On a relevance criterion

Logicians study logics – plural. There are different logics for different reasoning tasks. Classical logic, the flavour taught to undergraduate students of all persuasions, falls apart when confronted with the kinds of reasoning that people do effortlessly every day. My favourite way to break classical logic involves an innocent “if” and “or”.

Ponder the following sentence (based on an example by Alf Ross, 1944):

If Alex posted the letter [P], then Alex posted the letter [P] or Alex set fire to the letter [F].

If you think this sentence is true, then your interpretation and reasoning are compatible with translating it into classical logic using the material conditional (\(\Rightarrow\)) for the “if” and inclusive disjunction (\(\lor\)) for the “or”. You could write it like this and it’s trivially true: \(P \Rightarrow (P \lor F)\).

Some people are perfectly content with this interpretation, but many think the sentence is fishy and false.  There are a number of ways to explain what has happened.

One is to assume that the issue is language pragmatics rather than logic. Pragmatics studies the ways in which context and social conventions for communication affect people’s interpretation of language. According to one theory of communication (see Liza Verhoeven’s 2007 explanation), asserting that you posted the letter or burned it under the assumption that you posted it violates principles of cooperativeness. These principles affect the meaning of a sentence and its truth, so in this case the sentence is false.

Another way to make sense of what has gone wrong is using a relevance criterion devised by Gerhard Schurz (1991). The first step we need to take is to transform the “if” into an argument with a single premise and conclusion.

Premise: Alex posted the letter [P].
Conclusion: Alex posted the letter [P] or Alex set fire to the letter [F].

This is an uncontroversial step in classical logic, e.g., application of a rule for introducing an “if” in natural deduction.

Schurz introduces a criterion for a conclusion relevance that roughly goes as follows. The starting point is an argument that is valid according to classical logic. That’s the case for the argument above. If there are any terms in the conclusion that can be substituted with arbitrary alternatives without affecting the argument’s validity, then the conclusion is irrelevant. Otherwise the conclusion is relevant.

For our letter example, we can replace “Alex set fire to the letter” with anything and it has no effect on the validity of the argument. Alex opened the letter. Alex scribbled on the letter. Alex swallowed the letter. The letter was a surrealist painting. The letter was the size of house. And so on. No substitution in the second half of the conclusion can affect the validity of the argument, so the conclusion is irrelevant.

How about an argument where the conclusion is relevant? The trick is to ensure that everything in the conclusion is… relevant. That’s what I like about the criterion: it formalises (and the details are fiddly) an intuitive property of arguments. Here’s an easy example:

Premise: It’s raining and I left my umbrella at home
Conclusion: I left my umbrella at home and it’s raining

This is an example of the conjunction, “and”, being commutative in classical logic: the order of the conjuncts in the sentence (the parts on either side of “and”) doesn’t affect its truth. There are many ways to edit the conclusion so that the argument is no longer valid. For instance replace one or both of the conjuncts with “I posted a letter”. Then the conclusion doesn’t follow from the premise since the premise doesn’t tell us anything about a letter.

Colleagues and I explored people’s interpretations of these kinds of sentence about a decade ago in the context of an alleged paradigm shift in the psychology of reasoning. Read all about it. I was reminded of this again as Google Scholar dutifully notified me that Michał Sikorski recently cited it (thank you kindy Michał!).

Science for the half-wits

A bit from Jean Yves-Girard‘s latest rant, The phantom of transparency:

Still under the heading « science for the half-wits », let us mention non monotonic « logics ». They belong in our discussion because of the fantasy of completeness, i.e., of the answer to all questions. Here, the slogan is what is not provable is false : one thus seeks a completion by adding unprovable statements. Every person with a minimum of logical culture knows that this completion (that would yield transparency) is fundamentally impossible, because of the undecidability of the halting problem, in other terms, of incompleteness, which has been rightly named : it denotes, not a want with respect to a preexisiting totality, but the fundamentally incomplete nature of the cognitive process.

Logic and Reasoning: do the facts matter?

Had a read of Logic and reasoning: do the facts matter? by Johan van Benthem. Covers much ground in a short space but I found it thought provoking. Here’s a quick sketch of the bits I liked.

Van Benthem mentions the anti-psychologism stance, briefly the idea that human practice cannot tell us what correct reasoning is. He contrasts Frege’s view with that of Wundt; the latter, he argues, was too close to practice; Frege was too far. He argues that if logics were totally inconsistent with real practice then they’d be useless.

Much logic is about going beyond what classical logic has to offer and is driven by real language use. Van Bentham cites Prior’s work on temporal structure, Lewis and Stalnaker’s work on comparative orderings of worlds, work on generalised quantifiers which was driven by the mess of real language and for instance produced formalisations of quantifiers like most and few. Generally, van Bentham argues, “one needs to move closer to the goal of providing more direct and faithful mathematical renderings of what seem to be stable reasoning practices.” You want your logic to be more natural, closer to the phenomena. Conceptions of mathematical logic were driven by the terms that appeared in rigorous proofs, so the linguistic stuff is just widening the set of practices that are modelled.

Correctability in a logic is more important than correctness, he argues. This is consistent with the goals of the non-monotonic logic crowd I know and love. I find this most interesting when looking at individual differences in reasoning processes: perhaps a correctability dimension is out there somewhere, if only we could measure it and its correlates. I have some ideas—stay tuned.

Divergences from competence criteria, he argues, suggest new practices. I still see many papers in which people are scored against classical logic. Failure should cause an attempt to work out what practice is being followed by a person rather than the more common concern of what went wrong and how we could bring people back.

Much more in this little paper…