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Uva-DARE (Digital Academic Repository) UvA-DARE (Digital Academic Repository) Inference in action van Benthem, J. Publication date 2007 Document Version Final published version Published in Publications de l'Institut Mathématique - Nouvelle Série Link to publication Citation for published version (APA): van Benthem, J. (2007). Inference in action. Publications de l'Institut Mathématique - Nouvelle Série, 82(96), 3-16. http://www.emis.de/journals/PIMB/096/1.html General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date:24 Sep 2021 PUBLICATIONS DE L’INSTITUT MATHEMATIQUE´ Nouvelle s´erie, tome 82(96) (2007), 3–16 DOI 102298/PIM0796003B INFERENCE IN ACTION Johan van Benthem Abstract. Substructural logics arise whenever classical logic is put to new uses, and logicians from Serbia have been in the fore-front here. In this paper, we join the substructural tradition with another recent trend, viz. dynamic logic of information update. We show how these two approaches fit together, in particular, through a number of representation theorems concerning structural rules. The proper background for these results turn out to be modal and dynamic logics of cross-model relations. We connect this finding with recent accounts of generalized inference, interpolation, and preservation results. 1. Inference, structural rules, and information-producing actions Inference entangled with other information sources. In the 1980s, sev- eral interesting nonstandard notions of consequence P ⇒ C emerged, claiming to reflect features of our common sense reasoning. Circumscription in AI looked at conclusions C true only in minimal models of the premises P , with minimal- ity measured by some comparison order for model size or predicate interpretation. General non-monotonic logics followed up on this idea, high-lighting failures of classical principles much as those found earlier in conditional logic. Structural rules, i.e., abstract properties of an inference relation ⇒, seemed a natural focus for defining ‘styles of reasoning’, in terms of their basic mechanics. This idea was reinforced when it turned out that very different notions of consequence, such as the resource-conscious inferences found in categorial grammar (van Benthem 1991) have illuminating sets of structural rules setting them apart from others. Likewise, van Benthem 1996 showed how deviant structural rules emerge in a natural fashion when analyzing so-called dynamic semantics, emphasizing how inference and infor- mation change are intertwined in understanding and using language. In this paper I present some further thoughts on the notion of inference emerging from all this, and its entanglement with information update and general action. Before going to abstract structural rules and bare mechanics, however, con- sider an example. The Amsterdam Science Museum NEMO (http://www.nemo- amsterdam.nl/) organizes Kids’ Lectures ON Science for 8-year olds. While prepar- ing for this event, I wondered how to talk to such an audience? I came up with an example that goes back essentially to Antiquity: 2000 Mathematics Subject Classification: 03B45, 03B60. Key words and phrases: Modal logic, dynamic3 logic, non-classical logic, model theory. 4 VAN BENTHEM The Restaurant. “In a restaurant, your Father has ordered Fish, your Mother ordered Vegetarian, and you have Meat. Out of the kitchen comes some new person with the three plates. What will happen?” The children got excited, many little hands were raised, and one said: “He asks who has the Meat”. “Sure enough”, I said: “He asks, hears the answer, and puts the plate. What happens next?” Children said “He asks who has the Fish!” Then I asked once more what happens next. And now one could see the Light of Reason start shining in those little eyes. One girl shouted: “He does not ask!” Now, that is logic. In my view, the Restaurant is about the simplest realistic logical scenario (van Benthem 2007, van Benthem 2008). Several basic informational actions take place intertwined: questions, answers, and inferences, and the setting crucially involves more than one agent. Also, actions can be analyzed for their informational content after they have taken place, but they can also be planned beforehand. Thus there is no natural border-line here between inference and other actions that produce information. I would say that ‘logical analysis’ of even this basic scenario involves all of them – and a logical system should account explicitly for that interplay. Indeed, the same entanglement is found in Indian Logic, a tradition parallel to our western one, where various sources of obtaining information were treated on a par: including making an observation, drawing a conclusion,orasking someone! Dynamic inference over abstract transition models. Having said all this, let us first go to a very general abstract way of bringing actions into logic. We can view new propositions A dynamically as partial functions TA taking input states meeting the preconditions of update with A to output states: −−−−−−−−−TA → More generally, transition models M =(S, {TA | A ∈ Prop}) consist of infor- mation states S with a family of transition relations TA over these, one for each proposition A in some given set Prop. These suggest the following notion of infer- ence. A sequence of propositions P1,...,Pk dynamically implies conclusion C in transition model M, if any sequence of premise updates starting from any state in M ends in a fixed point for the conclusion: s T s ...T s s Cs whenever 1 p1 2 pk k+1, then k+1 k+1 We say the sequent P1,...,Pk ⇒ C is true in the model – M P1,...,Pk ⇒ C. Here P ,Q,R stand for finite sequences of propositions, and A, B, C for single ones. Dynamic inferential sequents lack the structural rules of classical consequence (van Benthem 1996, Chapter 7). Simple counter-examples refute Monotonicity, Con- traction, Permutation, or Reflexivity – and their general idea amounts to this: any sequential recipe for some desired effect may be disturbed by inserting instructions, deleting repeats of an instruction, permuting instructions, etc. Even the Cut Rule fails in its general form: if P ⇒ A and R,A,Q ⇒ C, then R, P , Q ⇒ C INFERENCE IN ACTION 5 But dynamic inference is not totally unprincipled – and some ‘substitute rules’ turn out to hold. Partial update functions validate the following structural rules: if P ⇒ C, then A, P ⇒ C, Left-Monotonicity if P ⇒ A and P ,A,Q ⇒ C, then P , Q ⇒ C Left-Cut if P ⇒ A and P , Q ⇒ C, then P ,A,Q ⇒ C Cautious Monotonicity Indeed, these structural rules are completely characteristic for dynamic inference with partial update functions. Take any set of propositions Prop as abstract objects – and any binary relation ⇒ between finite sequences of propositions and propositions. We repeat a result from van Benthem 1996, as it shows the flavour of the situation rather nicely: Theorem 1. The following are equivalent for any structure (Prop, ⇒): (a) ⇒ satisfies Left-Monotonicity, Left-Cut, Cautious Monotonicity, viewed as abstract conditions on relations of type ‘sequence-to-object’, (b) there is a transition model (S, {TA | A ∈ Prop}) with partial maps TA whose relation of dynamic inference coincides with the given ⇒. Proof. The direction from (b) to (a) is easy to check, as we suggested above. From (a) to (b), any abstract structure (Prop, ⇒) induces a transition model M with states are finite sequences X, Y of propositions. Each proposition A then defines a partial function over these states: TA = {(X, X) | X ⇒ A}∪{(X, X, A) | not X ⇒ A} We must check that the following equivalence holds: M P1,...,Pk ⇒ C iff P1,...,Pk ⇒ C is true in (Prop, ⇒) s T s ...T s T ‘If’. Suppose that 1 p1 2 pk k. By the definition of the functions A,each step in this sequence either adds a proposition at the end, or it just ‘pauses’. Here is a typical illustration: XT X, P X ⇒ P p1 1 (not 1) X, P T X, P X, P ⇒P 1 p2 1 ( 1 2) X, P T X, P ,P X, P ⇒P 1 p3 1 3 (not 1 3) We show that the end state X, P1,P3 is a fixed point for TC : i.e., X, P1,P3⇒C. First we have P1,P2,P3⇒C, and so by Left-Monotonicity X, P1,P2,P3⇒C. Following the transition steps, we suppress one proposition thanks to X, P1⇒P2, using Left-Cut to get X, P1,P3⇒C. This argument is general. ‘Pauses’ involve valid sequents used to cut out items in the sequence P1,...,Pk at the right places. ‘Only if’. This involves the remaining structural rule. Again, here is a simple example. Let P1,P2,P3 dynamically imply C in our transition structure M. Start with the empty sequence –. We choose three particular transitions for the premises. If – ⇒ P1 in Prop, the first transition is –, – ; otherwise, take an extended sequence P1 ; etc. Suppose this yields the following sequence of transformations: −, P1P1, P1 (where P1 ⇒!P2) P1, P1,P3 6 VAN BENTHEM By assumption, the final state is a fixed point for TC :thatis,P1,P3 ⇒ C is true in Prop.
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