First, it must incorporate an element of non-contradiction. The same thing cannot both be and not be at the same time and in the same respect. The same proposition cannot be both true and false simultaneously. This follows directly from the Principles of Existence and Identity, for if the subject of an assertion exists, then it is itself and not its opposite. The principle of non-contradiction assures that not all assertions are true.
Thus, if we define what is true we are also defining what is false. If A is true, then A* (A-Not) is false.
Second, just as non-contradiction assures that not everything can be simultaneously true, the Excluded Middle teaches that not all is false. A proposition either is or is not; there is no third possibility “between” contradictory claims. This also follows from the Principles of Existence and Identity; a middle assertion would not have independent identity in existence and thus would possess no meaning.
Thus our theory of truth must be exclusive and absolute: A=A, A*=A*, and A≠A*.
What does a subject mean by asserting that a statement is "true?"
There are a few theories about this:
Correspondence Theory of Truth: Also called Foundationalism. An assertion is true if it accurately represents Reality, or “the way things are.” A rejection of any sort of relativism about truth, the correspondence theory maintains that the truth or falsity of a statement is determined only by how it relates to the world, and whether it accurately describes (corresponds with) that world. This is the classical view espoused by Plato, Aristotle, G.E. Moore, Bertrand Russell, etc. The only weaknesses here are questions of defining “ correspondence,” issues that can be dealt with by a thorough examination of the term’s meaning.
Coherence Theory of Truth: This could be called the Theory of Correspondence of Facts in the Mind. An assertion is true if it is coherent with some specified set of statements. Usually the set is identified as the statements that make up what is the best-justified and most complete description of the world. A realist might define truth as correspondence with the facts, and argue that the only valid way to determine the truth of a proposition is to see if it corresponds to the facts. A Coherentist might also define truth as correspondence with mind-independent reality, but also maintain that the truth or falsity of a statement is determined by its cohering with the body of accepted scientific or empirical knowledge. Thus, there is only a subtle shade of difference between these two theories, the additional element of Coherence that man’s study is the means by which truth or falsehood is discovered. Yet the inherent weakness in this system is that people may have an entirely logical and coherent set of beliefs and yet still be wrong. Psychologists have discovered many delusional people who have complete, consistent, and rational belief systems that are still quite wrong, simply because of incorrect premises. For an excellent demonstration of this principle, read Isaac Asimov’s short story “Reason.”
Consensus Theory of Truth: A statement is true if all those who investigate it would agree it upon. A more extreme form of Coherence, this belief originally espoused by Charles Sanders Pierce refuses to acknowledge something as true unless it has been tested by majority consensus of the educated. Note that, since it is possible in principle for everyone to agree but be mistaken about the facts, the consensus theory implies that a statement can be true even if it fails to describe reality. For example, if all who investigate “The center of Venus is molten copper” are destined to accept it, then it is “true” according to the consensus theory even if they are all wrong about the fact of the matter. An objection to this theory is that it presupposes that for every possible statement, investigators are destined eventually to agree about it one way or the other. But this seems dubious, for many, many areas of knowledge are extremely controversial and probably will never come to consensus opinion. Furthermore, we know that the majority of the world has been wrong about facts at one time or another (ex., “humours” in the Middle Ages), thus invalidating the claim. This is not synonymous with subjectivism, but still has the practical applications of the same.
Pragmatic Theory of Truth: also called utilitarianism, the success of the practical consequences of an idea makes a statement true. Pragmatism describes an indicator or a sign of truth. It really cannot be regarded as a theory of the meaning of the word “true.” There's a difference between stating an indicator and giving the meaning. For example, when the streetlights turn at the end of a day, that's an indicator, a sign, that evening is coming on. It would be an obvious mistake to say that the word “evening” just means “the time that the streetlights turn on.” In the same way, while it might be an indicator of truth that a proposition is part of that perfect science at the ideal limit of inquiry, that just isn’t what “truth” means. Outside of the fact that this is not exactly a theory of truth as the others, it has quite a few problems. How does one determine what is “useful?” Sometimes, one could argue, it is pragmatic to lie. Surely a lie does not suddenly become truthful? And what if two opposing philosophies were both “useful?” How, then, could “truth” be determined? Neither is this logical or a theory of truth.
Social Constructivism Theory of Truth: Social processes construct Truth, and it represents the power struggles within a community. Critical constructionism recognizes that all our knowledge is “constructed;” it does not reflect any externally “transcendent” realities, but is contingent on convention, human perception, and social experience. It is believed that even physical and biological reality, including race, nation, gender, are socially constructed. Similar in practice to Subjectivism and Skepticism, this theory denies that man can objectively know almost anything, since we see through the artificial filter of our own perceptions and conditioning. It asserts that language and cognition do not actually apply to reality. Of course, this is self-contradictory, for to assert Social Constructivism is to espouse a human interpretation of reality that would be “constructed” and thus unreal. With such a philosophy it is impossible to come to any conclusions at all.
Redundancy Theory of Truth: Also called the Deflationary Theory of Truth. This proposes that to assert that a statement is true is just to assert the statement itself. Thus, to say "It is true that snow is white" is to say nothing more or less than that show is white. Of course, the same would apply to stating the Redundancy Theory of Truth. This is more so a theory of language than one of truth and does not convey a meaningful objection to any of the classical theories. Debates concerning this theory are more likely to have participants talking past each other.
Functionalism Theory of Truth: This purports that Truth should be classified into different “functions” or categories depending on which field of interest on examines. This means that metaphysical “truth” is different from ethical “truth” or aesthetic “truth.” We grant that these are different kinds of the larger concept of “truth,” yet they are not so independent as to be groupable into such distinct categories. All of life is interrelated, and reality is knit together to form the grand tapestry that is the universe; to slice different aspects of reality into different constituent parts is to severely hamper one’s understanding in other fields. Man can understand truth from different perspectives, but that does not change the objective facts of existence.
Semantic Theory of truth: This espouses that any assertion that a proposition is true can be made only as a formal requirement regarding the language in which the proposition itself is expressed. This basically means that statements concerning truth-claims cannot express a truth within the confines of the same language, i.e. that semantic limitations make it impossible for the term “true” to be part of a claim. This theory was originally advanced as a possible solution to the liar paradox (“This statement is false.”) and other linguistic problems in philosophy. It primarily concerns the nature of language and really has little to do with metaphysical “truth.”