Logic of Paradox: a (too) simple paraconsistent logic

by Stuart_Armstrong2 min read5th Apr 20125 comments

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The logic of paradox (LP) is the simplest, and one of the oldest, of the paraconsistent logics. Instead of assigning truths to statements A, it instead uses relationships binary relationships v(A,1) and v(A,0). "A is true" is encoded by v(A,1); "A is false" is similarly encoded by v(A,0). Each statement A is required to be true or false, but it can be both (in which case we could say that "A is undetermined"). "A is strictly true" means v(A,1) but not v(A,0); strict falsity is the converse.

The usual symbols →, ∨, ∧ and ¬, retain their standard meanings, and a compound statement takes all possible values it could take, seeing all the possible values its components could take. So, for instance if A is (strictly) true, then ¬A is (strictly) false. If A is undetermined, then ¬A is undetermined. If A is undetermined and B is strictly false, then A ∨ B is undetermined and A ∧ B is strictly false - though if B were strictly true, then A ∨ B would be strictly true and A ∧ B undetermined.

These properties make LP quite easy to work with, and one can determine the truths of many statements using truth tables. In fact, it can be seen that every tautology of classical logic is a tautology of LP. This derives from the fact that tautologies are true regardless of the truth values of their components; hence they remain true in LP whether we take undertermined statements to be true or false. Consequently, all of the following are true in LP:

  1. A  → (B → A)
  2. (A ∧ ¬A)  → B
  3. ((A ∨ B)  ∧ ¬A) → B
  4. A ∧ (A → B)  → B

But wait a second. Isn't the second line a statement of the principle of explosion - the fact that we can derive anything from a contradiction? Indeed it is. LP can state the principle of explosion as a (true) theorem - but it can't actually use it as a rule of deduction. Similarly, the third line is a statement of the disjunctive syllogism - a true theorem, but not a valid rule of deduction. That is easy to see: let A be undetermined, and B strictly false. Then A ∨ B is true, and so is ¬A - and yet we cannot deduce that B is true from this information.

So LP can accept contradictions without blowing up, has all the tautologies of classical logic, but lacks some of the rules of inference. "Some" of the rules of inference? LP even lacks modus ponens! As before, let A be undetermined and B strictly false; then A and (A → B) are both true, but B is not.

So while LP is a pleasant logic to play with, it isn't particularly useful. Another weakness is that is still defines the material conditional (A → B) as (¬A ∨ B): false statements still imply anything, and we haven't solved the Löbian problem for UDT. In the next post, I'll look at relevance logics, which have a more restricted use of →, and do allow modus ponens.

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Could you give the rules of inference (transformation) that are allowed? I'm guessing double negation introduction and elimination, conjunction introduction and elimination, and hypothetical syllogism all work. Is that right? Are there other valid rules? Does the logic have a standard axiomatization? (Another way to ask this is, "In what sense are the sentences you identified theorems of the logic?")

One worry. I'm not sure that what you say about explosion here is quite right. You say:

Isn't the second line a statement of the principle of explosion - the fact that we can derive anything from a contradiction? Indeed it is. LP can state the principle of explosion as a (true) theorem - but it can't actually use it as a rule of deduction.

The theorem is about a conditional sentence. The principle of explosion is about logical consequence. (Basically, this comes down to the difference between the conditional "->" and the turnstile "|-".)That is, insofar as the logic does not allow the sentence to function as a rule of inference, the sentence is not a statement of the principle of explosion in that logic. We might recognize it as explosive in classical logic. But in classical logic, we have modus ponens!

The theorem is about a conditional sentence. The principle of explosion is about logical consequence.

Yes, I'm not going to be making that confusion again :-)

The book I was reading refered to sentences like "A ∧ (A → B) → B" as modus ponens theorems; I've copied that usage, but I can change it if it's confusing.

I was under the impression that the only reason we use the syntax (A → B) in classical logic is that it mimics a rule of inference, and including the symbol makes it more intuitive (A ∧ (A → B) → B vs A ∧ (¬A ∨ B) → B). Given that there no longer is a modus ponens for → to refer to semantically, why bother using it as a symbol at all instead of restricting LP to the ¬A ∨ B syntax?

It would be interesting to see a comparison to Dodgson Logic Here is a reference: http://www.gutenberg.org/ebooks/4763

I thought that was just ordinary syllogistic logic, i.e. the first-order theory of the subset relation.