Online research in philosophy

We motivate and formalize the idea of sameness by default: two objects are considered the same if they cannot be proved to be different. This idea turns out to be useful for a number of widely different applications, including natural language processing, reasoning with incomplete information, and even philosophical paradoxes. We consider two formalizations of this notion, both of which are based on Reiter’s Default Logic. The first formalization is a new relation of indistinguishability that is introduced by default. We (...) prove that the corresponding default theory has a unique extension, in which every two objects are indistinguishable if and only if their non-equality cannot be proved from the known facts. We show that the indistinguishability relation has some desirable properties: it is reflexive, symmetric, and, while not transitive, it has a transitive “flavor.” The second formalization is an extension (modification) of the ordinary language equality by a similar default: two objects are equal if and only if their non-equality cannot be proved from the known facts. It appears to be less elegant from a formal point of view. In particular, it gives rise to multiple extensions. However, this extended equality is better suited for most of the applications discussed in this paper. (shrink)

In 1952, Heinrich Scholz published a question in The Journal of Symbolic Logic asking for a characterization of spectra, i.e., sets of natural numbers that are the cardinalities of finite models of first order sentences. Günter Asser in turn asked whether the complement of a spectrum is always a spectrum. These innocent questions turned out to be seminal for the development of finite model theory and descriptive complexity. In this paper we survey developments over the last 50-odd years pertaining to (...) the spectrum problem. Our presentation follows conceptual developments rather than the chronological order. Originally a number theoretic problem, it has been approached by means of recursion theory, resource bounded complexity theory, classification by complexity of the defining sentences, and finally by means of structural graph theory. Although Scholz' question was answered in various ways, Asser's question remains open. (shrink)

We show that the spectrum of a sentence ϕ in Counting Monadic Second Order Logic (CMSOL) using one binary relation symbol and finitely many unary relation symbols, is ultimately periodic, provided all the models of ϕ are of clique width at most k, for some fixed k. We prove a similar statement for arbitrary finite relational vocabularies τ and a variant of clique width for τ-structures. This includes the cases where the models of ϕ are of tree width at most (...) k. For the case of bounded tree-width, the ultimate periodicity is even proved for Guarded Second Order Logic GSOL. We also generalize this result to many-sorted spectra, which can be viewed as an analogue of Parikh's Theorem on context-free languages, and its analogues for context-free graph grammars due to Habel and Courcelle. Our work was inspired by Gurevich and Shelah (2003), who showed ultimate periodicity of the spectrum for sentences of Monadic Second Order Logic where only finitely many unary predicates and one unary function are allowed. This restriction implies that the models are all of tree width at most 2, and hence it follows from our result. (shrink)

We study the effects of Vopěnka's principle on properties of model theoretic logics. We show that Vopěnka's principle is equivalent to the assumption that every finitely generated logic has a compact cardinal. We show also that it is equivalent to the assumption that every such logic has a global Hanf number.

Finitary sketches, i.e., sketches with finite-limit and finite-colimit specifications, are proved to be as strong as geometric sketches, i.e., sketches with finite-limit and arbitrary colimit specifications. Categories sketchable by such sketches are fully characterized in the infinitary first-order logic: they are axiomatizable by σ-coherent theories, i.e., basic theories using finite conjunctions, countable disjunctions, and finite quantifications. The latter result is absolute; the equivalence of geometric and finitary sketches requires (in fact, is equivalent to) the non-existence of measurable cardinals.

We improve a general theorem of J. A. Makowsky which characterizes, for a wide class of languages, those sentences θ such that both $\operatorname{Mod}(\theta)$ and $\operatorname{Mod}(\neg\theta)$ are closed under unions of chains.

We associate with any abstract logic L a family F(L) consisting, intuitively, of the limit ultrapowers which are complete extensions in the sense of L. For every countably generated [ω, ω]-compact logic L, our main applications are: (i) Elementary classes of L can be characterized in terms of $\equiv_L$ only. (ii) If U and B are countable models of a countable superstable theory without the finite cover property, then $\mathfrak{U} \equiv_L \mathfrak{B}$ . (iii) There exists the "largest" logic M such (...) that complete extensions in the sense of M and L are the same; moreover M is still [ω,ω]-compact and satisfies an interpolation property stronger than unrelativized ▵-closure. (iv) If L = L ωω(Q α ) , then $\operatorname{cf}(\omega_\alpha) > \omega$ and $\lambda^\omega for all $\lambda . We also prove that no proper extension of L ωω generated by monadic quantifiers is compact. This strengthens a theorem of Makowsky and Shelah. We solve a problem of Makowsky concerning L κλ -compact cardinals. We partially solve a problem of Makowsky and Shelah concerning the union of compact logics. (shrink)

"This is a full original version of Makowski's work on Aristotelian dunamis (shortened & revised version has been previously published as "Metaphysics of Practical Philosophy" paper). The author presents the Aristotelian conception of capacity/potentiality (dunamis) – one of the most important in Aristotle’s metaphysics. A closer inspection allows to draw conclusion, that the concept of capacity is an important link between ‘theory’ and ‘practice’ (metaphysics on the one side, and practical – ethical, rhetorical, political – skills, on the other). A (...) picture of the connection between theory and practice is based on the most important parts of Metaphysics (books delta and theta), it relates metaphysical definitions to an essential element of Aristotelian practical philosophy – the concept of virtue (aretê). In the practical works of Aristotle, we can find different definitions of aretê: in Nicomachean Ethics Aristotle defines aretê using the notion of disposition (hexis), but in Rhetoric he formulates a definition based on the concept of capacity. Distinctive analysis of this inconsistency shows the significance of capacity in The Stagirite’s philosophy.". (shrink)

The paper is devoted to the interpretation of one of the most important passages in modern Anglophon philosophy: III.1.3 of Treatise of Human Nature by David Hume. The author considers the problem of its meaning at an angle of the standard interpretation, which can be summed up in a dictum: ‘no ought from is’ (so called “Hume’s Guillotine”). The author outlines four possible approaches to this putative meaning of the Treatise passage and weighs arguments for them. The investigation, based mainly (...) on the strategies by Arthur Prior, Charles Pigden and John Searle, allows to defend two approaches regarding Hume’s putative proposal of is/ought gap: /1/ the idea of logical autonomy of moral discourse, making strong rationality of deduction from is to (non-trivial) ought invalid, together with /2/ the possibility of weak rational (justified) inference from is to ought. However, the comparison of these two accounts with the guts of Humean philosophy as such (critical skepticism, fallibilism, ethical noncognitivism) makes the author prone to give a priority to the latter. (shrink)

"Traditional interpretations of Kantian idea of autonomy – based on the classical texts such as Kritik der praktischen Vernunft and Grundlegung zur Metaphysik der Sitten – stress basically one point: action is autonomous only when an agent obeys the law. In this paper, the author tries to introduce an interpretation of Kant’s practical philosophy, which covers a wider perspective, resulting in the idea of “radical autonomy”. Re-reading classical texts of Kant in connection with Religion innerhalb der Grenzen der bloßen Vernunft (...) and Tugendlehre I try to show, that Kant’s conception of autonomy also allows for acting, that is not determined by the law (i.e. categorical imperative). Roughly, to be radically autonomous you do not have to be necessarily moral (there are other conditions that make your autonomy valid). This paper adopts reasons of „historicistic enlightment” (Schnädelbach), which is rather contrary to the general anti-historicistic context of Kant’s philosophy. Although this approach seems to be against the Kantian „metaphysical hipotheque” (e.g.: pure Faktum der Vernunft, absolute justification/Grundlegung of morality), it provides a (re)interpretation that allows to defend the broadened idea of autonomy. And this ‘radical autonomy’ shows some explanatory attractiveness of Kant’s ethical theory in the contemporary critical horizon of historicism. ". (shrink)

The author presents the Aristotelian conception of capacity/potentiality (dunamis) – one of the most important in Aristotle’s metaphysics. A closer inspection allows to draw conclusion, that the concept of capacity is an important link between ‘theory’ and ‘practice’ (metaphysics on the one side, and practical – ethical, rhetorical, political – skills, on the other). A picture of the connection between theory and practice is based on the most important parts of Metaphysics (books delta and theta), it relates metaphysical definitions to (...) an essential element of Aristotelian practical philosophy – the concept of virtue (aretê). In the practical works of Aristotle, we can find different definitions of aretê: in Nicomachean Ethics Aristotle defines aretê using the notion of disposition (hexis), but in Rhetoric he formulates a definition based on the concept of capacity. Distinctive analysis of this inconsistency shows the significance of capacity in The Stagirite’s philosophy. (shrink)