# How many kinds of contradictions are there?

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?

• Re your "the statement ¬(0 = 1) could be thought of as a suppressed premise" Why are you not confident about this elementary arithmetic proposition even one never heard of PA or ACA?... Commented Aug 11 at 6:13
• A contradiction is a statement that, when taken together with a set of other statements, leads to a falsehood. Commented Aug 11 at 6:27
• @causative, that's a property, although not a uniquely identifying one, of a contradiction. Commented Aug 11 at 6:31
• There is only one "kind", a set of statements is said to be inconsistent or contradictory when both p and ¬p can be derived from it according to accepted rules of inference and background axioms. Often the word "contradiction" is loosely applied to any such set. 0 = 1 is contradictory in the context of standard (Peano) arithmetic because the latter derives ¬(0 = 1). Commented Aug 11 at 6:31
• What if there's another kind of always false statement? There was a question on Chosmky's "colorless green ideas sleep furiously". It has a contradiction viz. colorless green, but it seems possible to construct a Chomsky sentence without a contradiction e.g. "trees barked in the coughing night". Is this sentence true/false? If it's false, is it always false, like a contradiction? Commented Aug 11 at 6:35

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.

• Gracias, It seems I'm looking for a statement, other than a downright contradiction, that's false in all possible worlds. What is false about all possible worlds, but is not a contradiction? What would you say about the modal claim life is necessary. Commented Aug 12 at 1:55
• Life is necessary for what? Why wouldn't there be possible worlds where there is no life? There are different accounts of necessity and under some a proposition may be necessarily false without being a contradiction. For example, Kripke considers "water is H2O" to be necessary because 'water' is a rigid designator. So there is no PW in which water is not H2O, but, "water is not H2O" is not a contradiction. I say a little more in my answer to this question. Commented Aug 12 at 2:41
• A not-bad but alas, not perfect example. I'm looking for a statement P that's false in all possible worlds but is not a contradiction. If life isn't necessary in all possible worlds then to say it is would be false in all possible worlds, oui? This is distinct from saying life is necessary in all possible worlds & life is not necessary in all possible worlds, oui? Commented Aug 12 at 3:28

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.

• Ok, so is there a elemental/compound (atomic/molecular) statement that doesn't both assert and deny itself but is a contradiction? In a restrictive sense, say we take this world in which we inhabit. We know water is a metal = W is false and we know Lions are birds = L is false. So we know W and L by themselves or W ∧ L are/is false. These are not contradictory, but they are false. Commented Aug 11 at 8:09
• Well, I suppose one can make definitive statements here only provided one can formalize the notion of a "lion" mathematically :-) But I happen to think that there is indeed a contradiction here. Let p be the proposition that lions are not birds. We know that p is true. Therefore if you claim "lions are birds", in effect in your system you have "p and not-p". Commented Aug 11 at 8:12
• I see, that's a good point, but a contradiction is false in all possible worlds and it's possible for lions to be birds in some world (contingency as opposed to necessity). Commented Aug 11 at 8:16
• OK well I did mention that my comments are modulo being able to formalize lions :-) Incidentally, your point is very well taken: Leibniz wrote extensively on the difference between (accidental) impossibility and absolute impossibility (a.k.a. contradiction). If you are interested, I have some articles on the subject. Commented Aug 11 at 8:17
• @Speakpigeon, I admit you are a bigger expert on lions than myself :-) Commented Aug 11 at 11:26

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: xx}, 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).

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).

• Most of that ... over my noggin sir/madam. But ... interesting that for Frege the set of true contradictions is { }, which means he subscribed to absolute denial, the strong(est) form of negation there is I suppose. I didn't quite get your take on weak negation. For me weak ¬p would mean I'm not categorically ruling out p i.e. it's possible that p ∧ ¬p (pseudocontradiction?). Perhaps this is equivalent to ¬physical doesn't imply nonphysical. Commented Aug 11 at 10:56
• Just curious, is it possible for p ∧ q, in general for p ⊗ q (where ⊗ is some logical connective) to be a contradiction, where q ≠ ¬p and that q is itself not some combination of p ∧ ¬p? Commented Aug 11 at 11:11
• @Hudjefa don't sell yourself short, per your comments you do seem to understand this stuff aright. Now as for whether there are contradictions without negation, hmm, not quite BUT have you heard of the Curry paradox? Commented Aug 11 at 14:31

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.

• Good post. I don't think logicians are the ones stating/claiming p ∧ ¬p. It comes up when working with inconsistent premises. The question then is should we "accept" it? We can't (ex contradictione sequitur quodlibet). That's the point to assuming p ∧ ¬p, oui? Cogito Commented Aug 11 at 11:04
• @Hudjefa People who claim that (p ∧ ¬p) → q is true ipso facto claim implicitly that p ∧ ¬p is possibly true, and thereby contradict their posture that they are reasoning logically. Many if not most academic self-described logicians today insist that (p ∧ ¬p) → q is true. Ergo, they are not logicians. Commented Aug 11 at 11:14
• Ok cogito I can see where you're coming from. Can you answer my 1st question: Is there a "contradiction" that where it's not simply contra (denying) diction (saying). My examples (scattered in my posts) always have a component that is a contra-diction. Commented Aug 11 at 11:17
• @Hudjefa "My examples (scattered in my posts) always have a component that is a contra-diction." So what? p ∨ (r ∧ ¬r) is not a contraction although it contains a contradiction. A car is not a wheel just because there is a wheel in the car or because it has wheels. Commented Aug 11 at 11:34
• @speakpigeon You’ve never substantiated the claim that A->B implies that ‘A’ is possible, which is a major stretch. Commented Aug 11 at 12:28