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The day kicked off with Per Martin-Löf presenting ‘Is Logic about Consequence?’ I really enjoyed finally getting the opportunity to hear Per speak, as his view of logical consequence – a particular constructive type theory in which proof objects play a central role – has always fascinated me, but I’ve not looked into it in any depth. Per argued that logic is not primarily about consequence, but must be also about the act of inference, and that distinguishing content and force in the judgement helps us understand the difference.
I think that this is clearly right, and in my work since the Logical Pluralism book I’ve started to attend to these issues. In “Multiple Conclusions” I attended to the connection between valid sequents and the acts of assertion and denial. While the account I give there is not one that Per would endorse, the constellation of issues connecting content and force, and the correctness of the steps we make in inferring conclusions from premises, are certainly in the domain of logic.
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Then we had Dag Westerståhl with ‘Sources of Plurality.’ In this talk, Dag started off with our analysis of logical consequence as truth preservation in cases – and distinguished this from interpretational accounts, in which it is truth that counts, not hypothetical truth-in-some-case. Consequence differs from material consequence by the choice of logical vocabulary to keep fixed, and to vary the other vocabulary arbitrarily. Varying the choice of vocabulary gives you a range of different consequence relations. Dag had interesting results showing how to construct the vocabulary from the consequence relation rather than vice versa.
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After lunch, we had Agustín Rayo working ‘Towards a Trivialist Account of Logic.’ This wasn’t really about pluralism as such, but rather, was about one sort of account of logical truths as statements which are, in a certain sense, trivial, those statements whose denial is unintelligible. Agustín connected the notion of intelligibility with identity (he argued that if you take a to be identical to b, then the de re consideration of a case where a is not b will strictly speaking be unintelligible) with modality (understood as truth in all intelligible scenarios) and why questions.
This was an interesting trinity of considerations, and most interesting for me was the methodological points Agustín made about the kinds of choice we have in forming vocabulary. The more identities we endorse, then the more scenarios become unintelligible, and the more why questions we cannot answer. (According to Agustín there is no answer to the question of why water is H2O, or why Hesperus is Phosphorous.) On the other hand, the more identities we endorse, the more consequences are derivable.
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Agustín was followed by Marcus Rossberg with ‘Pluralism about Logic Proper,’ in which Marcus wanted to reconcile a Fregean view taking the topic-neutrality of logic seriously, with the pluralist sympathies of Carnap. Marcus developed the idea of a topic, for which different logics may be appropriate (say, constructive mathematics; quantum objects; etc.) and attempted to reconcile pluralism with a proper ’lower limit’ logic which is most general and applicable for all topics.
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Day 3 was wrapped up by Dag Prawitz who presented ‘Kinds of Pluralism in Logic’
Dag’s talk started off with an excellent summary of the kind of logical pluralism favoured by JC and me, and then he proceeded to inquire about what logic ought to say about an inference: a transition to a conclusion C from premises P, Q, etc. He did not think that either of the claims made in the book – (1) that if an argument is invalid then the inference from premises to conclusion makes a mistake or (2) that if an argument is valid, then it’s a mistake to assert the premises and deny the conclusion – sufficed to give an account of what is going on. Dag’s preferred story is that valid arguments provide a way to transform the grounds of the premises into a ground for the conclusion, and that if an argument is known to be valid, then the infer-er can use the argument to supply the ground for the conclusion of the argument.
There’s a lot to like about this view, I think. It has an account of where the difference lies in knowing that an argument is valid, by showing that this gives you a means to get from the premises to the conclusion, and there are very well developed theory of proofs (say, Per Martin-Löf’s proof objects or realisability semantics, or some other theory) which can give us accounts of the structures of such grounds and the way we can manipulate them (by taking them together in pairs; by applying one to another in modus ponens, etc).
Dag thought that this removed the need/desire for plurality, but I’m not sure about this. It seems to me that you could accept this, and think that there was a plurality, say, between deductive conseqence and formal consequence; and between classical formal consequence and intuitionist formal consequence; and perhaps also into relevant consequence — if one had different accounts of what were allowable techniques for transforming one ground into another. Mathematical reasoning, for example, uses recursion as a technique for constructing a new ground out of old grounds, and this does not play a role in purely formal logical consequence. You might also say that classical reasoning uses a technique for constructing grounds (continuations, in the double-negation translation, or something more complicated with positive and negative ‘grounds’ in a symmetric proof system) that are not permitted with intuitionist ground-formation. The pluralist view of the move from premise to conclusion remains, but translated here into a very different key.
I think there’s lots more to think about here, and I’m looking forward to attempting to work some of it out, to see if the sketches I’m making here actually bear fruit. If you do any thinking on these issues – and get anywhere with them – please let me know.
And that was it! It was a wonderful conference – Daniel, Marcus and Peter did a wonderful job organising, and we all enjoyed ourselves immensely. It was good to catch up with old friends and to make new ones. I’ve left with lots more things to think about, and some new ways to do that thinking. You can’t ask for anything more from a conference.
I’m writing this final report late in the evening in Tallinn. Tomorrow I take the long journey back to Melbourne, via Amsterdam and Singapore. Hopefully these conference reports have been useful if you weren’t able to come to the conference. Writing them helped me collect my thoughts.
