Abstract
In this paper, I discuss the analysis of logic in the pragmatic approach recently proposed by Brandom. I consider different consequence relations, formalized by classical, intuitionistic and linear logic, and I will argue that the formal theory developed by Brandom, even if provides powerful foundational insights on the relationship between logic and discursive practices, cannot account for important reasoning patterns represented by non-monotonic or resource-sensitive inferences. Then, I will present an incompatibility semantics in the framework of linear logic which allow to refine Brandom’s concept of defeasible inference and to account for those non-monotonic and relevant inferences that are expressible in linear logic. Moreover, I will suggest an interpretation of discursive practices based on an abstract notion of agreement on what counts as a reason which is deeply connected with linear logic semantics.
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Notes
Wittgenstein (1967), §202.
The point of view here adopted is closer to Brandom foundational considerations. A modelization of speech acts that keep track of their illocutionary force using linear logic has been developed in Bellin and Dalla Pozza (2002).
It would be interesting to take a closer look at the relationship between logic discursive practice and dialogic approach to logic, in particular to compare Brandom’s view with dialogic tradition of Lorenzen (Lorenz and Lorenzen 1978) and with the recent developments in game semantics. In Section “Incompatibility Semantics Based on Linear Logic”, I will take some insights inspired by the dialogic tradition to provide an interactive account of incompatibility detection.
In particular, Curry-Howard isomorphism between proofs in intuitionistic logic and terms in lambda calculus can be considered as an effective construction of witnesses for good inferences.
The point is that the relationship between classical and intuitionistic reasoning cannot be stated in terms of pragmatically mediated semantic relations. I claim that this relationship has a special interest for semantics since, as Dummett points out in Dummett (1991), classical and intuitionistic logic lead to two different theories of meaning: the first one states meaning in terms of truth conditions, and in general is committed with realism, while the second one gives a characterization of meaning in terms of proof, or reasons, and can be considered matching anti-realist insights.
An interesting aspect of linear logic is that many abstract machines can be encoded in linear logic (see Girard 2006, pp. 220–221). We cannot enter the details here, I simply remark that this feature provides a precise definition of Brandom’s idea of LX-relation: if the practice or abilities are represented, as in the formal language example, by means of automata, we can see how to define the notion of explication straightforwardly: we simply say that a vocabulary V explicates the practice P if the vocabulary is expressive enough to encode the machine implementing P.
This property doesn’t hold in the non-linear case. For example, Peirce law ((A → B) → A) → A, which does not contain the constant for absurd, it is provable in classical logic while it is not provable in intuitionistic logic.
The dialogical interpretation of logic and in particular the interpretation of sequent calculus in terms of dialog games goes back to the work of Lorenzen (Lorenz and Lorenzen 1978). A dialogic interpretation of linear logic is provided by Blass, see Blass (1992). Further refinements of Blass semantics lead to the definition of a game semantics for linear logic in Abramsky and Jagadeesan (1994). Game semantics can be considered a truly interactive account of the meaning of logical constants, which seems to be closer to Brandom’s point of view. Here, we decided to work with the algebraic semantics of linear logic in order to make the comparison with Brandom’s incompatibility semantics easier.
We can consider linear negation as a type of negation at least considering that it defines De Morgan dualities between conjunction and disjunction: \((A\otimes B)^{\bot}\) iff \((A^{\bot} \wp B^{\bot}\)).
The definitions are just an interpretation of those defining phase semantics for linear logic, see Girard (2006). We are going to define a classical phase space which provides models for classical linear logic; then we will point at some differences with intuitionistic linear logic discussing defeasible inferences.
The order of actions should matter too, we assume here commutativity for simplicity. Note that it is possible to consider non-commutative versions of linear logic; for an algebraic semantics, see Yetter (1990).
We could say that it is the property itself that let us speak of a same issue, namely, an issue is determined by the interaction in the dialog.
Remark that the technical framework I am presenting can be considered independently from the suggested interpretation. The interpretation here defined points at a purely pragmatic interpretation of propositional content of sentences, which is identified with sets of actions that count as reasons in a dialog. One could anyway take the technical definitions and state them as in Brandom (2008), see technical appendix p. 141–155. Define an incompatibility frame as a syntactic phase space \((\mathcal{P}, \bot)\), where P is a set of propositional formulas; example of syntactic phase spaces are involved in the completeness proof of sequent calculus for linear logic, see Girard (2006). The incompatibility-entailment relation can be defined as \(X \models_{\bot} Y\) iff \(X^{\bot} \subseteq Y\). In this way \(\models_{\bot}\) will define precisely the linear logic consequence relation.
It would be interesting to consider the relationship between logic and other form of discursive dynamics: if logic works where the agreement we defined holds, how communication can be formalized when that form of agreement is missing? Is it still a form of reasoning? Interesting insights on this issue can be found in Girard’s Ludics, a recent development of linear logic, see Girard (2007). The connection between the paradigm of computation defined in Ludics and Wittgenstein’s language games has been investigated in Pietarinen (2003).
In classical logic one consider sequents to be defined as sets of formulas. In linear logic, we have to consider multi-sets since repetitions matter.
I state the definition using the syntactic notion of provability since we will use it to prove the results. By completeness theorem, the same holds for semantic consequence relation.
The remark concerning two notions of absurdity constitutes a partial answer to the question raised by Brandom on the status of relevant logics, see Brandom (2008), pp. 173–175.
The reason why Eq. 17 is no provable in LL is that it should be obtained form an axiom C ⊢ C by means of monotonicity, which doesn’t hold at a global level in linear logic.
This sequent can be proved in LL. Intuitively it means that if C proves the ex falso quodlibet absurdity 0, then C together with any other formula proves 0. This shows a form of monotonicity implicit in 0.
This treatment of intuitionistic logic is still not fully satisfactory since it is a negative characterization that states that intuitionistic inferences are those that lacks defeasors. However, the approach proposed allow to place intuitionistic logic within a same framework, the one defined by linear logic. So we can state explicitly within the model that intuitionistic inferences lack defeasors, as the formulas restriction shows. We decided here to study the relationship between classical and intuitionistic logic with respect to defeasibility from a syntactic point of view, namely considering provability. We leave to a future work the comparison between classical and intuitionistic phase spaces with respect to incompatibility semantics. Algebraic investigations on the properties of intuitionistic and classical phase spaces are provided in Kanovich et al. (2006).
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Porello, D. Incompatibility Semantics from Agreement. Philosophia 40, 99–119 (2012). https://doi.org/10.1007/s11406-010-9259-4
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DOI: https://doi.org/10.1007/s11406-010-9259-4