Semantics without Toil? Brady and Rush Meet Halldén

The present discussion takes up an issue raised in Section 5 of Ross Brady and Penelope Rush’s paper ‘Four Basic Logical Issues’ concerning the (claimed) triviality – in the sense of automatic availability – of soundness and completeness results for a logic in a metalanguage employing at least as much logical vocabulary as the object logic, where the metalogical behaviour of the common logical vocabulary is as in the object logic. We shall see – in Propositions 4.5–4.7 – that this triviality claim faces difficulties in the face of Halldén incompleteness, for essentially the same reasons that Halldén thought this phenomenon raised seman- tic difficulties for the modal logics of C. I. Lewis exhibiting it. To counter any inclination to dismiss the phenomenon as providing at best a marginal range of counterexamples to the triviality claim, a Postscript assembles some reminders of the extent of – and the varied considerations favouring – Halldén incompleteness

Axiom (cc)0 and Verifiability in Two Extracanonical Logics of Formal Inconsistency

In the field of logics of formal inconsistency (LFIs), the notion of “consistency” is frequently too broad to draw decisive conclusions with respect to the validity of many theses involving the consistency connective. In this paper, we consider the matter of the axiom (cc)0—i.e., the schema ◦ ◦ϕ—by considering its interpretation in contexts in which “consistency” is understood as a type of verifiability. This paper suggests that such an interpretation is implicit in two extracanonical LFIs—Sören Halldén’s nonsense-logic C and Graham Priest’s cointuitionistic logic daC—drawing some interesting conclusions concerning the status of (cc)0. Initially, we discuss Halldén’s skepticism of this axiom and provide a plausible counterexample to its validity. We then discuss the interpretation of the operator in Priest’s daC and show the equivalence of (cc)0 to the intuitionistic principle of testability. These observations suggest that it may be fruitful for members of the LFI community to look outside the canon for evidence concerning the adoption of principles like (cc)0

Squeezing arguments and the plurality of informal notions

In this paper we argue that squeezing arguments à la Kreisel fail to univocally capture an informal or intuitive notion of validity. This suggests a form of logical pluralism, at a conceptual level, not only among but also within logical systems

Quine and Quantified Modal Logic

The textbook - like history of analytic philosophy is a history of myths, r e- ceived views and dogmas. Though mainly the last few years have witnessed a huge amo unt of historical work that aimed to reconsider our narratives of the history of an a- lytic philosophy there is still a lot to do. The present study is meant to present such a micro story which is still quite untouched by historians. According to the receiv ed view Kripke has defeated all the arguments of Quine against quantified modal logic and thus it became a respectful tool for philosophers. If we accept the historical interpret ation of the network between Quine, Kripke and modal logic, which is to be pre sented here, we have to conclude that Quine’s real philosophical animadversions against the modalities are still on the table: though Kripke has provided some important (formal - logical) answers, Quine’s animadversions are still viable and worthy of further consider ation

Quine and Quantified Modal Logic – Against the Received View

The textbook-like history of analytic philosophy is a history of myths, re-ceived views and dogmas. Though mainly the last few years have witnessed a huge amount of historical work that aimed to reconsider our narratives of the history of ana-lytic philosophy there is still a lot to do. The present study is meant to present such a micro story which is still quite untouched by historians. According to the received view Kripke has defeated all the arguments of Quine against quantified modal logic and thus it became a respectful tool for philosophers. If we accept the historical interpreta-tion of the network between Quine, Kripke and modal logic, which is to be presented here, we have to conclude that Quine’s real philosophical animadversions against the modalities are still on the table: though Kripke has provided some important (formal-logical) answers, Quine’s animadversions are still viable and worthy of further consideration

A LOGICAL MODELING OF SEVERE IGNORANCE

In the logical context, ignorance is traditionally defined recurring to epistemic logic $S_4$ \cite{Hintikka1962}. In particular, an agent ignores a formula $\varphi$ when s/he does not know neither $\varphi$ nor its negation $\neg\varphi$: \neg\K\varphi\land\neg\K\neg\varphi (where \K is the epistemic operator for knowledge). In other words, ignorance is essentially interpreted as ``lack of knowledge''. \textcolor{red}{This received view has - as we point out - some problems, in particular we will highlight how it does not allow to express a type of content-theoretic ignorance, i.e. an ignorance of $\varphi$ that stems from an unfamiliarity with its meaning.} Contrarily to this trend, in this paper, we introduce and investigate a modal logic having a primitive epistemic operator \I, modeling ignorance. Our modal logic is essentially constructed on the modal logics based on weak Kleene three-valued logic introduced by Krister Segerberg \cite{Segerberg67}. Such non-classical propositional basis allows to define a Kripke-style semantics with the following, very intuitive, interpretation: a formula $\varphi$ is ignored by an agent if $\varphi$ is neither true nor false in every world accessible to the agent. As a consequence of this choice, we obtain \textcolor{red}{a type of content-theoretic} notion of ignorance, which is essentially different from the traditional approach based on $S_4$. \textcolor{red}{We dub it \emph{severe ignorance}.} We axiomatize, prove completeness and decidability for the logic of reflexive (three-valued) Kripke frames, which we find the most suitable candidate for our novel proposal and, finally, compare our approach with the most traditional one

The Nature and Logic of Vagueness

The PhD thesis advances a new approach to vagueness as dispersion, comparing it with the main philosophical theories of vagueness in the analytic tradition