Table of Contents
 Definitions of Truth
 Knowledge & Justification
 Types of Truth
 Analytic vs. Synthetic Truth
 a priori vs. Emprical Truth
 Relations between Analytic/Synthetic and a priori/Empirical/Indeterminate Statements
 Justification
 Summary
 End Notes
Definitions of Truth
The definition of truth is, to most people, intuitive and obvious: It is a relationship between a statement or belief and the real world. If the statement or belief corresponds to the state of affairs as it exists (or “obtains”, in philosophical jargon), then that statement or belief is true; otherwise it is false. This is known in philosophy as the correspondence theory of truth. It has been formalized in a rigorous definition by mathematical logic, in which a statement is true if and only if it corresponds in a precise way to a “universe of discourse”^{1}.
Philosophers have invented other theories of truth. Here is a brief commentary on some of the more prevalent ones^{2}:
 Coherence Theory: A statement is true if it does not logically conflict with any other true statements. I see this as a necessary rather than a sufficient condition. While any two true (in the correspondence sense) statements must be coherent, basing truth entirely on coherence is rather like lifting oneself by one’s bootstraps. The statements that “Unicorns have horns” and “Unicorns can fly” are mutually coherent, but that does not make them true.
 Constructivist Theory: A statement is true because society believes it to be so. This means that if most people believe that ghosts exist, then they really do exist. To me, that this is preposterous goes without saying.
 Pragmatic Theory: A statement is true if believing that it is true is useful. Thus, it could be useful to you to believe that your wife is not sleeping around behind your back, because this belief saves you the anguish that you would feel if you believed otherwise (what you would believe using the correspondence theory). I find this theory equally as preposterous as the constructivist theory.
Knowledge & Justification
Since the time of Plato, “knowledge” has been defined as “justified, true belief”^{3,4}. For most of history philosophers have been concerned with knowledge in general; the process of forming a belief and justifying that belief were assumed to be one and the same. In modern times, there has been a shift in emphasis to investigation of the processes by which we justify beliefs, with the processes by which beliefs are formed being of less importance^{5}. I concur with this shift in emphasis: If your crystal ball produced a new “theory of everything” in physics that proved to be correct, why would it matter that the theory came from a crystal ball, as long as we were able to demonstrate that the theory was (or most likely was) true?
Obviously, if we can prove that a statement is true, then we are justified in believing it. We shall see below, however, that there are statements that most likely are true, but that cannot be proven. For these statements, we need some other form of justification. The only valid method of justification that has been found for these statements is the scientific method, which is discussed below and in detail a separate essay on that subject. We will also find that there are some statements which may be true, but that cannot be justified by any means.
Types of Truth
Both David Hume, in a An Inquiry Concerning Human Understanding, and later Immanuel Kant, in Critique of Pure Reason, proposed that true statements can be organized into two types. The first type are those that must be true, meaning that even in a fantasy world that was different from the real world, these statements would still hold. Hume called these “Relations of Ideas”, and Kant called them “necessary” or “analytic” truths. These are logically tautological statements, such as the law of the excluded middle (A or not A). All other truths are what Hume called “Matters of Fact”, and Kant called “contingent” or “synthetic” truths. Synthetic statements are true only because they happen to correspond to the real world, meaning that they could have been otherwise. Thus, the statement “unicorns do not exist” is a true statement, but we can imagine a fantasy world in which they do indeed exist.
Kant also believed that truths could be divided into two types in another way, depending on whether or not we learned them through experience. Those that we know without experience he called “a priori“, while those we learn through experience he called “a posteriori“, or “empirical”. By a priori, Kant meant that we are somehow, perhaps as a gift from God, able to innately know these truths; knowing them is part of our nature. Any other true statements, those that we must learn to be true, fall into the empirical category.
The fact that true statements can be cleanly and unambiguously categorized like this has come under fire in modern times, starting with W.V.O. Quine’s classic paper, “Two Dogmas of Empiricism”^{6}. While the main thrust of Quine’s argument can, I believe, be explained away as due to ambiguities in the use of natural language, these “dichotomies” are indeed not as neat and clean as has traditionally been assumed. I will investigate each of them in the following two sections.
Analytic vs. Synthetic Truth
Hume defined his “relations of ideas” as an “object of of human reason… [that is] discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe.”^{7} Kant defined analytic statements as “judgements in which the relation of a subject to the predicate is thought, …[and in which] the predicate B belongs to the subject A.”^{8} In modern times, there are two common definitions of analytic:
 Statements that are true in all possible worlds
 Statements that can be proven to be true using logic
Neither of these definitions is completely unambiguous, but with clarification they can be demonstrated to be equivalent, as follows.
In order to obtain a precise definition of analytic truth, I will restrict my discussion to languages of 1storder (predicate) logic, but the concepts here apply equally well to higherorder logics and modal logic. While some may claim that the language of logic is not broad enough to encompass all of the concepts of natural language, the former provides a necessary framework for the degree of rigor and precision required for philosophical analysis^{9}. If such a gap exists, a language that is as rigorous as logic but broad enough to encompass the alleged gap must be developed.
In 1storder logic, a “possible world” is defined rigorously as a “structure”, a function that maps the “wellformed formulas”^{10} of a particular formal language to a “universe of discourse”, consisting of a set of objects and a set of relations between those objects. Thus, to say that a statement is true in all possible worlds is to say that it is true in every possible structure for the language in which the statement was made. This means that a statement is analytic only in the context of a particular language. This is necessary because the language must be specified in order to know what predicate symbols are relevant, and thus what relations must be represented in the structure. We could define analytic as true in all possible structures in all possible languages, but by this definition there would be no analytic statements. I would conjecture that all analytic statements in any language can be formed by plugging atomic formulas^{11} into various tautological templates. For example, ∀xPx ⇒ ∃xPx is true for any (unary) predicate symbol P in any language. However, I have not attempted to prove this conjecture.
In logic, a theory is defined as a set of sentences^{12,13} that is “closed” under logical implication. This means that if T is a theory, for any sentence s, if s can be logically deduced or derived from the sentences in T, then it must be that s∈T. For every structure there is exactly one theory that is the set of all sentences that is true in the structure (although a theory can be true in more than one structure). Therefore, another (equivalent) definition of analytic truth is a sentence that is an element of all possible theories in a particular language.
It turns out that the second definition of analytic provided above is exactly equivalent to the first. In 1storder logic, a set of sentences S “logically implies” sentence s by definition if and only if s is true in every structure in which every sentence in S is true. A special case of this is when S is the null set, in which case s is true in every possible structure in the language. Two famous theorems in 1storder logic, the soundness theorem and the completeness theorem, proved that S logically implies s if and only if s can be logically derived or deduced from S. So saying that s is true in all possible structures in the language is exactly equivalent to saying that it can be proven to be true using logic. Such sentences are often called “tautologies”.
Traditionally the discussion of analytic and a priori statements (see next section) has been restricted to true statements. However, I wish to extend this scope to include all sentences, because in the next section I will propose that there are statements which cannot be justified at all. Therefore, I will include in the category “analytic” the negation of tautologies, also know as “contradictions”. For example, “Aristotle was both a genius and not a genius” is a contradiction, because it is impossible for something to both be and not be. Similar to tautologies, contradictions are false in all possible worlds. I will refer to tautologies as “analytic truths” and contradictions as “analytic falsehoods”.
I will define a synthetic statement as any sentence that is not analytic. Thus, any sentence in any language must be either analytic or synthetic. Note that some authors such as Quine (as described above) dispute a clear distinction between these two types of sentences. However, if we accept the rigorous definition of analytic presented here then the distinction is unambiguous.
Analytic truths are sometimes defined as truths that are as such “by definition”. However, these sentences are actually just a subset of all analytic sentences. Definitions are often cited as a source of ambiguity regarding a sentence’s analyticity, however, the alleged ambiguity actually derives from a lack of clear hierarchy of definitions rather than the definition of analyticity itself. Any formal language must include a nonempty set of predicate symbols (I treat equality as a predicate), and optionally a set of object and/or function symbols. These are the socalled primitive symbols, which are undefined. Any other symbols used in the language must be defined in terms of these primitives. To determine if a sentence is analytic or not, the defined symbols must be replaced by their definitions. For example, assume that in a particular language, “man” and “married” are primitives, and that “bachelor” is defined as being a man but being not married. Then the sentence “All bachelors are men” is analytic, because substituting the definition of “bachelor” into this sentence produces “All men that are not married are men”, which is a tautology. Using symbolic logic, let M be the predicate symbol for “man”, R be the symbol for “married”, and B be the symbol for “bachelor”. Then the definition of B can be written Bx ≡ Mx & ¬Rx. Thus, the original sentence is ∀x[Bx ⇒ Mx]. When we substitute the definition of B, this becomes, ∀x[ (Mx & ¬Rx) ⇒ Mx ], which can be demonstrated to be a tautology.
a priori vs. Empirical Truth
Kant defined a priori knowledge as “independent of experience and even of all impressions of the senses.”^{14} I will defined an a priori sentence as one whose truth can be justified without the need for any observational information. Note that unlike the definition of analytic, which is based entirely on logic, a priori is defined in the context of a rational being or consciousness, a “rational agent” in philosophical jargon. In particular, the definition of a priori is made with respect to sensory input to, or observations made by, the agent. I will assume that a rational agent has the ability to execute the predicate calculus (the process for deriving truths in 1storder logic). Therefore all analytic statements are a priori, since logic provides a mechanism for justifying them (indeed, proving them) or refuting them without sensory input.
For Kant, any knowledge that requires sensory input or observation to be justified was termed a posteriori, or “empirical”. Thus, empirical statements include scientific laws, but also include everyday observations, such as “the color of my automobile is dark grey”. There is in fact a process, known as the scientific method, that can be used to justify empirical statements. Historically science has been extremely successful in providing us with empirical knowledge, yet under the microscope of philosophy, many of the assumptions on which the scientific method is based are, surprisingly, tenuous at best. See my essay The Scientific Method for a detailed discussion of this process and its shortcomings. I also briefly touch on this process below. Despite its philosophical issues, science is the only known method to successfully justify empirical statements.
Note that Kant provided this definition for knowledge, so any a priori or empirical statements must be justifiable and true. However, it is crucial to point out that these two categories are not exhaustive for statements in general. There are statements s such that we cannot justify s (or ¬s) by any means. Such statements would be synthetic by the definition above, but would be neither a priori nor empirical. I will refer to these statements as “indeterminate”, to reflect that fact that we have no way to justify them.
Relations between Analytic/Synthetic and a priori/Empirical/Indeterminate Statements
Much of the focus of epistemology and the philosophy of science since Kant has been on the degree to which these categories overlap, or whether one category completely encompasses the other. Most notable, for example, was Kant’s claim that there are synthetic a priori truths. I will examine this subject in detail here to determine whether or not Kant was correct, and in general, what all the relationships are between these categories.
As stated above, we already know that all analytic statements are a priori; i.e., analytic ⇒ a priori for short. Thus, we can conclude that a statement cannot be both analytic and empirical, nor can it be both analytic and indeterminate.
The fact that analytic ⇒ a priori does not imply its converse, i.e., that a priori ⇒ analytic. There is the possibility that there are synthetic, a priori statements, as Kant maintained. However, by our definitions, this would require that there be some mechanism for justifying a priori statements other than logic. I know of no such mechanism. The branch of philosophy know as rationalism^{15} claims that such a mechanism exist, and I suspect that many philosophers, at least implicitly, believe this claim. However, “reason”, the process of rationalistic justification, seems to me to be so vaguely defined that it is hard to even grasp, much less analyze. Certainly it presents no welldefined process whereby the measured confidence of an a priori statement could systematically approach truth, as is the case with the scientific method. The claim that there are no synthetic a priori truths is often associated with the branch of philosophy of science known as empiricism^{16}.
Therefore, it appears, contrary to Kant’s belief, that there are no synthetic a priori statements. Indeed, the fact that there appear to be synthetic statements that cannot be justified by either of the known methods of justification, logic for analytic statements, or the scientific method for empirical statements, means that there are indeed synthetic indeterminate statements. I would maintain that this is the distinction between logic/mathematics, science, and philosophy. Logic and mathematics study and attempt to prove analytic, a priori statements, science studies and attempts to justify synthetic, empirical statements, and philosophy studies and speculates regarding synthetic, indeterminate statements.
Is mathematics purely analytic? It was Kant’s claim that it was not, and this was his justification for his belief in synthetic a priori truths. However, a detailed analysis shows that the answer is a complex one. Certainly, given any set of mathematical axioms A, and any theorem τ that can be derived from the axioms, the following is an analytic (and therefore a priori) statement: A ⇒ τ. However, the intent of the question most likely is whether any statement of mathematics alone is analytic. We also need to ask in what language must a statement be analytic. I will adopt the interpretation that a statement is analytic if it is so in at least one language.
Alas, not all statements of mathematics are analytic, because we know that the axioms of set theory are not^{17}. Are there any statements at all in mathematics that are analytic, such as 1+1=2? In a language where these symbols are primitive, it is not; axioms must be specified to describe the behavior of the symbols. Another approach might be to define the symbols of arithmetic in terms of more primitive concepts (e.g. set theory), and therefore prove the statement as analytic, but this may not work either. For example, if we define ‘1’ as the set of all sets with one element, then it may not even be meaningful, as ‘1’ by this definition does not even exist in some set theories! However, we can interpret the entire phrase “1+1=2” to mean that for any two disjoint sets each of whose members is a particular thing and nothing else, then the set that is the union of those two sets has as elements a thing and another thing and nothing else^{18}. This interpretation is indeed analytic.
We have established that there are analytic (and therefore a priori) statements in mathematics. What is the nature of those that are not analytic: Are they empirical or indeterminate or both? It would appear that set theory is indeterminate, as the only sets that we can observe empirically are sets of atoms, or real objects. For example, I can see that the set of fingers on my left hand contains five members, but we cannot observe the set of numbers from 1 to 5. Because of this, there is no way to empirically select one set theory over the other. On the other hand, the axioms for those branches of mathematics that have applications in the real world, such as arithmetic and geometry, can certainly be justified empirically.
It might be possible to prove that the axioms of mathematics (or at least the major branches like number theory and geometry) can be derived analytically in any set theory of sufficient strength. If possible, the requirement of “sufficient strength” would become important. One approach might be to define “sufficient strength” as the ability to provide a foundation for mathematical logic, in which case mathematics would indeed by analytic. As an alternative, “sufficient strength” might be defined as sufficient to model the world as we observe it, in which case all of mathematics would rest on an empirical foundation.
As mentioned above, any set theory is an example of a set of statements that cannot be justified either analytically or empirically. Therefore (synthetic) indeterminate statements do exist. Thus we have found that there are three types of true statements, and each type can be said to correspond to a particular branch of human study:
 Analytic & a priori: Logic and mathematics (proof)
 Synthetic & Empirical: Science
 Synthetic & Indeterminate: Philosophy
Note that just because a statement is indeterminate does not mean that it cannot be true. An example of indeterminate truth may be found in the subject of ethics. While there does not seem to be any way to justify one theory of ethics over another, once one theory (or rather, one set of definitions) has been selected, the specific statements of ethics can be said to true or false. (For more details on this, see my essay on Naturalism in Ethics.)
Justification
It is well established that 1storder logic is the means of proving analytic truths, although it was not until the late nineteenth century, with the invention of the predicate calculus by Gottlob Frege, that the details of modern mathematical logic were completely understood^{19}. 1storder logic has variables for objects only, but variables for predicates can be added by using either set theory or the closely related 2ndorder logic. The ability to make statements about possibility and necessity, or in other words, about possible worlds, requires an extension of logic called modal logic, the details of which were worked out in the twentieth century^{20}.
Up to this point we have implicitly assumed that 1storder logic is valid, that a proven analytic statement is indeed true. It it certainly reasonable to ask whether or not the use of logic itself is justified. The answer is yes, but the justification is circular, in that we must use 1storder logic to prove that 1storder logic works. Specifically, as described above, we adopt the correspondence theory of truth, and then can prove that our mechanism for deduction is equivalent to logical implication of truth; proof of this fact took the form of the soundness and completeness theorems. The mechanism for the proof of these theorems (like any proof), makes use of the very deductive calculus whose efficacy we are attempting to prove. This is why the justification of logic is circular^{21}. This fact about deductive logic is ironic, as the justification of empirical statements has long been criticized as circular, and often cited as a shortcoming of empirical knowledge as opposed to analytic truths^{22}.
Unfortunately, there is no apparent way to prove synthetic truths with absolute certainty. However, there is a means for demonstrating synthetic truth with a high degree of certainty: by use of the scientific method. We know that the scientific method works, because science is the foundation of all technological advances. Our understanding of the physical world continues to grow exponentially, and technological advances ride on science’s coattails. Each of us has so much confidence in the knowledge that science has provided that we bet our lives on the validity of that knowledge every day. For example, every time we board a jet airplane, we assume that Bernouli’s law of fluid flow is correct and will continue to be correct.
Despite its tremendous success at providing information about the external world, the scientific method is fraught with philosophical problems. For a discussion of these problems and justification of empirical truths in general, see my essay on the Scientific Method.
Summary
In summary, there are two modal distinctions for knowledge, analytic and synthetic, and three categories of justification of statements, a priori, empirical, and indeterminate, the latter meaning that justification is not possible. While there are a total of 2×3 = 6 different combinations of these two sets of categories, half of them have been ruled out. The table below gives a summary of all six possibilities, with examples and other information.
Table 1: Knowledge & Justification
Modality 
Justification 
Method 
Examples 
Subject 
Comments 
Analytic  a priori  Logic  Tautologies Mathematical Proof 
Logic Mathematics 

Analytic  Empirical  None  None  None  Analytic statements can only be justified by logic, which does not require sensory input. 
Analytic  Indeterminate  None  None  None  By definition, analytic sentences are either true or false (in all possible worlds). 
Synthetic  a priori  None  None  None  The only way to justify a priori truths is via logic, which makes them analytic. 
Synthetic  Empirical  Scientific Method  Everyday Knowledge Scientific Theories 
Science  
Synthetic  Indeterminate  None  Foundations of Set Theory Ethics 
Philosophy 
As described above, there are some unanswered questions in each of these areas, especially with respect to synthetic empirical statements and the scientific method. However, compared to other areas of philosophy like metaphysics and ethics, the subject of truth is fairly well understood.
 See any introductory textbook to mathematical logic or model theory, such as Enderton, Herbert B., A Mathematical Introduction to Logic, Academic Press, San Diego, CA, 1972.
 The Wikipedia page on Truth has a more detailed discussion of each of these theories of truth and others (“Truth”, Wikipedia, URL=<http://en.wikipedia.org/wiki/Truth>).
 Plato proposed that knowledge is justified, true belief in his dialog Theaetetus. See “Belief”, Wikipedia, URL=<http://en.wikipedia.org/wiki/Belief.
 In modern times this definition has been challenged, most notably in Gettier, Edmund L., “Is Justified True Belief Knowledge?”, Analysis 23 ( 1963): 121123, available online at http://www.ditext.com/gettier/gettier.html.
 See, for example, Salmon, Wesley C., The Foundations of Scientific Inference, University of Pittsburgh Press, Pittsburgh, PA (1966), pg. 7.
 Quine, W.V.O., “Two Dogmas of Empiricism”, The Philosophical Review 60 (1951): 2043; reprinted in Quine, W.V.O., From a Logical Point of View (Harvard University Press, 1953; available online at http://www.ditext.com/quine/quine.html.
 Hume, David, An Inquiry Concerning Human Understanding, Section IV, Part I. Available online at http://www.earlymoderntexts.com/he.html.
 Kant, Immanuel, Critique of Pure Reason, A6 (Introduction, IV). Available online at http://www2.hn.psu.edu/faculty/jmanis/kant/CritiquePureReason.pdf.
 For more on this see my essay on Philosophical Discourse. This article also includes a very high level overview of 1storder logic. A detailed discussion of logic can be found in many texts, such as Enderton, op. cit.
 Logic includes rules for forming statements from a set of symbols. A wellformed formula is a statement that conforms to these rules. One that does not conform to these rules is meaningless gibberish. For more information see Enderton, op. cit.
 Atomic formulas are strings or phrases containing a single predicate symbol and a string of object symbols or variables; i.e. predicate, subject, and objects in natural language.
 Formally, a sentence is a statement in which all object symbols are either constant symbols referring to a specific object or quantified variables; e.g. ∀x or ∃x. A sentence contains no “free” variables.
 Note that I will sometimes use “statement” in lieu of “sentence” below where it is clear from the context that I mean the latter.
 Kant, op. cit., B2 (Introduction, I).
 Once again, I refer the reader to “Rationalism”, Wikipedia, URL=<http://en.wikipedia.org/wiki/Rationalism for an introductory discussion of rationalism.
 Carnap, Rudolph., An Introduction to the Philoposophy of Science, Basic Books, New York, NY (1966), pg. 180.
 There are many set theories that are mutually exclusive. For example, the set of all sets with just one element exists in the theory called New Foundations, but it does not exist in the ZermeloFraenkel theory. The theory in which all definable sets exists can easily be shown to be inconsistent. For an excellent discussion of several leading set theories, see Quine, W.V.O., Set Theory and its Logic, Harvard University Press, Cambridge, MA, 1969 (Revised Edition).
 Symbolically, this would be ∀X∀Y∀Z{ [ ∃x∃y( ∀w(w∈X ⇔ w=x) & ∀w(w∈Y ⇔ w=y) & x≠y & Z=X∪Y ] ⇒ ∃x∃y∀w( w∈Z ⇔ w=x  w=y ) }
 For a brief history of logic, see my essay 18791931, in particular the section on The Predicate Calculus.
 See for example Hughes, G.E. and Cresswell, M.J, A New Introduction to Modal Logic, Routledge Books, London, 1996. While the different systems of modal propositional logic are well established, there is still some controversy concerning the predicate logic system, such as whether or not to include the Barcan formula.
 To the best of my knowledge, this fact was first pointed out in Haack, Susan, “The Justification of Deduction”, Mind, New Series, Vol. 85, No. 337 (Jan. 1976), pp. 112119. Available online at http://www.as.miami.edu/phi/haack/Justification%20of%20Deduction%20reprint%202010.pdf.
 See for example Vickers, John, “The Problem of Induction”, The Stanford Encyclopedia of Philosophy (Spring 2013 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/spr2013/entries/inductionproblem/>.
November, 2013
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