How many kinds of contradictions are there?

Question

I'm familiar with contradictions as the compound/molecular statement p ∧ ¬p. There's no truth-value combination of p (and ¬p) that can make p ∧ ¬p true. Stated differently, no possible world (as defined by different truth value combinations) exists where the statement p ∧ ¬p is true.

As you can see another way of defining a contradiction is a statement (atomic/molecular) for which all lines in its truth table evaluate to false.

A coupla questions:

1. Inconsistency is descriptive of a set of statements such that no line in its truth table has all of the individual component statements true. This results in the conjunction of all these statements evaluating to false for each line in its truth table. That means an inconsistent set of statements satisfies the truth-table definition of a contradiction (all lines evaluate to false). What's interesting/not about this is that we have the situation (where p, q, r, ... are the statements that form an inconsistent set of statements) p ∧ q ∧ r ∧ ... being false in all possible worlds but it is not of the form p ∧ ¬p. What does this mean I wonder? Do we have a species of contradiction different from p ∧ ¬p?

2. 0 = 1 is declared on many sites as a contradiction. I have trouble with this because it doesn't fit the mold of a contradiction p ∧ ¬p. Where's the other statement? 0 = 1 is atomic and contradictions are molecular (conjunction of a statement and its denial). Say you have a mathematical argument like below:
   Assumption A
   Ergo 0 = 1
   We can see that we've arrived at a falsehood. Using modus tollens we can then conclude the falsehood of the assumption A because the argument boils down to Assumption A ⇒ 0 = 1 and ¬(0 = 1). Ergo ¬Assumption A.
   How is 0 = 1 a contradiction? My best guess is that we know 0 = 1 is false and the (erroneous) conclusion is that 0 = 1 is true and just like that we have 0 = 1 ∧ ¬(0 = 1); the statement ¬(0 = 1) could be thought of as a suppressed premise. Correct/incorrect/both/neither?

Answer 1

In traditional logical terminology, i.e. ancient and medieval, a contradiction referred to a pair of propositions. Two propositions are contrary if they cannot both be true, subcontrary if they cannot both be false, and contradictory if they are both contrary and subcontrary, i.e. if the truth of one entails the falsity of the other and vice versa. So, P, not-P is a pair of propositions that are contradictory for any P.

In modern usage, the term contradiction has come to refer to any proposition that is a logical falsehood. We might express this proof-theoretically by saying it proves falsum (⊥), or model-theoretically by saying that it is false under all interpretations. So, P ∧ ¬P is a single sentence that is a contradiction in the modern sense. But it is only a special case of a contradiction. Any arbitrarily complex sentence that is false under all interpretations is also a contradiction. It is a little bit of stretch to call 0=1 a contradiction, since this is dragging arithmetic into the issue, and the symbols '0' and '1' have to be interpreted within a theory. Though it would be OK to say that ¬(0=0) is a contradiction, since this is false under any interpretation of '0'.

That said, there are dialetheists such as Graham Priest who hold that some contradictions are true. They are using contradiction in a more limited sense of a proposition of the form P and not-P, or similar.

Answer 2

At least in classical logic, once you reach a contradiction, you can prove every assertion, and in particular 0=1 (which is evidently false). Such a phenomenon is sometimes referred to as "explosion" by logicians. Thus "0=1" is simply the shortest way of signaling that a contradiction has been reached. As far as (p ∧ ¬p) ∧ (q ∧ ¬q) is concerned, it is not in the form "assertion and its negation", but it certainly implies p ∧ ¬p, which therefore implies 0=1.

Answer 3

We (who, though?) are nowadays more accustomed to inter-sentence talk of inconsistency, though Kant for example said something about predicates that contradict subjects in such a way as to "cancel out" the meanings of some sentences.

This pertains to the duality of identity and negation themselves. So in set theory, for example, the empty set is {x: x ≠ x}, since no x presumably satisfies that formula; and V is almost exactly the same as such, {x: x = x}, so that V contains everything that can be contained (has as elements all things that are in the way of being elements). (For a "quirky," at-an-angle-relevant article that mentions Frege's empty set as "the set of true contradictions," see here.) So per the string, "0 = 1," this can be seen as, "The False is the True," or then, "The negation of a thing is equal to that thing's identity," which is why they choose that representation, rather than 119 = 2223 (or even rather than 0 ≠ 0 or 1 ≠ 1).

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Incidentally, given how many and varied are the concepts or types of "not" out there, it remains to ask if A&~A is always wrong but dependent on which kind of ~ we mean? We might imagine "weak" and "strong" negations, for example, and claim that A and ~A can be weakly true together, but not strongly so, etc. Indeed, to some extent we already do this when we note that the absence-of-X is not necessarily the same as the presence-of-the-opposite-of-X ("not believing" is not the same as "believing not," for example).

Answer 4

How many kinds of contradictions are there?

Just one, namely, p ∧ ¬p, if we count ¬p ∧ p as the same thing.

Contradiction literally means "to say" (diction) something "against" (contra). In p ∧ ¬p, ¬p is said against p (and vice versa).

This is not the case with merely inconsistent terms, for example with p ∧ r and p ∧ ¬r, which like a contradiction cannot be both true but, unlike a contradiction, may well be both false (if p is false).

Thus, contradictions and inconsistencies don't behave logically in the same way and have therefore to be distinguished.

You don't need to pay much attention to today's self-described logicians, for the theory which they support is not consistent with the facts of logic.

Most of today's self-described logicians would for example claim that p ∧ ¬p implies anything whatsoever, and yet this is patently false. We cannot infer anything from p ∧ ¬p precisely because p ∧ ¬p is a contradiction and therefore cannot be logically assumed to be true. Thus, p ∧ ¬p just doesn't imply anything.

0 = 1 is declared on many sites as a contradiction.

It is not.

0 = 1 is only contradictory, to the standard assumption, namely ¬(0 = 1).

A contradiction would be 0 = 1 ∧ ¬(0 = 1), but 0 = 1 in itself is not a contradiction.

This is crucial, because the falsehood of 0 = 1 depends entirely on our prior (standard) assumption that ¬(0 = 1) is true. Without this prior assumption, there is literally no logical reason to conclude that 0 = 1 is false. In other words, 0 = 1 is only contradictory because of our prior assumption that ¬(0 = 1) is true.

Unfortunately, most academic logicians confuse the words "contradictory" and "contradiction". Apparently, Aristotle already did. Yet, if you read carefully the literature before George Boole, there is not doubt that people, including Aristotle, made the distinction. In other words, the confusion results from shoddy language if it doesn't result from having adopted the wrong theory of logic.