We explore the problems that confront any attempt to explain or explicate exactly what a primitive logical rule of inferenceis, orconsists in. We arrive at a proposed solution that places a surprisingly heavy load on the prospect of being able to understand and deal with specifications of rules that are essentiallyself-referring. That is, any rule$\rho $is to be understood via a specification that involves, embedded within it, reference to rule$\rho $itself. Just how we arrive at this position is explained by (...) reference to familiar rules as well as less familiar ones with unusual features. An inquiry of this kind is surprisingly absent from the foundations of inferentialism—the view that meanings of expressions (especially logical ones) are to be characterized by the rules of inference that govern them. (shrink)

An inference can be seen as a speech act, in which one passes from a number of assertions called premisses to another assertion, the conclusion, which is presented as supported or justified by the premisses. To justify the assertion that appears as conclusion is the characteristic aim of an inference. Here, we confine ourselves to deductive inferences where the justification is taken to be conclusive. A short, natural explanation of what it is for a (deductive) inference to be valid is (...) to say that it is valid if it succeeds in its aim, that is, if its premisses do (conclusively) justify the conclusion, given that premisses are justified. The paper is concerned with developing this explanation further. (shrink)

We define a class of formal systems inspired by Prawitz’s theory of grounds. The latter is a semantics that aims at accounting for epistemic grounding, namely, at explaining why and how deductively valid inferences have the power to epistemically compel to accept the conclusion. Validity is defined in terms of typed objects, called grounds, that reify evidence for given judgments. An inference is valid when a function exists from grounds for the premises to grounds for the conclusion. Grounds are described (...) by formal terms, either directly when the terms are in canonical form, or indirectly when they are in non-canonical form. Non-canonical terms must reduce to canonical form, and two terms may be said to be equal when they converge towards equivalent grounds. In our systems these properties can be proved through rules distinguished according to whether they concern types or logic. Type rules involve type introduction and elimination, equality for application of operational symbols, and re-writing equations for non-canonical terms. The logic amounts to a sort of intuitionistic system in a Gentzen format. To conclude, we show that each system of our class enjoys a normalization property. (shrink)

The lecture spells out the difference between the validity of inference (-figure)s and validity applied to demonstrations (‘proof acts’). The latter notion is not an ordinary characterizing one; in Brentano's terminology it is a modifying one. A demonstration lacking validity is not a real demonstration, just as a false friend is no true friend. Throughout, the treatment makes crucial use of an epistemological perspective that is cast in the first person. Furthermore, the difference between (logical) consequence among propositions and the (...) validity of inference from judgement to judgement is explained. Particular attention is paid to alleged issues of circularity in the definition of the validity of inference, and to the ‘explosion’ validity of inference from contradictory premisses. Drawing upon a version of the dialogical framework of Per Martin-Löf, namely, ‘When I say Therefore, I give others my permission to assert the conclusion’, while stressing also the importance of the first person perspective, both difficulties can be neutralized. (shrink)

We outline a class of term-languages for epistemic grounding inspired by Prawitz’s theory of grounds. We show how denotation functions can be defined over these languages, relating terms to proof-objects built up of constructive functions. We discuss certain properties that the languages may enjoy both individually and with respect to their expansions. Finally, we provide a ground-theoretic version of Prawitz’s completeness conjecture, and adapt to our framework a refutation of this conjecture due to Piecha and Schroeder-Heister.

This is a dialogue between Lisa and Max on Dag Prawitz's work concerning the concept of deductive validity. Lisa first explains Prawitz's criticisms of the presently prevailing non‐epistemic analyses of validity. Then Lisa describes three different ways in which Prawitz attempted to develop an epistemic concept of validity. Max asks questions for clarification, raises some objections and compares Prawitz's three approaches with other lines of thought. Two inference rules are specially discussed: disjunction introduction and ex contradictione quodlibet. Max and Lisa (...) view Prawitz's contribution as part of a variegated, ongoing research pursuing an explication of the concept of validity, which began with the Socratic and Platonic distinction between merely persuasive arguments and good arguments and with Aristotle's definition of syllogism. Throughout the history of philosophy, logicians have aimed at clarifying a pre‐theoretic idea of good deduction and have proposed, criticized, refined, adjusted precise theoretical concepts of validity. Prawitz has criticized current views about validity, has advanced new proposals and revised them. His work is an important chapter of the long pursuit of the concept of validity. (shrink)