drug_mentor
Bluelight Crew
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Some great background reading for this topic, for anyone who is interested, is Anderson & Belnap's 1962 article 'Tautological Entailments', which can be found here. For the most part, when I share articles in P&S I do try to make them accessible. This article is no different, however, it does presuppose some knowledge of classical logic, including concepts like Disjunctive and Conjunctive Normal Forms.
Classical logic validates a principle which is often referred to as 'the principle of explosion'. In a nutshell, this principle says that anything may be inferred from a contradiction, more formally it is stated thus: A&~A |- B - at first it may seem incredible that anyone has supposed this is a valid inference. Here is an argument in support of its validity:
Grant the following inferences are valid:
(1) From 'A and B' infer A
(2) From 'A and B' infer B
(3) From A infer 'A or B'
(4) From 'A or B' and 'not-A' infer B
P1. A and not-A - Assumption
P2. A - From P1 by inference (1)
P3. A or B - From P2 by inference (3)
P4. not-A - From P1 by inference (2)
C1. B - From P3 and P4 by inference (4)
It seems that if one is willing to grant that inferences (1) through (4) are valid then the argument goes through. In order to deny that the principle of explosion is valid, we have to deny that at least one of the inferences (1) through (4) is a deductively valid inference. I think inferences (1) and (2) are fairly uncontestable. Anderson & Belnap's strategy is to deny that (4), frequently called disjunctive syllogism, is a valid form of inference. They don't really mount what I take to be a thoroughgoing argument against the disjunctive syllogism, and mostly argue that it is invalid because it leads to explosion. Of course, there is a bit more to their (rather interesting) argument than this, but I won't hash out the details here (those interested should consult the link at the start of this post). Another strategy would be to deny (3), I am not sure that I find this strategy plausible, I think reflection on the meaning of 'or' indicates that whenever A is true, 'either A is true or B is true' is also true. I think the prima facie plausibility of denying the validity of (3) derives from the thought that it would be extremely peculiar for someone to assert in everyday argument 'A is true, therefore A or B is true'; however, the fact that it is peculiar does not necessarily suggest that it is invalid.
So what do people think, is the principle of explosion valid? If so, why? If not, which of (1)-(4) do you think is not a deductively valid inference?
Classical logic validates a principle which is often referred to as 'the principle of explosion'. In a nutshell, this principle says that anything may be inferred from a contradiction, more formally it is stated thus: A&~A |- B - at first it may seem incredible that anyone has supposed this is a valid inference. Here is an argument in support of its validity:
Grant the following inferences are valid:
(1) From 'A and B' infer A
(2) From 'A and B' infer B
(3) From A infer 'A or B'
(4) From 'A or B' and 'not-A' infer B
P1. A and not-A - Assumption
P2. A - From P1 by inference (1)
P3. A or B - From P2 by inference (3)
P4. not-A - From P1 by inference (2)
C1. B - From P3 and P4 by inference (4)
It seems that if one is willing to grant that inferences (1) through (4) are valid then the argument goes through. In order to deny that the principle of explosion is valid, we have to deny that at least one of the inferences (1) through (4) is a deductively valid inference. I think inferences (1) and (2) are fairly uncontestable. Anderson & Belnap's strategy is to deny that (4), frequently called disjunctive syllogism, is a valid form of inference. They don't really mount what I take to be a thoroughgoing argument against the disjunctive syllogism, and mostly argue that it is invalid because it leads to explosion. Of course, there is a bit more to their (rather interesting) argument than this, but I won't hash out the details here (those interested should consult the link at the start of this post). Another strategy would be to deny (3), I am not sure that I find this strategy plausible, I think reflection on the meaning of 'or' indicates that whenever A is true, 'either A is true or B is true' is also true. I think the prima facie plausibility of denying the validity of (3) derives from the thought that it would be extremely peculiar for someone to assert in everyday argument 'A is true, therefore A or B is true'; however, the fact that it is peculiar does not necessarily suggest that it is invalid.
So what do people think, is the principle of explosion valid? If so, why? If not, which of (1)-(4) do you think is not a deductively valid inference?