called propositional attitudes, toward what they think, say, or write, and may express their different stances in widely different linguistic modalities. Propositions can, for example, be accepted, asserted, believed, commanded, contested, declared, denied, doubted, enjoined, exclaimed, expected, imagined, intended, observed, proven, questioned, suggested, or wished to be true. Differentiating among the various attitudes and modalities that persons are capable of taking toward a proposition can be critical in evaluating truth. Due to the many factors involved, the analyses can be quite complex, and the philosophical discussions generally reflect this complexity.

Truthbearers
Truthbearer is used by a number of modern 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. Truthbearer, in the context of modern philosophical discussion, is never applied to a person or group of persons; rather, the term is applied to the kinds of entities above because they are deemed specific enough to reasonably be subjected to a close analysis of whether or not they are true. 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.

Truth predicates
Many 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. In ordinary parlance, the things that one says about a subject are expressed in predicates. If one says a sentence is true, then one is predicating truth of that sentence. Is this the same thing as asserting the sentence without the additional qualification that the sentence "is true"? This question serves as an important touchstone for sorting out some of the major theories of truth.

Major theories of truth
Questions about what is a proper basis upon which to decide whether, and to what extent, belief is in accordance with fact, and to what extent statements and the ideas they convey are in accord with real things, whether for a single person or an entire community or society, are among the many important questions dealt with by the theories introduced in this section.

Traditional theories of truth share the claim that truth is a property that certain types of things may have, perhaps in relation to other things, and so the assertion that something is true makes a substantive, significant claim about it. There have more recently arisen so-called deflationary or minimalist theories of truth that are based on the idea that the application of a term like true to a statement does not assert anything significant about it, for instance, anything about its nature, but that the label truth is a tool of discourse used to express agreement, to emphasize claims, or to form certain types of generalizations.

Traditional (substantive) theories
Traditional theories of truth treat truth as a meaningful concept, taking it to refer to significant properties of meaning-bearing entities, to relations that connect them to each other, and to relations that involve them with other things in the world. According to these theories, ascribing truth to meaning-bearing elements says something significant about them. Such theories analyze truth as a descriptive property with a nature that can be discovered through philosophical investigation and reflection. The task for such theories is to explain the alleged nature of truth.

Coherence theory
There is no single coherence theory of truth but rather an assortment of perspectives that are commonly collected under this title. A pervasive tenet is the idea that truth is primarily a property of whole systems of propositions and can be ascribed to individual propositions only derivatively according to their coherence with the whole. Where theorists differ is mainly on the question of whether coherence entails many possible true systems of thought or only a single absolute system. For coherence theories in general, truth requires a proper fit of elements within the whole system. Very often, though, coherence is taken to imply something more than simple logical consistency. For example, the completeness and comprehensiveness of the underlying set of concepts is a critical factor in judging the utility and validity of a coherent system.

Some variants of coherence theory are claimed to characterize the essential and intrinsic properties of formal systems in logic and mathematics. However, formal reasoners are content to contemplate axiomatically independent but mutually contradictory systems side by side, for example, the various alternative geometries. On the whole, coherence theories have been criticized as lacking justification in their application to other areas of truth, especially with respect to assertions about the natural world, empirical data in general, assertions about practical matters of psychology and society, especially when used without support from the other major theories of truth.

Coherence theories distinguish the thought of continental rationalist philosophers, particularly of Spinoza, Leibniz, and G.W.F. Hegel, along with the British philosopher F.H. Bradley. They have found a resurgence also among several proponents of logical positivism, notably Otto Neurath and Carl Hempel.

Correspondence theory
Correspondence theories claim that true beliefs and true statements correspond to the actual state of affairs. This type of theory, in essence, attempts to posit a relationship (a "truth relation") between thoughts or statements on the one hand, and things or objects on the other, as it might theoretically exist independently of the persons involved in the exchange and independently of other issues. It is a traditional model which goes back at least to some of the classical Greek philosophers. This class of theory holds that the truth or the falsity of a representation is determined in principle solely by how it relates to objective reality, by whether it accurately describes (that is, corresponds with) that reality.

Correspondence theory has traditionally operated on the assumption that there is an objective truth relation with which it is the human task to become properly aligned. In practice, however, more recent theorists have articulated that this ideal cannot be achieved independently of some analysis of additional factors. For example, analyses of correspondence that are cast within particular languages are forced to admit the particular language in question as an additional parameter at the outset of theoretical work, and only gradually construct a language-independent truth predicate by means of a careful theory of translation among different languages. There are strong theoretical limitations on the extent to which this can be done. Commentators and proponents of several of the theories introduced below also have widely asserted that the correspondence theory neglects the role of the persons involved in the "truth relation."

Constructivist theory
Social constructivism holds that truth is constructed by social processes, is historically and culturally specific, and that it is in part shaped through the power struggles within a community. Constructivism views all of our knowledge as "constructed," because it does not reflect any external "transcendent" realities (as a pure correspondence theory might hold). Rather, perceptions of truth are viewed as contingent on convention, human perception, and social experience. It is believed by constructivists that representations of physical and biological reality, including race, sexuality, and gender are socially constructed (Hegel, Garns, and Marx were among the first to suggest such an ambitious expansion of social determinism).

Consensus theory
The consensus theory holds that truth is whatever is agreed upon, or in some versions, might come to be agreed upon, by some specified group. The label "consensus theory" of is variously attached to a number of otherwise very diverse philosophical perspectives. Some variants of "pragmatic theory of truth" have been included as consensus theory, though the range of "pragmatic theories" is sufficiently broad that it merits its own classification. Among the current advocates of consensus theory as a useful accounting of the concept of "truth" is Jürgen Habermas. Among its current strong critics is the philosopher Nicholas Rescher.

Pragmatic theory
Three influential versions of the pragmatic theory of truth are due to Charles Peirce, William James, and John Dewey.

Peirce defines truth as follows: "Truth is that concordance of an abstract statement with the ideal limit towards which endless investigation would tend to bring scientific belief, which concordance the abstract statement may possess by virtue of the confession of its inaccuracy and one-sidedness, and this confession is an essential ingredient of truth." This statement emphasizes Peirce's view that ideas of approximation, incompleteness, and partiality, what he describes elsewhere as fallibilism and "reference to the future", are essential to a proper conception of truth. Although Peirce uses words like concordance and correspondence to describe one aspect of the pragmatic sign relation, he is also quite explicit in saying that definitions of truth based on mere correspondence are no more than nominal definitions, which he accords a lower status than real definitions.

William James's version of the pragmatic theory, while complex, is often summarized by his statement that "the 'true' is only the expedient in our way of thinking, just as the 'right' is only the expedient in our way of behaving." By this, James meant that truth is a quality the value of which is confirmed by its effectiveness when applying concepts to actual practice (thus, "pragmatic"). James's pragmatic theory is in part a synthesis of correspondence and coherence theory, with an added dimension. Truth is verifiable to the extent that thoughts and statements correspond with actual things, as well as "hang together", or cohere, and these are in turn verified by the observed results of the application of proposed truths to actual practice  James extended his pragmatic theory well beyond the scope of scientific verifiability, and even into the realm of the mystical:  "On pragmatic principles, if the hypothesis of God works satisfactorily in the widest sense of the word, then it is 'true'". John Dewey, less broadly than James but more broadly than Peirce, held that inquiry, whether scientific, technical, sociological, philosophical or cultural, is self-corrective over time if  openly submitted for testing by a community of inquirers in order to clarify, justify, refine and/or refute proposed truths.

Deflationary (minimalist) theories
Other philosophers reject the idea that truth is a robust concept in the sense discussed in the previous section. From this point of view, to say "2 + 2 = 4 is true" is to say no more than to say "2 + 2 = 4", and that there is no more to say about truth than this. These positions are broadly called "deflationary" theories of truth (because the concept has been "deflated" of importance) or "disquotational" theories (to draw attention to the mere "disappearance" of the quotation marks in cases like the above example), or alternately, "minimalist" views. Whichever term is used, deflationary theories can be said to hold in common that "[t]he predicate 'true' is an expressive convenience, not the name of a property requiring deep analysis."

In addition to highlighting this formal feature of the predicate "is true", some deflationists point out that the concept enables us to express things that might otherwise require infinitely long sentences. For example, one cannot express confidence in Michael's accuracy by asserting the endless sentence:
 * Michael says, 'snow is white' and snow is white, or he says 'roses are red' and roses are red or he says ... etc.

But it can be expressed succinctly by saying:
 * Whatever Michael says is true.

Once we have identified the truth predicate's formal features and utility, deflationists argue, we have said all there is to be said about truth. The primary theoretical concern of these views is to explain away those special cases where it appears that the concept of truth does have peculiar and interesting properties. (See Semantic paradoxes, and below.)

Performative theory of truth
Attributed to P. F. Strawson is the performative theory of truth which holds that to say "'Snow is white' is true" is to perform the speech act of signalling one's agreement with the claim that snow is white (much like nodding one's head in agreement). The idea that some statements are more actions than communicative statements is not as odd as it may seem. Consider, for example, that when the bride says "I do" at the appropriate time in a wedding, she is performing the act of taking this man to be her lawful wedded husband. She is not describing herself as taking this man. In a similar way, Strawson holds: "To say a statement is true is not to make a statement about a statement, but rather to perform the act of agreeing with, accepting, or endorsing a statement. When one says 'It's true that it's raining,' one asserts no more than 'It's raining.'  The function of [the statement] 'It's true that...' is to agree with, accept, or endorse the statement that 'it's raining.'"

Redundancy and related theories
According to the redundancy theory of truth, asserting that a statement is true is completely equivalent to asserting the statement itself. For example, asserting the sentence " 'Snow is white' is true" is equivalent to asserting the sentence "Snow is white". Redundant theorists infer from this premiss that truth is a redundant concept, that is, a mere word that is conventional to use in certain contexts of discourse but not a word that points to anything in reality. The theory is commonly attributed to Frank P. Ramsey. Ramsey held that the use of words like fact and truth was nothing but a roundabout way of asserting a proposition, and that treating these words as separate problems in isolation from judgment was merely a "linguistic muddle".

A variant of redundancy theory is the disquotational theory which uses a modified form of Tarski's schema: To say that '"P" is true' is to say that P. Yet another version of deflationism is the prosentential theory of truth, first developed by Dorothy Grover, Joseph Camp, and Nuel Belnap as an elaboration of Ramsey's claims. They argue that sentences like "That's true", when said in response to "It's raining", are prosentences (see pro-form), expressions that merely repeat the content of other expressions. In the same way that it means the same as my dog in the sentence My dog was hungry, so I fed it, That's true is supposed to mean the same as It's raining &mdash; if you say the latter and I then say the former. These variations do not necessarily follow Ramsey in asserting that truth is not a property, but rather can be understood to say that, for instance, the assertion "P" may well involve a substantial truth, and the theorists in this case are minimalizing only the redundancy or prosentence involved in the statement such as "that's true."

Deflationary principles do not apply to representations that are not analogous to sentences, and also do not apply to many other things that are commonly judged to be true or otherwise. Consider the analogy between the sentence "Snow is white" and the person Snow White, both of which can be true in a sense. To a minimalist, saying "Snow is white" is true is the same as saying "Snow is white", but to say "Snow White is true" is not the same as saying "Snow White."

Semantic theory of truth
The semantic theory of truth has as its general case for a given language:
 * 'P' is true if and only if P

where 'P' is a reference to the sentence (the sentence's name), and P is just the sentence itself.

Logician and philosopher Alfred Tarski developed the theory for formal languages (such as formal logic). Here he restricted it in this way: no language could contain its own truth predicate, that is, the expression is true could only apply to sentences in some other language. The latter he called an object language, the language being talked about. (It may, in turn, have a truth predicate that can be applied to sentences in still another language.) The reason for his restriction was that languages that contain their own truth predicate will contain paradoxical sentences like the Liar: This sentence is not true. See The Liar paradox. As a result Tarski held that the semantic theory could not be applied to any natural language, such as English, because they contain their own truth predicates. Donald Davidson used it as the foundation of his truth-conditional semantics and linked it to radical interpretation in a form of coherentism.

Bertrand Russell is credited with noticing the existence of such paradoxes even in the best symbolic formalizations of mathematics in his day, in particular the paradox that came to be named after him, Russell's paradox. Russell and Whitehead attempted to solve these problems in Principia Mathematica by putting statements into a hierarchy of types, wherein a statement cannot refer to itself, but only to statements lower in the hierarchy. This in turn led to new orders of difficulty regarding the precise natures of types and the structures of conceptually possible type systems that have yet to be resolved to this day.

Kripke's theory of truth
Saul Kripke contends that a natural language can in fact contain its own truth predicate without giving rise to contradiction. He showed how to construct one as follows:


 * Begin with a subset of sentences of a natural language that contains no occurrences of the expression "is true" (or "is false"). So The barn is big is included in the subset, but not ' The barn is big is true', nor problematic sentences such as "This sentence is false".


 * Define truth just for the sentences in that subset.


 * Then extend the definition of truth to include sentences that predicate truth or falsity of one of the original subset of sentences. So ' The barn is big is true' is now included, but not either This sentence is false nor "' The barn is big is true' is true".


 * Next, define truth for all sentences that predicate truth or falsity of a member of the second set. Imagine this process repeated infinitely, so that truth is defined for The barn is big; then for ' The barn is big is true'; then for "' The barn is big is true' is true", and so on.

Notice that truth never gets defined for sentences like This sentence is false, since it was not in the original subset and does not predicate truth of any sentence in the original or any subsequent set. In Kripke's terms, these are "ungrounded." Since these sentences are never assigned either truth or falsehood even if the process is carried out infinitely, Kripke's theory implies that some sentences are neither true nor false. This contradicts the Principle of bivalence: every sentence must be either true or false. Since this principle is a key premise in deriving the Liar paradox, the paradox is dissolved.

Types of truth
Metaphysical subjectivism holds that the truth or falsity of all propositions depends, at least partly, on what we believe. In contrast, metaphysical objectivism holds that truths are independent of our subjective beliefs. Except for propositions that are actually about our beliefs or sensations, what is true or false is independent of what we think is true or false.

Relative truths are statements or propositions that are true only relative to some standard, convention, or point-of-view, such as that of one's own culture. Many would agree that the truth or falsity of some statements are relative: That the fork is to the left of the spoon depends on where one stands. Relativism is the doctrine that all truths within a particular domain (say, morality or aesthetics) are of this form, and entails that what is true varies across cultures and eras. For example, moral relativism is the view that moral expectations are socially determined.

Relative truths can be contrasted with absolute truths. The latter are statements or propositions that are taken to be true for all cultures and all eras. For example, for the microeconomist, that the laws of supply and demand determine the value of any consumable in a market economy is true in all situations; for the Kantian, "act only according to that maxim by which you can at the same time will that it should become a universal law" forms an absolute moral truth. They are statements that are often claimed to emanate from the very nature of the universe, God, human nature, or some other ultimate essence or transcendental signifier.

The concept of absolute truth, as understood in philosophy, should not be confused with the concept of absolute truth as it is used in religious traditions.

Absolutism in a particular domain of thought is the view that all statements in that domain are either absolutely true or absolutely false: none is true for some cultures or eras while false for other cultures or eras.

Truth in law
Truth, as a concept, is a central issue in law, though the actual use of the word "truth" in legal practice tends to be limited to certain very specific contexts. Witnesses who swear under oath to testify truthfully in courts of law are not expected to make infallibly true statements, but to make a good faith attempt to accurately recount prior events from memory, or provide expert testimony. Triers-of-fact are then charged with the responsibility to determine the credibility or veracity of a witness's testimony. Differing accounts from separate witnesses testifying in good-faith are commonplace. Errors are not attributed to perjury unless a very stringent standard of proof is met when charging a witness with lying under oath.

Beyond the standard oath taken by witnesses, the word "truth" is seldom officially used. Courts, judges and juries are never referred to, for instance, as "finders of truth." In addition to procedural and substantive legal rulings, courts designate a finder of fact (jury or judge) to decide what the facts are in a case given often conflicting sets of information. In a very specific and limited context, "truth" is also a legal term of art referring to a standard affirmative defense to a charge of defamation, this use of the term in this specific manner deriving from common law dating back at least seven centuries. William J. Shields summarizes the weight which the law gives to the word "truth" by noting that Black’s Law Dictionary devotes just a few lines to it: "1. A fully accurate account of events, factuality. 2. Defamation. An affirmative defense by which the defendant asserts that the alleged defamatory statement is substantially accurate." Shields goes on to say: "By contrast, the definition of “fact” covers about two full pages, with three principal meanings and 42 definitions of specific types of fact. “Evidence” covers five full pages with four principal meanings and 93 definitions of specific types. Even the definition of the crime of perjury avoids the use of “truth”: “the act or an instance of a person’s deliberately making material false or misleading statements while under oath.”  Perusal of other legal references (treatises, dictionaries, periodicals, etc.) will yield the same results.  Where “truth” or “true” is defined, it will be to the effect of “that which is a fact” or “that which is verifiable,” leading back to the legal concepts of evidence and proof."

Truth in mathematics
There are two main truth theories in mathematics. They are the model theory of truth and the proof theory of truth.

Historically, with the 19th century development of Boolean algebra mathematical models of logic began to treat "truth", also represented as "T" or "1", as an arbitrary constant. "Falsity" is also an arbitrary constant, which can be represented as "F" or "0". In propositional logic, these symbols can be manipulated according to a set of axioms and rules of inference, often given in the form of truth tables.

In addition, from about the time of Hilbert's program in the late 19th century to the proof of Gödel's theorem and the development of the Church-Turing thesis in the early 20th century, true statements in mathematics were generally assumed to be those statements which are provable in a formal axiomatic system.

The works of Kurt Gödel and Alan Turing shook this assumption, with the discovery of statements that are true but cannot be proven, and of statements whose truth or falsity is undecidable. Two examples of the latter can be found in Hilbert's problems. Work on Hilbert's 10th problem led in the late 20th century to the construction of specific Diophantine equations for which it is undecidable whether they have a solution, or even if they do, whether they have a finite or infinite number of solutions. More fundamentally, Hilbert's first problem was on the continuum hypothesis. Gödel and Paul Cohen showed that this hypothesis cannot be proved or disproved using the standard axioms of set theory. In the view of some, then, it is equally reasonable to take either the continuum hypothesis or its negation as a new axiom.

Truth in religion
Most religious traditions hold a body of truths that are part of the particular tradition. Such truths may be considered to be spiritually revealed, or may be developed through tradition, or may be a combination of both. Whatever these truths are for the particular religious tradition, they can be called part of the doctrine of that tradition.

Christianity
Assertions of truth based upon revelation and testimony set forward in the Bible is central to Christian beliefs. Some denominations have asserted additional authorities as sources of doctrinal truth &mdash; for instance, in Roman Catholicism the Pope is asserted to be infallible on matters of church doctrine.

Biblical inerrency
Some Christian traditions hold a doctrine called Biblical inerrancy, which asserts that the Bible is without error, that is, it can be said to be true as to all issues contained within, whether Old Testament or New. Various interpretations have been applied, depending on the tradition. According to some interpretations of the doctrine, all of the Bible is without error, i.e., is to be taken as true, no matter what the issue. Other interpretations hold that the Bible is always true on important matters of faith, while yet other interpretations hold that the Bible is true but must be specifically interpreted in the context of the language, culture and time that relevant passages were written.

"Double truth" theories
In thirteenth century Europe, the Roman Catholic Church denounced what it described as theories of "double truth," i.e. theories to the effect that although a truth may be established by reason, its contrary ought to be believed as true as a matter of faith. The condemnation was aimed specifically at a "Latin Averroist" (see Averroës), Siger of Brabant, but it was more broadly an attempt to halt the spread of Aristotle's ideas, which the reconquest of Spain and, accordingly, access to the libraries of the Moors had re-introduced into the Latin literate world. At the time, much of the doctrine of the Roman Catholic Church was based upon neoplatonic ideas, and Aristoteleanism struck many as heresy. Siger and others seem to have conceded this, and to have used the sharp reason/faith distinction that came to be known as "double truth" as a way of legitimizing discussion of Aristotle despite that concession.

Eastern religious perspectives
The two truths doctrine of Buddhism distinguishes between statements which reflect ultimate reality and relatively true statements. This doctrine attempts to reconcile relative and absolute truth(s), placing the concepts in a perspective in which "common sense" truths and empirical truths are "relative," in contrast to religious doctrine and perspective. The Jain doctrine of anekantavada (Sanskrit for "non-onesidedness") dictates that statements be considered from many points of view.The contemporary writer Ken Wilber has used these ideas to develop a four-fold conception of truth.

Additional observations about truth
Honest intentions play a unique role in the ethics of epistemology. Jurgen Habermas understands truthfulness to be one of the dimensions of valid speech. The moral importance of honest intent is underscored by the remarks of Buddha: “Herein someone avoids false speech and abstains from it. He speaks the truth, is devoted to truth, reliable, worthy of confidence, not a deceiver of people. Being at a meeting, or amongst people, or in the midst of his relatives, or in a society, or in the king's court, and called upon and asked as witness to tell what he knows, he answers, if he knows nothing: "I know nothing," and if he knows, he answers: "I know"; if he has seen nothing, he answers: "I have seen nothing," and if he has seen, he answers: "I have seen." Thus he never knowingly speaks a lie, either for the sake of his own advantage, or for the sake of another person's advantage, or for the sake of any advantage whatsoever.” In its most extreme form, the obligation to tell the truth may manifest itself as a strong form of evidentialism, which holds that "It is wrong always, everywhere, and for anyone, to believe anything on insufficient evidence".

Truth in logic

 * Logic
 * Logical value


 * Modal logic
 * Truth conditions


 * Truth function
 * Truth table

Theories of truth

 * Coherentism
 * Coherence theory of truth
 * Consensus theory of truth
 * Correspondence theory of truth
 * Deflationary theory of truth


 * Epistemic theories of truth
 * Indefinability theory of truth
 * Pragmatic theory of truth
 * Redundancy theory of truth
 * Semantic theory of truth

Major theorists

 * Aristotle
 * Thomas Aquinas
 * J.L. Austin
 * Brand Blanshard
 * John Dewey
 * Hartry Field
 * Jürgen Habermas


 * Paul Horwich
 * William James
 * Harold Joachim
 * Saul Kripke
 * Charles Sanders Peirce
 * Plato
 * Karl Popper


 * W.V. Quine
 * Frank P. Ramsey
 * Bertrand Russell
 * Socrates
 * P.F. Strawson
 * Alfred Tarski
 * Ludwig Wittgenstein