Rough guidelines for revising logic
What are our rough guidelines for revising logic? Typically, when the issue arises -- say, because of semantic paradoxes, vagueness, or quantum considerations -- the majority answer is: Don't do it! Even today, most philosopher (I conjecture) are endowed with some sort of instinct of preserving logic in the face of pressure to revise their theories.
Now, my polemic is not against preserving logic in and for itself, and not even against the (partly historical) fact that what is being preserved is classical logic (although I find the claim that our logic is classical logic somewhat puzzling). There is a lot to say about both of these issues: first, Quinean considerations could lead one to have a more relaxed view about giving up ones logic; second, precisely what we mean when we say that logic is being revised is ambiguous between revising a practice and revising a set of beliefs (see Priest's Doubt Truth to be a Liar, ch. 10).
So, setting aside the anti-revisionary crowd for the purpose of this post, let me introduce a second group, the moderates. The moderates are characterised by the following slogan: Save as much of classical logic as possible. This idea started interesting me after reading Maudlin and Field on the semantic paradoxes this spring. Both of them advocate some sort of moderate view where we give up just enough of classical logic to avoid the paradoxes. (Note, however, that Field describes Maudlin in a different way, as preserving classical logic and revising the T-inferences. See Field's review of Maudlin's book.) Maudlin and Field have in common that they prefer the paracomplete approach where the diagnosis is that LEM (together with reductio) is the culprit (I set issues about the Löb-Curry paradox aside). Others, such as McGee (1990), locates the problem with proof by cases (disjunction elimination), but neither Maudlin nor Field is tempted by this. (Again, note that Maudlin's view is twofold: Reductio is not truth-preserving, but there is a further issue of preservation of permissible assertion. This is why Field takes Maudlin as a proponent of a classical approach.)
The interesting question is what the underlying guidelines for dropping one inference rule rather than another. Everyone in the debate seems to acknowledge that there are more than one option, but they differ as to the analysis of why one rather than another rule ought to be revised (take LEM to be a zero-premise rule). The problem is this: On the backdrop of the general desideratum 'Save as much of classical logic as possible', what is the most rational revision?
For simplicity, let us assume that we're in a debate where the options are (i) weaken LEM or (ii) weaken proof by cases. Any sufficient revision (weakening) of these rules will involve abandoning classical logic in favour of some weaker system. But is there a sense in which revising one is worse than the other? There are two broad approaches here.
Perhaps one should stoop down to the level of intuitions and argue that, e.g., weakening proof by cases leaves our reasoning powers dead in the water, whereas LEM was always a bit dubious anyway (pardon the parody). Needless to say, there will philosophers whose intuitions pull in the opposite direction.
On the other hand, perhaps there is some metric in the calculus one might use to measure how damaging the revision is. The two options might yield different results for the inference system at large, e.g., in terms of which other principles fail (although then these other principles must be weighted somehow too), or in terms of complexity issues (how will the revisions affect proof lengths).
Both approaches have their disadvantages. Not only do the intuitions differ widely, they are largely untested outside the philosophical community. True, there are results in experimental psychology showing how even basic logical principles are dissented from in certain contexts. But, allowing such data to dictate our choices of revision might lead us astray. Famously, even principles we don't want to give up (modus ponens -- McGee notwithstanding) come under pressure in experimental testing.
Similarly, any attempt at giving a formal metric for the revision faces the problem that facts about interrelations between rules, which rules are immediate/mediate, derived/primitive, and facts about proof length, varies with the particular formalization we pick. Unless one wants to defend a particular version of, say, classical natural deduction over another, there's no straightforward way of measuring the results of the candidate revisions.
Now, my polemic is not against preserving logic in and for itself, and not even against the (partly historical) fact that what is being preserved is classical logic (although I find the claim that our logic is classical logic somewhat puzzling). There is a lot to say about both of these issues: first, Quinean considerations could lead one to have a more relaxed view about giving up ones logic; second, precisely what we mean when we say that logic is being revised is ambiguous between revising a practice and revising a set of beliefs (see Priest's Doubt Truth to be a Liar, ch. 10).
So, setting aside the anti-revisionary crowd for the purpose of this post, let me introduce a second group, the moderates. The moderates are characterised by the following slogan: Save as much of classical logic as possible. This idea started interesting me after reading Maudlin and Field on the semantic paradoxes this spring. Both of them advocate some sort of moderate view where we give up just enough of classical logic to avoid the paradoxes. (Note, however, that Field describes Maudlin in a different way, as preserving classical logic and revising the T-inferences. See Field's review of Maudlin's book.) Maudlin and Field have in common that they prefer the paracomplete approach where the diagnosis is that LEM (together with reductio) is the culprit (I set issues about the Löb-Curry paradox aside). Others, such as McGee (1990), locates the problem with proof by cases (disjunction elimination), but neither Maudlin nor Field is tempted by this. (Again, note that Maudlin's view is twofold: Reductio is not truth-preserving, but there is a further issue of preservation of permissible assertion. This is why Field takes Maudlin as a proponent of a classical approach.)
The interesting question is what the underlying guidelines for dropping one inference rule rather than another. Everyone in the debate seems to acknowledge that there are more than one option, but they differ as to the analysis of why one rather than another rule ought to be revised (take LEM to be a zero-premise rule). The problem is this: On the backdrop of the general desideratum 'Save as much of classical logic as possible', what is the most rational revision?
For simplicity, let us assume that we're in a debate where the options are (i) weaken LEM or (ii) weaken proof by cases. Any sufficient revision (weakening) of these rules will involve abandoning classical logic in favour of some weaker system. But is there a sense in which revising one is worse than the other? There are two broad approaches here.
Perhaps one should stoop down to the level of intuitions and argue that, e.g., weakening proof by cases leaves our reasoning powers dead in the water, whereas LEM was always a bit dubious anyway (pardon the parody). Needless to say, there will philosophers whose intuitions pull in the opposite direction.
On the other hand, perhaps there is some metric in the calculus one might use to measure how damaging the revision is. The two options might yield different results for the inference system at large, e.g., in terms of which other principles fail (although then these other principles must be weighted somehow too), or in terms of complexity issues (how will the revisions affect proof lengths).
Both approaches have their disadvantages. Not only do the intuitions differ widely, they are largely untested outside the philosophical community. True, there are results in experimental psychology showing how even basic logical principles are dissented from in certain contexts. But, allowing such data to dictate our choices of revision might lead us astray. Famously, even principles we don't want to give up (modus ponens -- McGee notwithstanding) come under pressure in experimental testing.
Similarly, any attempt at giving a formal metric for the revision faces the problem that facts about interrelations between rules, which rules are immediate/mediate, derived/primitive, and facts about proof length, varies with the particular formalization we pick. Unless one wants to defend a particular version of, say, classical natural deduction over another, there's no straightforward way of measuring the results of the candidate revisions.
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