Correspondence Theory of Truth

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The term "correspondence theory of truth" has circulated among modern philosophical writers largely through the influence of Bertrand Russell, who sets the view (which he himself adopts) that "truth consists in some form of correspondence between belief and fact" against the theory of the absolute idealists that "truth consists in coherence," that is, that the more our beliefs hang together in a system, the truer they are.

Ancient and Scholastic Versions of the Theory

The origins of the word correspondence, used to denote the relation between thought and reality in which the truth of thought consists, appear to be medieval. Thomas Aquinas used correspondentia in this way at least once, but much more often he used other expressions and preferred most of all the definition of truth that he attributed to the ninth-century Jewish Neoplatonist Isaac Israeli: Veritas est adaequatio rei et intellectus (truth is the adequation of things and the intellect). At one point he expanded this to adaequatio intellectus et rei, secundum quod intellectus dicit esse, quod est, vel non esse, quod non est. This is an echo of Aristotle's "To say of what is that it is not, or of what is not that it is, is false; while to say of what is that it is, and of what is not that it is not, is true." Other Scholastics sometimes said that a proposition is true when and only when ita est sicut significat ("the thing is as signified"); this too is in line with the Aristotelian account, in which "is" is not restricted to the meaning "exists"the definition also covers the point that to say of what is so that it is not so, or of what is not so that it is so, is false; while to say of what is so that it is so, and of what is not so that it is not so, is true. This simple statement is the nerve of the correspondence theory; we shall continually return to it.


Aristotle did not originate the correspondence theory but took it over from Plato's Sophist. There it was developed with an eye on a rejected alternativenot the coherence theory, which is a comparatively late invention (G. E. Moore is probably correct, in his "Truth" article in Baldwin's Dictionary, in tracing its vogue to Immanuel Kant), but one that we may call the existence theory, which also crops up in the Theaetetus. In this latter dialogue Socrates tries to find what differentiates true from erroneous belief, and the first suggestion he considers is that whereas true belief is directed toward what is, false belief is directed toward what is not. This view is rejected on the ground that just as to see or hear what is not is to see or hear nothing, and to see or hear nothing is just not to see or hear at all, so to "think what is not" is to think nothing, and that is just not to think at all, so that erroneous thought, on this view, would just not be thinking at all.

The same theory is considered in the Sophist, but here an alternative is put forward. Thought is compared with speech (it is the soul's dialogue with itself), and the important thing about speech is that in order to be true or false it must be complexonly complete statements are true or false, and these must consist of both nouns and verbs. (These points are also stressed by Aristotle.) As simple examples of complete statements, Plato gives "Theaetetus is-sitting-down" and "Theaetetus is-flying." The first of these is true because Theaetetus is sitting down, and the second is false because he is not flying. This escapes the difficulties of the existence theory because it abandons the suggestion that thinking is a simple direction of the mind toward an objectif it were that, its verbal expression would not have to be a complete sentence but could be just a nameand so opens up the possibility for thinking to be erroneous even though what is thought about, such as Theaetetus, is perfectly real.

The existence theory, however, dies hard and has continued to maintain itself, not merely as a rival to the correspondence theory but even more as something that theory is in constant danger of becoming. (The two views continually oscillate, for example, in the early work of Russell and Moore.) It is easy to equate the complexity of thinking with its having a complex objectfor instance, Theaetetus's-sitting-down or Theaetetus's-flyingwhich exists if the thought is true and does not if it is not.


There is no trace of the above slide or degeneration in Aristotle, nor even of a conscious resistance to it, but he has passages that have some bearing on it and that in any case develop a little further the correspondence theory itself. For example, having said that the distinguishing mark of a substance or individual thing is that it may have opposite qualities at different times, he resists a suggestion that statements and opinions would count as things by this criterion, since they may be at one time true and at another time falsefor example, the statement or opinion that a person is sitting down will be true while he is doing so but false when he stands up. This, Aristotle suggests, is unfair, because what is in question here is not any genuine alteration in the statement or opinion itself, but rather in the facts outside it by which its truth or falsehood is measured. "For it is by the facts of the case, by their being or not being so, that a statement is called true or false."

Sometimes Aristotle represents the verification of statements by facts as a kind of causation. Causation, he says, differs from implication because even where implication is reciprocal we can distinguish the cause from the effect:

The existence of a man, for instance, implies the truth of the statement in which we assert his existence. The converse is also the case. For if he exists, then the statement in which we assert his existence is true, and conversely, if the statement in which we assert his existence is true, he exists. But the truth of the statement is in no way the cause of his existence, though his existence is in a way the cause of the truth of the statement. For we call the statement true or false according as he exists or not.
(categories 14b1520)

(What Aristotle calls a cause here is perhaps something more like a criterion.)

megarian "liar" paradox

The PlatonicAristotelian correspondence theory was not long formulated when a distressing consequence, or apparent consequence, of it was pointed out by Eubulides, a member of the school of Megara, which seems to have conducted constant warfare against various basic PlatonicAristotelian positions. Eubulides invited his hearers to consider a man who says "I am lying" or "What I am now saying is false." According to the PlatonicAristotelian view, this is true if what the man is saying is falseit is true if it is itself falseand false if what he is saying is not falsefalse if it is true. Therefore, in at least this one case, that view leads to the position that whatever we say about the truth or falsehood of an utterance entails its own opposite. We may note, too, that in this instance the Aristotelian one-sided dependence of the truth or falsehood of a proposition on the related matter of fact does not hold, since the related matter of fact in this instance is precisely the truth or falsehood of the proposition. This "paradox of the liar" was much discussed by both ancient and medieval writers and still presents a serious problem to anyone attempting to give a satisfactory general account of truth and falsehood.


What is substantially the PlatonicAristotelian account of truth is also found among the Stoics, but with modifications. The Stoics held that truth in the primary sense is a property of statements or axiomata, not in the sense of sentences but in the sense of what the sentences state or mean. These axiomata exist independently of their being expressed by sentences, and the "meanings" of false sentences exist just as much as the meanings of true onesthat is to say, axiomata include objective falsehoods as well as objective truths. (This is not, therefore, the existence theory.) Describing the Stoics' account of truth from this point on, Diogenes Laertius says that the axioma expressed by "It is day" is true if it is day and false if it is not. This is an example rather than a general theory; Sextus Empiricus says that the kind of axioma called simple and definitethe kind that would be expressed by a sentence of the form "This X Y 's" (for instance, "This bat flies")is true when the predicate belongs to the object denoted by the demonstrative. This, however, only defines "true" for the simplest type of proposition. For other types we know that the Stoics laid down such rules as that an axioma of the form "Some X Y 's" is true if and only if there is some true axioma of the form "This X Y 's," and one of the form "p and q " is true if and only if both of its components are; but we do not know whether they regarded such rules as actually defining "true" for these forms. It is scarcely likely that they saw them as parts of a single "recursive" definition of truth, such as is found in Alfred Tarski, but they laid the foundations for such a development.

medieval logicians

Such statements of truth conditions, as we now call them, were also laid down and discussed by the logicians of the later Middle Ages, although they generally treated truth as a property not of abstract axiomata but of spoken and written sentences. Besides the truth conditions of sentences containing "not," "or," and "some," they considered those of sentences containing such expressions as "possibly" and verbs in the past and future tenses. They observed, for example, that while in general a past-tense statement is true if and only if the corresponding present-tense statement was true, and a statement that something could have been so is true if and only if the statement that it is so could have been true, there are exceptions to such rules. For example, "Something white was black" is true, but "Something white is black" was never true. The rule here is that a past-tense predication is true if the corresponding predication was true of the individuals to which the subject term now applies; for instance, "Something white was black" is true if "It is black" could have been truly said in the past of a thing that is now white. "It could have been that no proposition is negative" is true, since God might have annihilated all negative propositions; but "No proposition is negative" could in no circumstances have been true, since the mere existence of this sentence (which is itself negative) falsifies it. The rule is rather that a sentence de possibili is true if and only if things could have been as the corresponding unqualified sentence says they are.

The later medieval logicians also implicitly modified the PlatonicAristotelian theory in order to cope with the "liar" and similar paradoxes. John Buridan, for example, although he preserved the formula that a sentence is true when ita est sicut significat, gave a somewhat un-Aristotelian twist to the meaning of significat. According to Buridan, the man who says "I am saying something false," and says nothing else, really is saying something false, not because things are otherwise than as his sentence significat formaliter but because they are otherwise than as his sentence significat virtualiter. A sentence "virtually" signifies whatever follows from itself together with the circumstances of its utterance, and what follows from this particular sentence together with the circumstances of its utterance is that it is both true and false; since this is never the case, things are not as it "virtually" says they are, and it is false.

Moore's Correspondence Theory

In the twentieth century a particularly extended and fruitful discussion of the correspondence theory is found in a series of lectures given by G. E. Moore in 19101911. Here truth and falsehood first appear as properties of what are called propositions. Moore uses the term proposition to mean not an indicative sentence but what such a sentence means, an axioma in the Stoic sense. When we both hear and understand a spoken sentence, and both see and understand a written one, there is something apprehended by us over and above the sentence, and while this apprehension or understanding is the same kind of act in all cases, what is apprehended is in general different when different sentences (such as "Twice two are four" and "Twice four are eight") are involved and therefore is distinguishable from the act of apprehending. We also "constantly think of and believe or disbelieve, or merely consider, propositions, at moments when we are neither hearing nor seeing any words which express them"; for example, when we "apprehend a proposition, which we desire to express, before we are able to think of any sentence which would express it." In this lecture Moore is quite confident that "there certainly are in the Universe such things as propositions," and that it is propositions rather than sentences or acts of belief that are true or false in the primary sense. We often say that beliefs are true or false, but this is only because the word belief is often used not for an act of believing but for what is believed; for instance, if we say that two different people have the same belief, we mean to identify what they believe rather than their respective acts of believing, and what is believed is simply a proposition in Moore's sense. Acts of believing and sentences could, however, be said to be true or false in a secondary sense, when what is believed or expressed is a true or false proposition.

moore's later position

Moore returns to the subjects of true and false beliefs and the nature of propositions in later lectures in the series, but now he seems to move somewhat away from the position outlined above. He leads up to them with a problem he states as follows: "Suppose a man believes that God exists; then to say that his belief is true seems to be exactly equivalent to saying that it is a fact that God exists or that God's existence is a fact" (Some Main Problems in Philosophy, p. 250). Quite generally it seems that "the difference between true and false beliefs is that where a belief is true, there what is believed is a fact; whereas where a belief is false, there what is believed is not a fact" (ibid.). Even where a belief is false, however, there does seem to be something that is believed.

A man believes in God's existence and it seems quite plain that he is believing in somethingthat there is such a thing as what he believes in, and that this something is God's existence. It seems quite plain, therefore, that there is such a thing as God's existence, whether his belief is true or false. But we have just seen that if his belief is false, then God's existence is not a fact. And what is the difference between saying that there is such a thing as God's existence and (saying) that God's existence is a fact? (ibid.)

This is the problem of the Theaetetus all over againif a false belief has no real object, how can it be a belief at all?

denial that propositions exist

Moore raises the above question with regard to a more certainly false proposition, namely, that the hearers of his lecture were at that time hearing the noise of a brass band; and he then restates, but no longer with conviction, his earlier theory. We could say that there was indeed such a thing as their hearing a brass band then but that this was a proposition, not a fact. But, Moore argues, this theory admits in the case of the phrase "the fact that they are hearing a brass band" that what looks like the name of a real object of a possible belief is not one, so why should we not say this also of the phrase "the proposition that they are hearing a brass band"? Moore is thus led to the view that the subjectverbobject form of assertions about beliefs is misleading. His new theory, he says, "may be expressed by saying that there simply are no such things as propositions. That belief does not consist in a relation between the believer, on the one hand, and another thing which may be called the proposition believed" (ibid., p. 265). He cannot give any satisfactory alternative analysis of belief statements to supplant the one he has abandoned, but he thinks he can give an account of the truth and falsehood of beliefs without one.

false belief and facts

In developing the account of truth and falsehood of beliefs, Moore considers the case of a friend believing that he has gone away for his holidays, and begins in a thoroughly Aristotelian vein. "If this belief of his is true then I must have gone away and, conversely, if I have gone away, then this belief of his certainly is true" (ibid., p. 274). And similarly, "if this belief is false, then I can't have gone away and conversely, if I have not gone away, then the belief that I have gone away certainly must be false" (ibid., p. 275). However, this statement of necessary and sufficient conditions does not constitute a definition of truth and falsehood, for "when we assert: 'The belief that I have gone away is true,' we mean to assert that this belief has some property, which it shares with other true beliefs," whereas "in merely asserting 'I have gone away,' we are not attributing any property at all to this belief" (ibid., p. 276). For "Plainly I might have gone away without my friend believing that I had; and if so, his belief would not be true, simply because it would not exist." This objection, however, suggests that Moore's having gone away would not after all be a sufficient condition, but only a necessary one, of his friend's belief being true; and it could be met by defining the truth of his friend's belief, not simply as Moore's having in fact gone away but as this together with his friend's believing it.

The problem remains, however, of generalizing this to cover all cases, which Moore goes about solving as follows: "We can see quite plainly," he says, "that this belief, if true, has to the fact that I have gone away a certain relation which that belief has to no other fact," a relation which cannot be defined in the sense of being analyzed, but with which we are all perfectly familiar and which "is expressed by the circumstance that the name of the belief is 'the belief that I have gone away,' while the name of the fact is 'that I have gone away'" (ibid.). Moore proposes to call this relation correspondence, and "To say that this belief is true is to say that there is in the Universe a fact to which it corresponds, and to say that it is false is to say that there is not in the Universe any fact to which it corresponds" (ibid., p. 277).

facts rather than propositions

It is essential to Moore's final account that although there are no propositions, there are facts. A belief, even if true, does not consist in a relation between a person and a fact, but the truth of a belief does. He is also at pains to insist that facts "are" or exist in the very sense in which, say, chairs and tables do. He concedes that as a matter of usage we find it natural to say "It is a fact that bears exist," while we do not find it at all natural to say "That bears exist, is" (or "That bears exist, exists", or even "The existence of bears exists"), but he thinks this simply reflects our acute sense of the difference in kind between facts and other thingsthey are real objects but objects of a very special sort. We also express their character by calling them truths, or by prefixing "It is true that" to them as an alternative to "It is a fact that." This property of being a truth or fact is to be carefully distinguished from the "truth" which is possessed by some beliefs and which consists, as previously explained, in correspondence to a truth or fact.

Russell's Correspondence Theory

In Moore's account of truth and falsehood, it will be seen, there are two elements that are a little mysterious and that he is reluctantly compelled to leave in that conditionthe correct analysis of belief statements and the nature of the correspondence that entitles us to use the same form of words in describing the content of a belief and in asserting the fact to which, if true, it corresponds.

Shortly before Moore gave these lectures, Russell had made an attempt to elucidate just these points. In the concluding section of a paper he gave before the Aristotelian Society in 1906, there is a hint of this explanation, which is more fully developed in various writings of the period 19101912. He suggests in the 1906 paper that a belief may differ from an idea or presentation in consisting of several interrelated ideas, whose objects will be united in the real world into a single complex or fact if the belief is true, but not otherwise, so that a false belief is indeed "belief in nothing, though it is not 'thinking of nothing,' because it is thinking of the objects of the ideas which constitute the belief." In the later versions this is expanded to the view that a belief consists in a many-termed relation, the number of terms always being two more than that occurring in the fact to which, if true, the belief corresponds. For example, if it is a fact that Desdemona loves Cassio, then in this fact the two terms Desdemona and Cassio are "knit together" by the relation of loving, while if it is a fact that Othello believes that Desdemona loves Cassio, then the four terms Othello, Desdemona, the relation of loving, and Cassio are "knit together" in this fact by the relation of believing. The correspondence between the belief and the fact, when the fact exists and the belief is therefore true, consists in a certain characteristic semiparallelism between the ordering of the last terms of the belief relation and the ordering of the terms by their ordering relation in the fact. Knowing and perceiving, on the other hand, really are relations between the knower or perceiver and the fact known or perceived (which of course must be a fact for knowledge or perception to occur).


The above theory is open to a number of objections, some of which have been particularly well stated by P. T. Geach, and one of which, due in essence to Ludwig Wittgenstein, had already led Russell to abandon the theory, in a course of lectures on logical atomism delivered in 1918.

Belief and what is the case

Russell's 19061912 theoryand indeed even Moore's more vague theory, of which it is a possible filling outmakes it altogether too mysterious that the very same words should be used to express what is believed and what is actually the case if the belief is true. (At most, there is in some languages a slight but regular formal alteration when the latter is put into oratio obliqua to give the former.) As Wittgenstein puts it (Philosophical Investigations, Para. 444), "One may have the feeling that in the sentence 'I expect he is coming' one is using the words 'he is coming' in a different sense from the one they have in the assertion 'He is coming.' But if it were so how could I say that my expectation had been fulfilled?"that the very thing I expected had come to pass?

This severance of the senses of the oratio obliqua and oratio recta forms of the same sentence is exacerbated in Russell's account, as Geach points out, by its consequence that believing is not one relation but several, since the number of terms it requires differs with the number of terms required by the relation that occurs among its objects (for instance, while Othello's believing that Desdemona loves Cassio is a 4-termed relation, his believing that Desdemona gave Cassio a certain ring would be a 5-termed one). This difference arises even when we are only considering beliefs of which the apparent objects are simple relational propositions; still more radical differences would have to be admitted with believings apparently directed toward compound and general propositions. This point was, indeed, stressed by Russell himself from the outset and seems never to have been regarded by him as a serious objection to the theory, since in his 1918 lectures, even when he had abandoned the view that necessitated it, we find him saying that "belief will really have to have different logical forms according to the nature of what is believed" (Logic and Knowledge, p. 226), so that "the apparent sameness of believing in different cases is more or less illusory."

There is here, it seems, a remnant of the ramified theory of types that Russell at first thought necessary to deal with such paradoxes as that of the "liar." According to this theory, propositions are not only of different logical forms but also of different logical types, and "truth" and "falsehood" must be differently defined for each type; indeed, even such ordinary logical functions as negation and conjunction must be understood differently according to the types of propositions to which they are attached. Even by the time he was exercising the influence acknowledged in Russell's 1918 lectures, Wittgenstein had definitely abandoned this theory: "Any proposition can be negated. And this shews that 'true' and 'false' mean the same for all propositions (in contrast to Russell)" (Notebooks 19141916, p. 21).

Verbs in judgments

What Russell did successfully assimilate from Wittgenstein at this period was that in such judgments as that Othello believes that Desdemona loves Cassio, "both verbs have got to occur as verbs, because if a thing is a verb it cannot occur otherwise than as a verb." He also says:

There are really two main things that one wants to notice in this matter that I am treating of just now. The first is the impossibility of treating the proposition believed as an independent entity, entering as a unit into the occurrences of the belief, and the other is the impossibility of putting the subordinate verb on a level with its terms as an object term in the belief. That is a point in which I think that the theory of judgment which I set forth once was a little unduly simple, because I did then treat the object verb as if one could put it as just an object like the terms. (Logic and Knowledge, p. 226)

"Every right theory of judgment," as Wittgenstein puts it, "must make it impossible for me to judge that 'this table penholders the book' (Russell's theory does not satisfy this requirement)" (Notebooks 19141916, p. 96).

Propositions in judgments

Russell's objection ties up in two ways with Wittgenstein's that "a proposition itself must occur in the statement to the effect that it is judged." In the first place, it is by inserting the "proposition itself" into the "statement to the effect that it is judged" that we enable the subordinate verb to occur as a verb and not disguised as an abstract noun. (It looks, in fact, as if these "two main things" that Russell says we must notice cannot be observed together.) We might put the two objections together thus: Because the use of abstract nouns is always something to be explained, it is more illuminating to say that "Othello ascribes unfaithfulness to Desdemona" (where "ascribes" is apparently a 3-termed relation with "unfaithfulness" as one of its terms) means exactly what is meant by "Othello believes that Desdemona is unfaithful" than it is to say that the second means exactly what is meant by the first.

Facts as objects

The second way in which the two objections come together is more complicated, and it can be gathered from an extended discussion of what may at first seem another point: that Russell's 19061912 theory, like Moore's of 1910, still takes "facts" seriously as a special sort of object. On this point Russell's 1918 view is a little obscure. He seems not to have changed at all on this subject, and describes it as one of those truisms that "are so obvious that it is almost laughable to mention them," that "the world contains facts, and that there are also beliefs, which have reference to facts, and by reference to facts are either true or false" (Logic and Knowledge, p. 182). He sharply contrasts facts with propositions in this respect. "If we were making an inventory of the world, propositions would not come in. Facts would, beliefs, wishes, wills would, but propositions would not" (ibid., p. 214). This last remark occurs in a criticism of an attempt by Raphael Demos to eliminate the negative fact that a certain piece of chalk is not red from the "inventory of the world" by equating it with the fact that the chalk has some other positive but incompatible color. "Even if incompatibility is to be taken as a sort of fundamental expression of fact," Russell says to this, "incompatibility is not between facts but between propositions. It is clear that no two facts are incompatible" (ibid.). And since propositions do not have being independently, this "incompatibility of propositions taken as an ultimate fact of the real world will want a good deal of treatment, a lot of dressing up before it will do." However, Russell's own alternative, that there are irreducibly negative factsfor instance, the fact that it is not the case that this piece of chalk is redequally involves the consequence that there are facts that contain real falsehoods as constituents. This Russell himself pointed out in his 1906 paper, and it led him then to be more hesitant than he was later about dismissing the notion of objective falsehoods. Even if, he says in this paper, we can remove the suggestion that false beliefs have objective falsehoods for their objects:

There is another argument in favour of objective falsehood, derived from the case of true propositions which contain false ones as constituent parts. Take, e.g., "Either the earth goes round the sun, or it does not." This is certainly true, and therefore, on the theory we are considering, it represents a fact, i.e. an objective complex. But it is, at least apparently, compounded of two (unasserted) constituents, of which one must be false. Thus our fact seems to be composed of two parts, of which one is a fact, while the other is an objective falsehood. ("On the Nature of Truth," pp. 4748)

The real moral of all this is surely that if propositions must go, facts must go, too; but Russell seems to shrink from this step.

Elsewhere in the 1918 lectures, however, he says that facts, although apparently real in a way in which propositions are not, have the extraordinary property that they cannot be named. In the first place, they are not named by propositions (sentences). For this he has a rather strange argument, taken from Wittgenstein. Whereas Moore thought of a false belief as one that corresponds to no fact at all, Wittgenstein held that a false statement does correspond to a fact, but in the wrong way. Hence, to quote Russell's exposition of the theory:

There are two propositions corresponding to each fact. Suppose it is a fact that Socrates is dead. You have two propositions: "Socrates is dead" and "Socrates is not dead." And those two propositions corresponding to the same fact, there is one fact in the world, that which makes one true and one false. There are two different relations that a proposition can have to a fact: the one the relation that you may call being true to the fact, and the other being false to the fact. (Logic and Knowledge, p. 187)

This means that a proposition does not name a fact, since in the case of a name, there is only one relation that it can have to what it names. Further,

You must not run away with the idea that you can name facts in any other way; you cannot. You cannot properly name a fact. The only thing you can do is to assert it, or deny it, or desire it, or will it, or wish it, or question it. You can never put the sort of thing that makes a proposition to be true or false in the position of a logical subject. (ibid., p. 188)

Ramsey and the Later Wittgenstein

Russell's whole position, as it stands, is difficult to maintain. If there are really individual objects to which the common noun "fact" applies, and we can sometimes actually perceive them (Russell continued to hold this in the 1918 lectures), then if at the time of our perceiving one our language has no name for it, why can we not invent one and christen the thing on the spot? However, there is not just superstition, but something true and important, behind the statement of Russell and Wittgenstein that facts cannot be named, and they both identify it in the end. "When I say 'facts cannot be named,'" Russell admitted in 1924, "this is, strictly speaking, nonsense. What can be said without falling into nonsense is: 'The symbol for a fact is not a name.'" Or better, perhaps: to state a fact is not to name an object. Whatever may be the case with "that" clauses, sentences aren't names of anything; just as, whatever may be the case with abstract nouns, verbs are not names of anythingthey are not names at all, but have other functions; naming is one thing, saying or stating another. Even Plato saw that this distinction was important.

But can we not name what a sentence says, for instance, by the corresponding "that" clause? Not really"what a sentence says," although a good sense can be given to it, is a misleading expression; when it means anything, it means "how a sentence says things are" or, better, "how we say things are" when we use the sentence in question. To name what we are saying is to say what we are saying, and to name what we are thinking or wishing is similarly to say what we are thinking or wishing. "I think that bears exist" is, therefore, not to be analyzed as "I think (that bears exist)," which suggests that "that bears exist" is one term of the relation expressed by "think" but rather as "I think that (bears exist)," where "bears exist" does not even look like a name (it looks like, and is, a sentence) and "think that" does not look like the expression of a relation. If Othello thinks that Desdemona loves Cassio, there is indeed a 3-termed relation between Othello, Desdemona, and Cassio (not, as Russell thought, a 4-termed one between Othello, Desdemona, Cassio, and loving), but this relation consists in his thinking that she loves him, that is, the relation is expressed by the whole complex verb "thinks thatloves," not by the simple "thinks that," which does not express any relation at all, since its second gap is not filled by the name of an object but by a sentence, which does not name but says what he thinks (how he thinks things are). The plain "thinks," without the "that," means nothing at all. I may, indeed, use forms of expression like "I think something that Jones doesn't think" or "Something that Jones thinks is not true," but the "thing" in this "something" is no more to be taken seriously than the "what" in "what I say"these sentences, respectively, mean simply "For some p, I think that p and Jones does not think that p " and "For some p, Jones thinks that p but it is not the case that p." The correspondence theory can now assume the simple form: "X says (believes) truly that p " means "X says (believes) that p, and p "; and "X says (believes) falsely that p " means "X says (believes) that p, and not p."


The above position was very lightly sketched in 1927 by Frank Plumpton Ramsey, who says in effect that the words fact and true in their primary use are inseparable parts of the adverbial phrases "truly," "in fact," "it is a fact that," and "it is true that"; and these, attached to some sentence, say no more than this sentence says on its own. "It is false that p " or "That p, is contrary to fact" similarly says no more than the simple "Not p." Thus there are not only no falsehoods but no facts or truths either, any more than there is an entity called "the case" involved in the synonymous phrase "It is the case that." This part of Ramsey's view has led some writers to set it in opposition to the correspondence theory as a "no truth" theory, but Ramsey also discusses more complex uses of "true" in which there is something more like a juxtaposition of what a man says and what is so. In particular he considers the statement "He is always right""Whatever he says is true"and renders this as "For all p, if he says that p, it is true that p," and this in turn as "For all p, if he says that p, then p." This may seem to require a further verb in its second clause, but there is already a "variable verb" implicit in the variable p.


We may expand Ramsey's discussion of the more complex uses of "true" by taking up a suggestion of the later Wittgenstein (which, indeed, we have already used a bit). In the Tractatus, Wittgenstein says that "the general form of propositions is: this is how things are." In the Investigations, criticizing this identification, he reminds us that "This is how things are" is itself a proposition, an English sentence applied in everyday language, as in "He explained his position to me, said that this was how things were, and that therefore he needed an advance." "This is how things are" can be said to stand for any statement and can be employed as a propositional schema, but only because it already has the construction of an English sentence. Wittgenstein continues, "It would also be possible here simply to use a letter, a variable, as in symbolic logic. But no one is going to call the letter 'p' the general form of propositions."

"This is how things are," although a genuine proposition, is nevertheless being employed only as a propositional variable. "To say that this proposition agrees or does not agree with reality would be obvious nonsense." "This is how things are" is a propositional variable in ordinary speech in much the same way that a pronoun is a name variable in ordinary speech. In Wittgenstein's example, the "value" of this "variable" is given by a specific sentence uttered earlier, much as the denotation of a pronoun may be fixed by a name occurring earlier. "I'm desperatethat's how things are" is like "There's Joneshe's wearing that hat again." "This (that) is how things are" is a pro-sentence. But we may also obtain a specific statement by "binding" this variable, as in "However he says things are, that's how they are," that is, Ramsey's "For all p, if he says that p, then p." We speak truly whenever things are as we say they are, and falsely when they are not. There was a hint of this way of putting things when the later Scholastics equated est vera, said of a sentence, with ita est sicut significat or qualitercumque significat, ita est"however the sentence signifies (that the case is), thus it is"avoiding the possibly misleading "What the sentence says is so."

These "misleading" forms, however, need not mislead us, once the whole picture has been spread out, and we can soften our earlier skepticism by agreeing that after all there are facts, and that there are falsehoods, if all that is meant by "There are facts" is "For some p, p " and by "There are falsehoods" "For some p, not p." We can say, too, that there are both facts and falsehoods that have never been either thought or asserted, that is, we can insist on the objective or mind-independent character of propositions, if by this we mean that for some p, both p and it has never been thought or said that p. (We cannot, of course, give examples of such facts or falsehoods, for to do so would be to state them, and then they would not be unstated; but this is no more strange than that there should be peopleas there certainly arewhose names we do not know, although we cannot in the nature of the case name any specific examples.) It is significant that Moore in his last years contrived to assimilate a Ramsey-like account of truth without losing any of his earlier sense of the mind-independent and speech-independent character of what is so. Propositions about propositions, he said in effect, are not propositions about sentences precisely because the words proposition, true, and false are eliminablejust because "The proposition that the sun is shining is true" is equivalent to and perhaps identical with the plain "The sun is shining," it neither says anything about sentences nor entails that there are such things, since the sun could obviously be shining even if no one ever said so.

Tarski's Semantic Theory

In the theories of Ramsey and the later Moore, truth is a quasi property of a quasi object. What is really defined in them is not a property of anything, but rather what it is to say with truth that something is so; it is an account of the adverbial phrase "with truth" rather than of the adjective "true." The late medieval treatment of "true" as a straightforward adjective applying to straightforward objectssentenceswas revived in the twentieth century, and developed with extraordinary precision, elegance, and thoroughness, in a paper by Alfred Tarski that is one of the classics of modern logic.

"true" as a metalinguistic adjective

A sentence, Tarski points out, is true or false only as part of some particular language. The Schoolmen were sensitive to this point also; Buridan, for example, observed that if we neglect it, we will be trapped by such arguments as the following: "A man is a donkey" is a true sentence if and only if a man is a donkey; but "A man is a donkey" could have been a true sentence (since we could have used it to mean what we now mean by "White is a color"); ergo it could have been that a man is a donkey. Moore was fond of making similar points.

Further, Tarski argues, a sentence asserting that some sentence S is a true sentence of some language L, cannot itself be a sentence of the language L, but must belong to a metalanguage in which the sentences of L are not used but are mentioned and discussed. He is led to this view by the paradox of the "liar" which he presents, after Jan Łukasiewicz, as follows: He uses the letter c as an abbreviation for the expression "the sentence printed on page 158 [of his paper], line 5 from the top," and the sentence printed there is "c is not a true sentence." By the ordinary Aristotelian criterion for the truth of sentences, we may say "'c is not a true sentence' is a true sentence if and only if c is not a true sentence." But "c is not a true sentence" is precisely the sentence c, so we may equate the preceding with "c is a true sentence if and only if c is not a true sentence," which is self-contradictory. The contradiction is eliminated if we put "of L " after "true sentence" throughout and deny the principle "'c is not a true sentence of L ' is a true sentence of L if and only if c is not a true sentence of L " on the ground that c is not a true sentence of L under any conditions whatever, because it is not a sentence of the language L at all but of its metalanguage M.

Similar paradoxes lead to similar conclusions about such terms as "is a name of" or "signifies"in fact, all terms that concern the relations between the expressions of a language and the objects which this language is used to describe or talk about. All such semantic terms must occur, not in the language that they concern, but in the associated metalanguage. This metalanguage must contain names for expressions in the object language and may also contain descriptions of the structure of such expressions; for instance, we might be able to say in it that one sentence is the negation of another, meaning by this that it is formed from that other by prefixing the expression "It is not the case that" to it. Tarski is attempting to state the conditions under which, for a given language L, we can define the term "true sentence" (and perhaps other semantic expressions) in terms of this basic metalinguistic apparatus, and in such a way as to entail all sentences, in the metalanguage M, of the form "x is a true sentence if and only if p," where x is a name of some sentence of L (we need not write "sentence of L " in the formula, since in M this is what "sentence" means), and p is the translation into M of this same sentence. (M could include L as a part of itself, in which case the sentences of L would be their own translations into M.) Note that this criterion of a satisfactory definition of truth, which Tarski calls the Convention T, is not itself such a definition in M of truth in L, since it talks about expressions of M, and about their relation to what they mean (they "name" sentences of L ), and so is itself not in the metalanguage M but in the meta-metalanguage.

Since in many (meta)languages we form the name of an expression by putting that expression in quotation marks, the following might seem to meet Tarski's criterion: "For all p, 'p ' is a true sentence if and only if p." This, one might think, would immediately yield such individual cases as "'Snow is white' is a true sentence if and only if snow is white" (given, of course, that "Snow is white" is a sentence of L ). This will not do, however, for by enclosing the fourteenth letter of the alphabet in quotation marks (however we use that letter elsewhere) we simply form the name of the fourteenth letter of the alphabet. Hence, what we get by instantiation of the proposed definition are, for example, the sentences "The fourteenth letter of the alphabet is a true sentence if and only if snow is white" and "The fourteenth letter of the alphabet is a true sentence if and only if snow is not white," which together entail that snow is white if and only if snow is not white, a contradiction.

"recursive" definition of truth

If the language L contained only the two simple sentences "Snow is white" and "Grass is green," plus such compounds as could be formed by prefixing "It is not the case that" to a sentence and by joining two sentences by "or," we might offer the following "recursive" definition of "true sentence":

  1. "Snow is white" is a true sentence if and only if snow is white, and "Grass is green" if and only if grass is green.
  2. The sentence formed by prefixing "It is not the case that" to a given sentence S is true if and only if S is not true.
  3. The sentence formed by placing "or" between the two sentences S 1 and S 2 is true if and only if either S 1 or S 2 is true.

There is a mathematical device for turning such "recursive" definitions into ordinary ones, so this feature of the above need not worry us; but we are clearly not very far along if we have to begin by listing all elementary sentences and defining "true" for each of them.

Suppose we enrich L by adding "Snow is green" and "Grass is white" to the elementary sentences, and enrich M by calling "snow" and "grass" names and "is green" and "is white" predicates, and defining an elementary sentence as a name followed by a predicate. We may then alter (1) above to "For any name X and predicate Y, the sentence XY is true if and only if the predicate Y applies to the object named by X." This, however, assumes that the metalanguage already contains the semantic expressions "names" and "applies to"; if it does not, we can only "define" them by saying that "snow" names snow, "grass" names grass, "is white" applies to X if and only if X is white, and "is green" applies to X if and only if X is green.

This is still not very satisfactory, but it is Tarski's basic procedure, except that for his simplest L he takes a language in which there is only one predicate, the relative or two-place predicate "is included in," and no names at all, but only variables standing for names of classes; sentences are formed from "sentential functions" by prefixing a sufficient number of universal quantifiers ("for all x," "for all y ") to "bind" all the variables in the function. That is, the sentences in this language are ones like "For all x, x is included in x," and ones in which "not" and "or" are used either inside or outside the quantifiers or both"For all x and y, either x is included in y or y is included in x " or "It is not the case that for all x, it is not the case that x is included in x " (this last can be abbreviated to "For some x, x is included in x "). Tarski so defines "sentential function" as to cover sentences as special cases (they are simply those sentential functions in which all the variables are bound by quantifiers) and defines the "satisfaction" of a sentential function by a class or classes (a notion very like that of a predicate's "applying to" an object) in such a way that the truth of a sentence becomes the satisfaction by all classes and groups of classes of the function which "is" the sentence in question.

To develop this in a little more detail: Tarski defines "sentential function" recursively, by saying that a variable followed by "is included in" followed by a variable is a sentential function, and so are expressions formed by joining sentential functions by "or" or by prefixing "It is not the case that" or a universal quantifier to a sentential function. "Satisfaction" is more complicated, for Tarski wishes to run together such cases as that the function "x is included in x " is satisfied by the class A if and only if A is included in A ; "For all y, x is included in y " is satisfied by A if and only if for all y, A is included in y ; "x is included in y " is satisfied by the pair of classes A and B if and only if A is included in B ; "x is included in y or y in z " by the trio of classes A, B, and C if and only if A is included in B or B in C ; and so on. To cover all such cases he introduces the notion of an infinite numbered sequence of classes, numbers his variables, and says that the sequence ƒ satisfies the function "v m is included in v n" if and only if the m th member of ƒ is included in the n th; the rest is done recursivelyƒ satisfies the negation of a function Φ if and only if it does not satisfy Φ itself, the disjunction of Φ and Ψ if and only if it satisfies either Φ or Ψ, and the universal quantification of Φ with respect to the n th variable if and only if Φ is satisfied both by ƒ itself and by all sequences that are like ƒ except in having a different n th term.

This last part of the definition is difficult but crucial. How it works can best be seen by considering a simple example. The function "v 1 is included in v 2" is satisfied by all sequences such that the first member is included in the second. The function "For any v 2, v 1 is included in v 2" is satisfied by a sequence ƒ if and only if ƒ is one of the sequences satisfying the preceding function and the preceding function is still satisfied if we replace ƒ by any sequence otherwise like it but with a different second term. This means, in view of what sequences satisfy the first function, that a sequence will satisfy the second function if and only if its first member is included in its second, whatever class that second member may be. Finally, consider the function "For all v 1, for all v 2, v 1 is included in v 2." This is satisfied by a sequence ƒ if and only if ƒ satisfies the preceding function (the second) and if the preceding function is still satisfied if we replace ƒ by any sequence otherwise like it but with a different first term. This means, in view of what sequences satisfy the second function, that a sequence will satisfy the third only if its first member is included in its second, whatever class either of them may be, that is, if and only if every class is included in every class. It is clear that if this function were satisfied by any sequence at all, it would be satisfied by every sequence whatever. (In fact, of course, it is not satisfied by all, and therefore not by any.) In some cases a sentential function will be satisfied by any sequence whatever, even though it contains free variablesas is the case with "v 1 is included in v 1"but if it is thus satisfied and has all its variables boundthat is, is not merely a sentential function but a sentenceit will be, in Tarski's sense, "true."

truth and correspondence

Tarski goes on to consider a more complicated language in which there are variables of two logical types, and an ingenious extension of the notion of a sequence enables him to define "true sentence" for this language also; but when he comes to consider "languages of infinite order," in which there are variables of an infinity of logical types, he has a proof (very similar to Gödel's proof of the incompletability of arithmetic) that any definition of either "truth" or "satisfaction" in terms of the basic material he allows himself would result in the provability of some sentence contravening his Convention T, that is, of the negation of some sentence of the form "x is a true sentence if and only if p," in which x is a name in the metalanguage of a sentence in the language studied and p is the translation into the metalanguage of the same sentence. Even with such a language, however, it is possible to introduce into the metalanguage the undefined semantic expression "true sentence" and so to axiomatize the metalanguage, thus enriched, that all sentences of the form indicated in the Convention T will be provable in it, and also desirable general theorems about truth, such as that "For any sentence x, either x is a true sentence or the negation of x is a true sentence." "Truth," introduced in this way, has something of the mysteriousness of the "correspondence" introduced without analysis by Moore, but Tarski has not merely a suspicion but a proof that, where "truth is understood as a property of sentences of the language in question, such acceptance of a semantic term without definition is inevitable.

See also Modal Logic; Negation.


Aristotle's definition of truth is in his Metaphysics 1011b26 ff., and there are further discussions in Categories 4a104b19 and 14b1223, and De Interpretatione 16a1019. As to Aristotle's Platonic sources, Plato states in the Theaetetus 188e189a the problem solved in the Sophist 240d and 260c263d.

Thomas Aquinas's echoes of Aristotle are in his De Veritate, Q. 1, A. 1, and Summa contra Gentiles, Book I, Ch. 59. Buridan's Sophismata unfortunately has not been reprinted in modern times, but there are accounts of his treatment of the "liar" and similar paradoxes in E. A. Moody, Truth and Consequence in Mediaeval Logic (Amsterdam: North-Holland, 1953), and A. N. Prior, "Some Problems of Self-Reference in John Buridan," in Proceedings of the British Academy 48 (1962): 115126.

G. E. Moore's 19101911 lectures were published under the title Some Main Problems of Philosophy (London: Allen and Unwin, 1953); see especially Chs. 6 and 1316. Bertrand Russell's multiple relation theory of truth is developed in his "On the Nature of Truth," in PAS (19061907): 2849; Philosophical Essays (London: Longmans, Green, 1910), Ch. 7; the introduction to the first edition of Principia Mathematica (Cambridge, U.K.: Cambridge University Press, 1910), pp. 4245; and Problems of Philosophy (New York: Henry Holt, 1912). P. T. Geach's criticisms of the multiple relation theory are in his Mental Acts (London: Routledge and Paul, 1957), Sec. 13; Wittgenstein's, in his Notebooks 19141916 (Oxford: Blackwell, 1961), pp. 21 and 9697. Russell's recantation is in his 1918 lectures on The Philosophy of Logical Atomism (edited by R. C. Marsh), which is included in the collection Logic and Knowledge, also edited by R. C. Marsh (London: Allen and Unwin, 1956); see especially pp. 182, 187188, 214, 218, 225226. His "double correspondence" theory is in The Analysis of Mind (London: Allen and Unwin, 1921), Lecture 13. The more subtle views of the later Russell and Wittgenstein on propositions and facts appear in Russell's 1924 paper "Logical Atomism," included in his Logic and Knowledge and in Wittgenstein's Blue and Brown Books (Oxford: Blackwell, 1958), pp. 3038, and Philosophical Investigations (Oxford: Blackwell, 1953), Paras. 134 and 444.

The key passage from F. P. Ramsey is in his paper "Facts and Propositions," included in The Foundations of Mathematics, edited by R. B. Braithwaite (London: K. Paul, Trench, Trubner, 1931), pp. 142143. For later views of G. E. Moore, exhibiting some influence from Ramsey, see R. B. Braithwaite, ed., The Commonplace Books of G. E. Moore (London, 1962), pp. 228231, 319, and especially 374377. For the use of a Ramsey-type definition in the handling of paradoxes, see A. N. Prior, "On a Family of Paradoxes," in Notre Dame Journal of Formal Logic 2 (1961): 1632, and J. L. Mackie, "Self-RefutationA Formal Analysis," in Philosophical Quarterly (July 1964): 112.

Alfred Tarski's "The Concept of Truth in Formalized Languages" was first printed as a paper in 1931, then modified and enlarged in a German translation in 1935, of which there is an English version in his Logic, Semantics, Metamathematics (Oxford: Clarendon Press, 1956), Paper 7. Tarski states its main ideas less formally in "The Semantic Conception of Truth and the Foundations of Semantics," in Philosophy and Phenomenological Research 4 (1944): 341376. Notable comments on it include Max Black, "The Semantic Definition of Truth," in Analysis 9 (March 1948): 4963, and J. F. Thomson, "A Note on Truth," in Analysis 9 (April 1949): 6772.

other recommended titles

Alston, William. A Realist Conception of Truth. Ithaca, NY: Cornell University Press, 1996.

Armstrong, D. M. A World of States of Affairs. Cambridge, U.K.: Cambridge University Press, 1997.

Austin, J. L. "Truth." In Philosophical Papers, edited by J. O. Urmson and G. J. Warnock. Oxford: Oxford University Press, 1950.

Bealer, George. Quality and Concept. Oxford: Clarendon Press, 1982.

Blackburn, Simon, and K. Simmons, eds. Truth. Oxford: Oxford University Press, 1999.

Coffa, Alberto. The Semantic Tradition from Kant to Carnap. Cambridge, U.K.: Cambridge University Press, 1991.

David, Marian. Correspondence and Disquotation: An Essay on the Nature of Truth. Oxford: Oxford University Press, 1994.

Davidson, Donald. "A Coherence Theory of Truth and Knowledge." In Reading Rorty, edited by Alan R. Malachowski. Cambridge, MA: Blackwell, 1990.

Davidson, Donald. "The Structure and Content of Truth." Journal of Philosophy 87 (1990): 279328.

Devitt, M. Realism and Truth, 2nd ed. Oxford: Blackwell, 1991.

Forbes, Graham. "Truth, Correspondence, and Redundancy." In Fact, Science and Morality: Essays on A. J. Ayer's Language, Truth & Logic, edited by G. Macdonald and C. Wright. Oxford: Blackwell, 1986.

Kirkham, R. L. Theories of Truth: A Critical Introduction. Cambridge, MA: MIT Press, 1992.

Moser, Paul K. Philosophy after Objectivity. New York: Oxford University Press, 1993.

Pitcher, G., ed. Truth. Englewood Cliffs, NJ: Prentice Hall, 1964.

Rorty, Richard. Philosophy and the Mirror of Nature. Princeton, NJ: Princeton University Press, 1979.

Schmitt, F. F. Truth: A Primer. Boulder, CO: Westview Press, 1995.

Soames, Scott. Understanding Truth. Oxford: Oxford University Press, 1999.

Strawson, P. F. "Truth." In Truth, edited by G. Pitcher. Englewood Cliffs, NJ: Prentice Hall, 1964.

A. N. Prior (1967)

Bibliography updated by Benjamin Fiedor (2005)