This is also where we begin investigating lattices of logics and varieties, rather than particular examples.

In "Variations on a theme of Curry," Humberstone conjectured that a certain logic, intermediate between BCI and BCK, is none other than monothetic BCI—the smallest extension of BCI in which all theorems are provably equivalent. In this note, we present a proof of this conjecture.

Monomodal logic has exactly two maximally normal logics, which are also the only quasi-normal logics that are Post complete, and they are complete for validity in Kripke frames. Here we show that addition of a propositional constant to monomodal logic allows the construction of continuum many maximally normal logics that are not valid in any Kripke frame, or even in any complete modal algebra. We also construct continuum many quasi-normal Post complete logics that are not normal. The set of extensions (...) of S4.3 is radically altered by the addition of a constant: we use it to construct continuum many such normal extensions of S4.3, and continuum many non-normal ones, none of which have the finite model property. But for logics with weakly transitive frames there are only eight maximally normal ones, of which five extend K4 and three extend S4. (shrink)

Monomodal logic has exactly two maximally normal logics, which are also the only quasi-normal logics that are Post complete, and they are complete for validity in Kripke frames. Here we show that addition of a propositional constant to monomodal logic allows the construction of continuum many maximally normal logics that are not valid in any Kripke frame, or even in any complete modal algebra. We also construct continuum many quasi-normal Post complete logics that are not normal. The set of extensions (...) of S4.3 is radically altered by the addition of a constant: we use it to construct continuum many such normal extensions of S4.3, and continuum many non-normal ones, none of which have the finite model property. But for logics with weakly transitive frames there are only eight maximally normal ones, of which five extend K4 and three extend S4. (shrink)

We prove that certain natural sequent systems for bi-intuitionistic logic have the analytic cut property. In the process we show that the (global) subformula property implies the (local) analytic cut property, thereby demonstrating their equivalence. Applying a version of Maehara technique modified in several ways, we prove that bi-intuitionistic logic enjoys the classical Craig interpolation property and Maximova variable separation property; its Halldén completeness follows.

The paper intends to provide an algebraic framework in which subluminal causation can be analysed. The framework merges Belnap's 'outcomes in branching time' with his 'branching space-time' (BST). it is shown that an important structure in BST, called 'family of outcomes of an event', is a boolean algebra. We define next non-stochastic common cause and analyse GHZ-Bell theorems. We prove that there is no common cause that accounts for results of GHZ-Bell experiment but construct common causes for two other quantum (...) mechanical setups. Finally, we investigate why some setups allow for common causes whereas other setups do not. (shrink)

We prove that all semisimple varieties of FL ew-algebras are discriminator varieties. A characterisation of discriminator and EDPC varieties of FL ew-algebras follows. It matches exactly a natural classification of logics over FL ew proposed by H. Ono.

The paper has two parts preceded by quite comprehensive preliminaries.In the first part it is shown that a subvariety of the variety ${\cal T}$ of all tense algebras is discriminator if and only if it is semisimple. The variety ${\cal T}$ turns out to be the join of an increasing chain of varieties ${\cal D}_n$, which are discriminator varieties. The argument carries over to all finite type varieties of boolean algebras with operators satisfying some term conditions. In the case of (...) tense algebras, the varieties ${\cal D}_n$ can be further characterised by certain natural conditions on Kripke frames.In the second part it is shown that the lattice of subvarieties of ${\cal D}_0$ has two atoms, the lattice of subvarieties of ${\cal D}_1$ has countably many atoms, and for $n>1$, the lattice of subvarieties of ${\cal D}_n$ has continuum atoms. The proof of the second of the above statements involves a rather detailed description of zero-generated simple algebras in ${\cal D}_1$.Almost all the arguments are cast in algebraic form, but both parts begin with an outline describing their contents from the dual point of view of tense logics. (shrink)

In this paper we show that a variety of modal algebras of finite type is semisimple iff it is discriminator iff it is both weakly transitive and cyclic. This fact has been claimed already in [4] (based on joint work by the two authors) but the proof was fatally flawed.

Am(B m B). Specifically I was wondering whether for every BCI-provable formula A there is a B for which the inset formula was provable. If you want to read about this issue, which I..

We show that the variety of residuated lattices is generated by its finite simple members, improving upon a finite model property result of Okada and Terui. The reasoning is a blend of proof-theoretic and algebraic arguments.

We exhibit a simple inference rule, which is admissible but not derivable in BCK, proving that BCK is not structurally complete. The argument is proof-theoretical.

Humberstone asks whether every theorem of BCI provably implies $\phi\to\phi$ for some formula $\phi$. Meyer conjectures that the axiom $\mathbf{B}$ does not imply any such "self-implication." We prove a slightly stronger result, thereby confirming Meyer's conjecture.

Varieties like groups, rings, or Boolean algebras have the property that, in any of their members, the lattice of congruences is isomorphic to a lattice of more manageable objects, for example normal subgroups of groups, two-sided ideals of rings, filters (or ideals) of Boolean algebras.algebraic logic can explain these phenomena at a rather satisfactory level of generality: in every member A of a τ-regular variety ������ the lattice of congruences of A is isomorphic to the lattice of deductive filters on (...) A of the τ-assertional logic of ������. Moreover, if ������ has a constant 1 in its type and is 1-subtractive, the deductive filters on A ∈ ������ of the 1-assertional logic of ������ coincide with the ������-ideals of A in the sense of Gumm and Ursini, for which we have a manageable concept of ideal generation. However, there are isomorphism theorems, for example, in the theories of residuated lattices, pseudointerior algebras and quasi-MV algebras that cannot be subsumed by these general results. The aim of the present paper is to appropriately generalise the concepts of subtractivity and τ-regularity in such a way as to shed some light on the deep reason behind such theorems. The tools and concepts we develop hereby provide a common umbrella for the algebraic investigation of several families of logics, including substructural logics, modal logics, quantum logics, and logics of constructive mathematics. (shrink)

A syntactic derivation of Cornish identity from the axioms of HBCK is presented which amounts to a syntactic proof of Wronski's conjecture that naturally ordered BCK-algebras form a variety.

We show the admissibility for BCI of a rule form of the characteristic implicational axiom of abelian logic, this rule taking us from →β to α. This is done in Section 8, with surrounding sections exploring the admissibility and derivability of various related rules in several extensions of BCI.

It is shown that the pure implication fragment of the modal logic [3], pp. 385--387) has finitely many non-equivalent formulae in one variable. The exact number of such formulae is not known. We show that this finiteness result is the best possible, since the analogous fragment of S4, and therefore of [3], in two variables has infinitely many non-equivalent formulae.

A sequent system is used to give alternative proofs of two well known properties of free lattices: Whitman’s condition and semidistributivity. It demonstrates usefulness of such proof systems outside logic.

It is proved that free dynamic algebras superamalgamate. Craig interpolation for propositional dynamic logic and superamalgamation for the variety of dynamic algebras follow.

It is proved that free dynamic algebras superamalgamate. Craig interpolation for proposi- tional dynamic logic and superamalgamation for the variety of dynamic algebras follow.

It is shown that there are precisely two pretabular varieties of equivalential algebras.

In this paper we investigate computable isomorphisms of Boolean algebras with operators (BAOs). We prove that there are examples of polymodal Boolean algebras with finitely many computable isomorphism types. We provide an example of a polymodal BAO such that it has exactly one computable isomorphism type but whose expansions by a constant have more than one computable isomorphism type. We also prove a general result showing that BAOs are complete with respect to the degree spectra of structures, computable dimensions, expansions (...) by constants, and the degree spectra of relations. (shrink)

The paper extends the framework of outcomes in branching space-time (Kowalski and Placek [1999]) by assigning probabilities to outcomes of events, where these probabilities are interpreted either epistemically or as weighted possibilities. In resulting models I define the notion of common cause of correlated outcomes of a single event, and investigate which setups allow for the introduction of common causes. It turns out that a deterministic common cause can always be introduced, but (surprisingly) only special setups permit the introduction (...) of truly stochastic common causes. I analyse next the Bell-Aspect experiment and derive the Bell-CH inequalities. I observe that we postulate there not a common cause for outcomes of a single event but rather a common common cause that accounts for outcomes of many events, where 'events' mean 'measurements with (different) directions of polarization'. Since the inequalities are violated, I claim that no causal story can be told about the Bell correlations, where causality is subliminal and restricted by screening-off condition. Similarly, given certain intuitive principles, no deterministic story can be told about these correlations. (shrink)

Since the validity of Bell's inequalities implies the existence of joint probabilities for non-commuting observables, there is no universal consensus as to what the violation of these inequalities signifies. While the majority view is that the violation teaches us an important lesson about the possibility of explanations, if not about metaphysical issues, there is also a minimalist position claiming that the violation is to be expected from simple facts about probability theory. This minimalist position is backed by theorems due to (...) A. Fine and I. Pitowsky.Our paper shows that the minimalist position cannot be sustained. To this end,we give a formally rigorous interpretation of joint probabilities in thecombined modal and spatiotemporal framework of `stochastic outcomes inbranching space-time' (SOBST) (Kowalski and Placek, 1999; Placek, 2000). We show in this framework that the claim that there can be no joint probabilities fornon-commuting observables is incorrect. The lesson from Fine's theorem is notthat Bell's inequalities will be violated anyhow, but that an adequate modelfor the Bell/Aspect experiment must not define global joint probabilities. Thus we investigate the class of stochastic hidden variable models, whichprima facie do not define such joint probabilities. The reasonwhy these models fail supports the majority view: Bell's inequalities are notjust a mathematical artifact. (shrink)

Since the validity of Bell's inequalities implies the existence of joint probabilities for non-commuting observables, there is no universal consensus as to what the violation of these inequalities signifies. While the majority view is that the violation teaches us an important lesson about the possibility of explanations, if not about metaphysical issues, there is also a minimalist position claiming that the violation is to be expected from simple facts about probability theory. This minimalist position is backed by theorems due to (...) A. Fine and I. Pitowsky.Our paper shows that the minimalist position cannot be sustained. To this end,we give a formally rigorous interpretation of joint probabilities in thecombined modal and spatiotemporal framework of `stochastic outcomes inbranching space-time' (SOBST) (Kowalski and Placek, 1999; Placek, 2000). We show in this framework that the claim that there can be no joint probabilities fornon-commuting observables is incorrect. The lesson from Fine's theorem is notthat Bell's inequalities will be violated anyhow, but that an adequate modelfor the Bell/aspect experiment must not define global joint probabilities. Thus we investigate the class of stochastic hidden variable models, whichprima facie do not define such joint probabilities. The reasonwhy these models fail supports the majority view: Bell's inequalities are notjust a mathematical artifact. (shrink)

We prove that the theory of fields with a derivation of Frobenius has the model companion which is stable and admits elimination of quantifiers up to the level of the λ-functions. Along the way, we give new geometric axioms of DCFp.

David Lewis’s latest theory of causation defines the causal link in terms of the relation of influence between events. It turns out, however, that one event’s influencing another is neither a necessary nor sufficient condition for its being a cause of that event. In the article one particular case of causality without influence is presented and developed. This case not only serves as a counterexample to Lewis’s influence theory, but also threatens earlier counterfactual analyses of causation by admitting a particularly (...) troublesome type of preemption. The conclusion of the article is that Lewis’s influence method of solving the preemption problem fails, and that we need a new and fresh approach to the cases of redundant causation if we want to hold on to the counterfactual analysis of causation. (shrink)

We cannot escape Indiana Jones! (Not that we would want to, of course.) Harrison Ford deserves credit for the character's popularity. His ability to subtly play up Indy's foibles while playing down the character's heroism, makes Indiana Jones relatable. Of course, Lucas and the screenwriters are also responsible, as they magnificently depict Indy battling antagonists seeking to possess mystical objects for world domination. But Indy is no mere action hero. He also struggles with unrequited love that lingers for decades, an (...) estrangement from his over-bearing father that lasts just as long, and a life-long obsession with regaining an important object unfairly taken from him. As we know, Indy doesn't always choose wisely, and if the Young Indiana Jones Chronicles are to be trusted, he made many missteps in his youth on the way to becoming the hero we know and love. Indy loses a fight as often as he wins, and when he wins, he doesn't always fight fairly. In fact, he stumbles so often, one might wonder why he persists decade after decade toward his goals. He's not getting any younger. What's the point? And, as he inexplicably trudges on, he-in the four films and various graphic novels-has many alleged encounters with the supernatural. He's not a religious person (at all), so what should he make of those? Should he take the leap of faith and become a believer? While we often see him conducting his highly unconventional fieldwork, spending an unusual amount of time in caves, he cannot completely ignore his "day job" teaching undergraduates and aiding their pursuit of knowledge. But why should archaeologists strive to put artifacts in a museum, or search for "fact," but not "truth" (and who is Dr. Tyree, anyway)? The Indiana Jones character is expressive of the human condition, vividly illuminating our struggles and issues on the big screen. True, we don't have John Williams's orchestra filling us with pride when we manage to do the right thing or play the part of the hero. Still, no one would blame you if you imagined Indy's theme blaring in the background while rising to meet a challenge! (shrink)

In the late 2010s, various international committees, expert groups, and national strategy boards have voiced the demand to ‘open’ the algorithmic black box, to audit, expound, and demystify artificial intelligence. The opening of the algorithmic black box, however, cannot be seen only as an engineering challenge. In this article, I argue that only the sort of transparency that arises from critique—a method of theoretical examination that, by revealing pre-existing power structures, aims to challenge them—can help us produce technological systems that (...) are less deceptive and more just. I relate the question of AI transparency to the broader challenge of responsible making, contending that future action must aim to systematically reconcile design—as a way of concealing—with critique—as a manner of revealing. (shrink)

There are books that debate math, science, and history; there are books that help you build walls or even pyramids; there are even books that discuss Neanderthals with tools and autotrophs that drool. This book discusses philosophy. But you don't need an IQ of 187 to enjoy it. I swear to cow! As you'll see, the philosophy is theoretical, but the fun is real.