According to the Gospels, there was an inscription above Christ on the cross which said (in English translation):
Matthew (27.37): “This is Jesus, the King of the Jews. ” (ABD)
Mark (15.27): “The King of the Jews.” (D)
Luke (23.38): “This is the King of the Jews.” (AD)
John (19.19): “Jesus of Nazareth, the King of the Jews. ” (BCD)
Note that the versions are composed of these phrases which appear in this order: (A) “This is”, (B) “Jesus”, (C) “of Nazareth, (D) “the King of the Jews.” Hence the capital letters in parentheses above.
What did the inscription say? If we insist that every true statement must tell “the truth, the whole truth, and nothing but the truth,” then at most one of these versions is true. If we expect every true statement to be consistent with the others though perhaps incomplete, then we would conclude that their union is the complete (or more complete) truth: “This is Jesus of Nazareth, the King of the Jews” (ABCD). If we expect every true statement to contain the truth but may be partially inconsistent with others, then we would conclude that their intersection is the whole truth: “The King of the Jews,” (D) the version Mark has.
We often need to analyze statements from different sources and then determine which ones are correct or how correct statements can best be extracted or reconstructed from the sources. So we could analyze the statements into four categories:
(1) The statements we are confident are exactly true, that is, they are consistent and complete representations of what is the case. The reason may be because they are from sources that are fully trusted, or we have knowledge which confirms them, etc. The conclusion should be the maximally consistent and complete set of statements.
(2) The statements that are consistent with statements in (1) and with each other but are partial statements of what is the case. They may be from reliable sources which aren’t expected to express full knowledge of what is the case. They are consistent but incomplete representations of what is the case. The conclusion should be the maximally consistent set of statements.
(3) The statements that contain the complete truth but may be partially inconsistent with each other. Perhaps they are different views of the same subject or they’re garbled messages but still reflect an authentic source. They contain a common truth that is consistent with (1) and with each other and so can be safely accepted. The conclusion should be the maximally complete set of statements.
(4) The statements that are rejected, that is, considered false.
In category (1) the default value for statements is false. If there’s anything wrong with a statement, it should be considered false. Only if it is found to be completely and consistently true should it be accepted as true.
Standard (Fregean) logic deals with category (1). If a statement in category (1) is accepted as true but turns out to be false in any way, it would be disastrous. Material implication would “explode” and any statement could be inferred. So statements in (1) must be exactly true or else rejected and the conclusion must be a tautology.
In categories (2) and (3) the default value for statements is true. If there’s anything true in a statement, it should be considered usable for determining conclusions. Only if it is found to be totally contradictory (2) or completely false (3) should it be rejected as false.
In standard logic, contradiction is concerned with whether or not a statement can be both true and false. The standard Principle of Non-Contradiction asserts that no statement can be both true and false. The standard Principle of Excluded Middle also applies to any statement. It asserts that every statement is either true or false.
In Aristotelian logic, contradiction is a relation between two statements. Given a statement and its negation: The Principle of Non-Contradiction asserts that at most one is true; both can be false. The Principle of Excluded Middle asserts that at least one is true; both can be true.
In standard logic, the Principle of Double Negation (p equals not not p) implies both these principles so they tend to be combined into one Principle of Bi-Valence: every statement is either true but not false or false but not true and no other values are allowed.
The principles of Aristotelian logic are better suited to deal with the statements in (2) or (3). In (2), the Principle of Non-Contradiction is upheld to protect consistency. In (3) the Principle of Excluded Middle holds. Both principles hold in categories (1) and (4).
Categories (2) and (3) complement one another. By exchanging consistency and completeness they can be mapped to one another. This gives us a clue as to how to solve the case of category (3): transform them into category (2) and solve them.