# A Discussion of “Truth” **URL:** **By** H. Raymond Strong **Published:** July 1, 1971 --- “The dismissal of a paradox is incidental. What is important is that the paradox arose from an invalid extension and misuse of concepts.” The purpose of this paper is to discuss an essential part of the Objectivist approach to the analysis of concepts. The technique with which we are concerned was introduced by Ayn Rand in _Introduction to Objectivist Epistemology_. (1) It consists of asking what facts “gave rise” to the concept in question (and of not permitting use of a “concept” for which an answer cannot be found). It is based on the recognition that valid concepts classify things (in reality). I will demonstrate that this technique is new and significant to epistemology by considering a well-known “paradox” and comparing various past approaches to its resolution with an objective approach. Note that I am presenting my own interpretation of Miss Rand’s work so that this work is not a part of the Objectivist Epistemology (where “Objectivist” is viewed as a trademark of the work of Ayn Rand and her associates). When I refer to an “objective approach”, I am referring to my own interpretation of Objectivism. The paradox in question is an argument is in which a contradiction is derived from “plausible” premises concerning the concepts “proposition” and “true”. It assumes a familiarity with elementary logic. The premises are: (1) At any given time, any _proposition_ is either _true_ or _not true_ (but not both). (2) Let P be a _proposition_, let Q be a _proposition_ stating that P is _true_, and let R be a _proposition_ stating that P is _not true_. Then P is _true_ if, and only if, Q is _true_; and P is true if, and only if, R is _not true_. (3) The following entities named (A) and (B) are _propositions_: > (A): Proposition (B) is true. > > (B): Proposition (A) is not true. It is not hard to conclude from the above premises that (A) is true if, and only if, (A) is not true; but (A) is either true or not true — a contradiction. I will consider five typical approaches to a resolution of this paradox. The first two reject (or modify) premise (3). The rest reject the usual meaning of “not” ## I. The Type Theory Approach Bertrand Russell’s type theory handles such paradoxes by imposing a hierarchical structure of types on concepts like “proposition”. (Eliminating the complexities of the theory of types which are not relevant to our problem) we could say that “proposition”, unmodified, is taken to mean “type 1 proposition”. A proposition about type 1 propositions is a “type 2 proposition”. A proposition concerning type 2 propositions is type 3, etc. The idea of “all propositions” is taken to be meaningless, at least as subject-matter for some proposition. (A major criticism of type theory is that its subject-matter includes all propositions: What is the type of the propositions of type theory?) For our paradox, type theory translates (A) and (B) to type 2 propositions: > (A): The type 1 proposition (B) is true. > > (B): The type 1 proposition (A) is not true. Thus it finds that neither (A) nor (B) is true. ## II. The “Meaningless” Approach The most widely accepted approach handles the paradox by claiming that (A) and (B) are not meaningful and that only meaningful propositions satisfy premise (1). Meaningful propositions are not allowed the kind of circularity (similar to “self-reference”) exhibited by (A) and (B). Of course, the argument begins when we ask which propositions are meaningful. I call this approach “most widely accepted” because those who ignore the paradox implicitly accept something like it. ## III. The “Both” Approach Some “resolutions” of the paradox are effected by “correcting” premise (1) to say that a proposition may be both true and not true. ## IV. The “Neither” Approach The “neither” approach “corrects” premise (1) to say that a proposition may be neither true nor not true. This approach is more appealing when the paradox is presented in terms of “true” and “false” with “false” substituted for “not true” throughout. One discusses “three-valued logics” with values such as “true”, “false”, and “maybe”, or “true”, “false”, and “unknown”. This approach is sometimes indistinguishable from the “meaningless” approach. The “both” approach also sounds slightly better when “false” is substituted for “not true”. ## V. The “So What” Approach Related to the “both” approach and the “meaningless” approach, the “so what” approach concludes: “Contradictions exist — so what?” The purpose of discussing these approaches has not been to set up straw men. Doubtless, there are other approaches which could not be rejected in a few words. The purpose has been to exhibit the epistemological chaos caused by treating the concepts “proposition” and “true” as out-of-context abbreviations for (verbally stated) properties called “definitions” or as “ideal” existents which are “intuitively” seen to satisfy the premises. Almost uniformly, no matter what the result, the philosophers’ approach to the paradox has been a an approach to a game with a fault in its design. The “language game” has produced a nasty paradox; so we must simply change the “rules”. Contrast this attitude toward concepts with the view that they are classifications of things for which definitions may be sought. By the method of analysis of concepts introduced by Ayn Rand, we could isolate valid concepts “proposition” and “true”. But we would _not_ be “explicating intuitive concepts”; we would be classifying things. Many such classifications are possible. There is nothing magic about the word “true” that requires that it be used for a certain classification. If there had never been _valid_ concepts concretized by the words “true” and “proposition”, it would make no sense to “search for their meanings”. However, it is not enough to find an historically valid definition for a concept. Justification must be supplied on two levels: (1) the concept defined must be a valid and _useful_ classification, and (2) the definition must be given in terms of essentials in the context of _present_ knowledge. (2) The “meaningless” approach is probably the most historically accurate: when the concept “true” was first being isolated, propositions like (A) and (B) were not considered (nor, at that time, was it necessary to consider them). Most people are astonished by the paradox because it seems to challenge their certainty that they have grasped a valid concept (“true”). It requires a great deal of “psycho-epistemological” confidence to put aside the paradox with the unshaken (and usually correct) belief that one has grasped a valid concept “true” which is an adequate classification for one’s own purposes. The fact is that the historically valid concept “true” is not an adequate classification of propositions in the context of a general theory of logic. It is incomplete and was never intended to classify propositions in isolation or propositions with the kind of subject-matter of (A) and (B). It was certainly never intended to classify arbitrary strings of symbols, although, in some systems of “symbolic logic”, the word “true” is now used for a valid classification of strings of symbols which is almost completely independent of of any meanings for the symbols. When we ask what facts gave rise to the concepts “true” and “proposition”, we will find that, for purposes of epistemology, the old (valid) concepts are inadequate. The above paradox is a fact leading to this conclusion. Such facts together with those originally discovered now give rise to more careful distinctions. For a long time logicians have recognized the need to distinguish propositions “in thought” from strings of verbal or written symbols. (When this distinction is made, the latter are often called “sentences”.) Moreover, even Aristotle recognized that the concept “true” which he had grasped was inadequate for a treatment or predictions in propositional form: when the concept which apparently satisfied premise (1) was extended to predictions, it imposed a paradoxical determinism on reality. (3) Facts which gave rise to the concepts “true” and “proposition” follow: 1. Man operates at the conceptual level and holds and communicates knowledge in conceptual form 2. There is a specific form in which man learns to concretize concepts for examination or communication as concepts: this form is the _word_ (written, spoken, or thought). 3. There is a specific form in which man learns to concretize his conceptual knowledge: this form is the _proposition_ (written, spoken, or thought). 4. Man is neither infallible nor omniscient. 5. Only some of the propositions which man examines _should_ be added to knowledge. propositions concretize knowledge of facts of reality, while others have only the _form_ of the concretization of knowledge of a fact. These facts were known in Aristotle’s time and a proposition was classed as _true_ when it represented knowledge of a fact. But the relation of the true propositions to knowledge and to fact was left ambiguous: a proposition which did not at one time represent a _known_ fact could at some later time be discovered to be, _and to have been_, a true proposition. Many examples of true propositions were known. The concept was certainly valid. But it was not validly extended to the domain of hypotheses and arbitrary assertions. For this extension we must elaborate further facts: 1. Those propositions should be added to knowledge are products of reason: facts are observed (perceived); knowledge of facts is obtained directly by perception or inferred. 2. The “correspondence” between a proposition in knowledge and a fact involves the method of inference of the proposition. This correspondence cannot be treated as a property intrinsic to the proposition and independent of the method (for proof see the above paradox). The concept “truth” which can now be isolated by an objective approach identifies the product of a process of inference. a proposition is _true_ if it is a conceptual-level statement of the directly observed or has been obtained from this base by proper rules of inference. The property “true” applies to the proposition _together with its method of inference_. If a proposition is true, it is true by virtue of its method of inference. The case of hypotheses can be handled as a part of the general case of answering the question, “Is it true that X?” where X is some proposition. If it had been the case that the property “true” were possessed by certain propositions independent of any method of inference, then this question could have been answered “yes”, “no”, or “unknown”, meaning “X is known to be true”, “X is known to be not true”, or “The truth-value of X is unknown but exists independently of our knowledge” respectively. However, this set of responses is inconsistent with the interpretation of “true” being discussed, even though this is generally the set of responses expected. The questioner who expects these responses is presuming to judge the answer against a standard of omniscience. In order to answer the question, “Is it true that X?” properly, one must examine it in context. A “yes” answer may indicate that X has been inferred and is known immediately to the answerer or that after some thought the answerer has inferred X or that X is “common knowledge” although the answerer has not followed the details of its derivation, depending on the context in which the question is asked. In this objective approach, anything but a negative answer indicates some evidence for X. Thus the negative answer with the least ambiguity is “There is no evidence for X” rather than “No”. A valid distinction can be made between propositions which are contradicted by present knowledge and those for which there is simply _no_ evidence. However, there is little evidence for the cognitive usefulness of such a distinction (in spite of widespread acceptance). The Intuitionist school of mathematics (and philosophy) makes this distinction and classifies propositions as “true”, meaning “known to be true”, “false”, meaning “known to be false”, or “unknown” meaning “as yet unproved” with no commitment to an intrinsic truth or falsity. Unfortunately, intuitionists also equal “false” with “not true” so they are forced to reject the “law of the excluded middle” (premise (1) above. The objective approach under discussion does not bother to make the distinction in question. The objective approach to the paradox discussed at the beginning of this paper would reject premise (2). A particular exception to premise (2) would be taken in the case of arbitrary assertions: if P is an arbitrary assertion, then “P is true” and “P is not true” can be viewed as arbitrary assertions. No arbitrary assertion is true. No proposition is true solely by virtue of its form (including “A is A” when it is viewed as an arbitrary assertion). In the context presented, both proposition (A) and proposition (B) of the paradox are arbitrary assertions. The fact that neither proposition is true _in this context_ does not change the truth-value of proposition (B) _in this context_. Now there are many approaches to this paradox which give roughly the same answer (e.g. type theory or the “meaningless” approach). But remember that we are not “changing or explicating the language in order to remove a paradox.” We are simply classifying things (in this case, propositions) in a cognitively useful way. The dismissal of a paradox is incidental. What is important is that the paradox arose from an invalid extension and misuse of concepts and that there is a correct way of handling concepts from which such paradoxes do not arise. **EDITOR’S NOTE: The following is excerpted from a letter from Dr. Strong regarding “The Liar Is A Thief”:** I found Ronn Neff’s analysis of “This sentence is false” (TSIF) interesting and creative. However, I believe it still accepts too many rules from a game which is stacked against reason. The effect of his analysis of TSIF is a rejection of what I called premise (2) in the above essay, in favour of the proposition that a sentence is logically equivalent to the conjunction of itself with the assertion of its truth. Thus TSIF is logically equivalent to a contradiction and can be judged false, the falsehood of TSIF being logically equivalent to a disjuction which involves no contradiction and therefore no paradox. The analysis implicitly accepts the principle that truth-values are intrinsic to sentences. The difficulty with this principle is most easily seen by attempting to ascertain the truth-value of > TSIT: This sentence is true. It does not help (or hinder) to assert that TSIT is logically equivalent to (TSIT and (TSIT is true.)) The difficulty may also be seen in the reasoning applied in extending the analysis to the example > G: Sentence H is false. > > H: Sentence G is true. If “…it does not follow from TSIF’s being false that TSIF is true,” then from “(b) that H is false” it does _not_ follow “(c) that G is false.” The analysis does work for H, however, so H can be judged false, leaving us in the unfortunate position of concluding G’s truth. I agree that the Liar paradoxes involve the fallacy of the stolen concept. I maintain that concepts are being stolen at a much greater depth than is indicated by the above analysis. We have in English a valid concept “true” (not the modern logicians’). The application of this concept to arbitrary assertions is simply inappropriate since it is not a property intrinsi to assertions. The concept “inference” is epistemologically prior to the concept “true”. To assert the truth of a sentence is to assert the existence of some method by which it can be inferred. _H. Raymond Strong has a PhD in the foundations of mathematics from the University of Washington and is currently doing research in the theory of programming for IBM Research._