Most Gentzen systems arising in logic contain few axiom schemata and many rule schemata. Hilbert systems, on the other hand, usually contain few proper inference rules and possibly many axioms. Because of this, the two notions tend to serve different purposes. It is common for a logic to be specified in the first instance by means of a Gentzen calculus, whereupon a Hilbert-style presentation ‘for’ the logic may be sought—or vice versa. Where this has occurred, the word ‘for’ has taken (...) on several different meanings, partly because the Gentzen separator ⇒ can be interpreted intuitively in a number of ways. Here ⇒ will be denoted less evocatively by ⊲.In this paper we aim to discuss some of the useful ways in which Gentzen and Hilbert systems may correspond to each other. Actually, we shall be concerned with thededucibility relationsof the formal systems, as it is these that are susceptible to transformation in useful ways. To avoid potential confusion, we shall speak of Hilbert and Gentzenrelations. By aHilbert relationwe mean any substitution-invariant consequence relation onformulas—this comes to the same thing as the deducibility relation of a set of Hilbert-style axioms and rules. By aGentzen relationwe mean the fully fledged generalization of this notion in whichsequentstake the place of single formulas. In the literature, Hilbert relations are often referred to assentential logics. Gentzen relations as defined here are their exactsequentialcounterparts. (shrink)

Relatively congruence regular quasivarieties and quasivarieties of logic have noticeable similarities. The paper provides a unifying framework for them which extends the Blok-Pigozzi theory of elementarily algebraizable (and protoalgebraic) deductive systems. In this extension there are two parameters: a set of terms and a variable. When the former is empty or consists of theorems, the Blok-Pigozzi theory is recovered, and the variable is redundant. On the other hand, a class of ‘membership logics’ is obtained when the variable is the only (...) element of the set of terms. For these systems the appropriate variant of equivalent algebraic semantics encompasses the relatively congruence regular quasivarieties. (shrink)

It is proved that the relevance logic \ has no structurally complete consistent axiomatic extension, except for classical propositional logic. In fact, no other such extension is even passively structurally complete.

Logics that do not have a deduction-detachment theorem (briefly, a DDT) may still possess a contextual DDT —a syntactic notion introduced here for arbitrary deductive systems, along with a local variant. Substructural logics without sentential constants are natural witnesses to these phenomena. In the presence of a contextual DDT, we can still upgrade many weak completeness results to strong ones, e.g., the finite model property implies the strong finite model property. It turns out that a finitary system has a contextual (...) DDT iff it is protoalgebraic and gives rise to a dually Brouwerian semilattice of compact deductive filters in every finitely generated algebra of the corresponding type. Any such system is filter distributive, although it may lack the filter extension property. More generally, filter distributivity and modularity are characterized for all finitary systems with a local contextual DDT, and several examples are discussed. For algebraizable logics, the well-known correspondence between the DDT and the equational definability of principal congruences is adapted to the contextual case. (shrink)

Hereditary structural completeness is established for a range of substructural logics, mainly without the weakening rule, including fragments of various relevant or many-valued logics. Also, structural completeness is disproved for a range of systems, settling some previously open questions.

The variety DMM of De Morgan monoids has just four minimal subvarieties. The join-irreducible covers of these atoms in the subvariety lattice of DMM are investigated. One of the two atoms consisting of idempotent algebras has no such cover; the other has just one. The remaining two atoms lack nontrivial idempotent members. They are generated, respectively, by 4-element De Morgan monoids C4 and D4, where C4 is the only nontrivial 0-generated algebra onto which finitely subdirectly irreducible De Morgan monoids may (...) be mapped by noninjective homomorphisms. The homomorphic preimages of C4 within DMM constitute a proper quasivariety, which is shown to have a largest subvariety U. The covers of the variety within U are revealed here. There are just ten of them. In exactly six of these ten varieties, all nontrivial members have C4 as a retract. In the varietal join of those six classes, every subquasivariety is a variety—in fact, every finite subdirectly irreducible algebra is projective. Beyond U, all covers of [or of ] within DMM are discriminator varieties. Of these, we identify infinitely many that are finitely generated, and some that are not. We also prove that there are just 68 minimal quasivarieties of De Morgan monoids. (shrink)

We characterize, in syntactic terms, the ranges of epimorphisms in an arbitrary class of similar first-order structures. This allows us to strengthen a result of Bacsich, as follows: in any prevariety having at most \ non-logical symbols and an axiomatization requiring at most \ variables, if the epimorphisms into structures with at most \ elements are surjective, then so are all of the epimorphisms. Using these facts, we formulate and prove manageable ‘bridge theorems’, matching the surjectivity of all epimorphisms in (...) the algebraic counterpart of a logic \ with suitable infinitary definability properties of \, while not making the standard but awkward assumption that \ comes furnished with a proper class of variables. (shrink)

The logic RM and its basic fragments (always with implication) are considered here as entire consequence relations, rather than as sets of theorems. A new observation made here is that the disjunction of RM is definable in terms of its other positive propositional connectives, unlike that of R. The basic fragments of RM therefore fall naturally into two classes, according to whether disjunction is or is not definable. In the equivalent quasivariety semantics of these fragments, which consist of subreducts of (...) Sugihara algebras, this corresponds to a distinction between strong and weak congruence properties. The distinction is explored here. A result of Avron is used to provide a local deduction-detachment theorem for the fragments without disjunction. Together with results of Sobociski, Parks and Meyer (which concern theorems only), this leads to axiomatizations of these entire fragments — not merely their theorems. These axiomatizations then form the basis of a proof that all of the basic fragments of RM with implication are finitely axiomatized consequence relations. (shrink)

Relatively congruence regular quasivarieties and quasivarieties of logic have noticeable similarities. The paper provides a unifying framework for them which extends the Blok-Pigozzi theory of elementarily algebraizable (and protoalgebraic) deductive systems. In this extension there are two parameters: a set of terms and a variable. When the former is empty or consists of theorems, the Blok-Pigozzi theory is recovered, and the variable is redundant. On the other hand, a class of membership logics is obtained when the variable is the only (...) element of the set of terms. For these systems the appropriate variant of equivalent algebraic semantics encompasses the relatively congruence regular quasivarieties. (shrink)

A full separation theorem for the derivable rules of intuitionistic linear logic without bounds, 0 and exponentials is proved. Several structural consequences of this theorem for subreducts of (commutative) residuated lattices are obtained. The theorem is then extended to the logic LR+ and its proof is extended to obtain the finite embeddability property for the class of square increasing residuated lattices.

A quasivariety of algebras has the joint embedding property (JEP) if and only if it is generated by a single algebra A. It is structurally complete if and only if the free ℵ0‐generated algebra in can serve as A. A consequence of this demand, called ‘passive structural completeness’ (PSC), is that the nontrivial members of all satisfy the same existential positive sentences. We prove that if is PSC then it still has the JEP, and if it has the JEP and (...) its nontrivial members lack trivial subalgebras, then its relatively simple members all belong to the universal class generated by one of them. Under these conditions, if is relatively semisimple then it is generated by one ‐simple algebra. We also prove that a quasivariety of finite type, with a finite nontrivial member, is PSC if and only if its nontrivial members have a common retract. The theory is then applied to the variety of De Morgan monoids, where we isolate the sub(quasi)varieties that are PSC and those that have the JEP, while throwing fresh light on those that are structurally complete. The results illuminate the extension lattices of intuitionistic and relevance logics. (shrink)

A full separation theorem for the derivable rules of intuitionistic linear logic without bounds, 0 and exponentials is proved. Several structural consequences of this theorem for subreducts of (commutative) residuated lattices are obtained. The theorem is then extended to the logic LR+ and its proof is extended to obtain the finite embeddability property for the class of square increasing residuated lattices.

Correspondence and texts by Fichte and Schelling illuminate their thought and the trajectory of their philosophical falling out.

One might legitimately ask what reasons other than vanity could prompt an author to issue a collection of his previously published essays. The best reason, I think, is the belief that the essays hang together in such a way that, as a book, they produce a whole which is in a sense greater than the sum of its parts. When this happens, as I hope it does in the present case, it is because the essays pursue related themes in such (...) a way that, together, they at least form a start toward the development of a systematic theory on the common foundations supporting the particular claims in the particular articles. With respect to this collection, the essays can all be read as particular ways of pursuing the following general pattern of thought: that a commitment to justice and a respect for rights (and not social utility) must be the foundation of any morally acceptable legal order; that a social contractarian model is the best way to illuminate this foundation; that a retributive theory of punish ment is the only theory of punishment resting on such a foundation and thus is the only morally acceptable theory of punishment; that the twentieth century's faddish movement toward a "scientific" or therapeutic response to crime runs grave risks of undermining the foundations of justice and rights on which the legal order ought to rest; and, finally, that the legitimate worry about the tendency of the behavioral sciences to undermine the values of. (shrink)

A number of cases involving self-locating beliefs have been discussed in the Bayesian literature. I suggest that many of these cases, such as the sleeping beauty case, are entangled with issues that are independent of self-locating beliefs per se. In light of this, I propose a division of labor: we should address each of these issues separately before we try to provide a comprehensive account of belief updating. By way of example, I sketch some ways of extending Bayesianism in order (...) to accommodate these issues. Then, putting these other issues aside, I sketch some ways of extending Bayesianism in order to accommodate self-locating beliefs. Finally, I propose a constraint on updating rules, the "Learning Principle", which rules out certain kinds of troubling belief changes, and I use this principle to assess some of the available options. (shrink)

The reception history of the French Revolution in France and England is well documented among Anglophone scholars; however, the debate over the Revolution in Germany is much less well known. Fichte's Contribution played an important role in this debate. Presented here for the first time in English, Fichte's work provides a distinctive synthesis of Locke's "possessive individualism," Rousseau's general will, and Kant's moral philosophy. This eclectic blend results in an unusual rights theory that at times veers close to a form (...) of anarchism. Written in 1792–93, just before Fichte moved to Jena to develop his philosophical system in a series of works—above all the Wissenschaftslehre of 1794—the Contribution provides invaluable insight into Fichte's early development. In addition, Fichte's work predates much of Kant's political philosophy, and can shed light on the rich dialogue in German political thought in the 1790s. (shrink)

This essay is written on the following premises and argues for them. “Enlightenment” is a word or signifier, and not a single or unifiable phenomenon which it consistently signifies. There is no single or unifiable phenomenon describable as “the Enlightenment,” but it is the definite article rather than the noun which is to be avoided. In studying the intellectual history of the late seventeenth century and the eighteenth, we encounter a variety of statements made, and assumptions proposed, to which the (...) term “Enlightenment” may usefully be applied, but the meanings of the term shift as we apply it. The things are connected, but not continuous; they cannot be reduced to a single narrative; and we find ourselves using the word “Enlightenment” in a family of ways and talking about a family of phenomena, resembling and related to one another in a variety of ways that permit of various generalizations about them. We are not, however, committed to a single root meaning of the word “Enlightenment,” and we do not need to reduce the phenomena of which we treat to a single process or entity to be termed “the” Enlightenment. It is a reification that we wish to avoid, but the structure of our language is such that this is difficult, and we will find ourselves talking of “the French” or “the Scottish,” “the Newtonian” or the “the Arminian” Enlightenments, and hoping that by employing qualifying adjectives we may constantly remind ourselves that the keyword “Enlightenment” is ours to use and should not master us. (shrink)

In Deep Space Nine's Chief of Security Odo is one who finds his identity in his job, which is a sufficient burden to keep him plenty occupied. In a more benign laboratory than what Odo experienced, philosophical instead of exobiological, he presents an ideal case study of George Herbert Mead's theory of how the self is formed and transformed. According to Mead, language starts with meaningful gestures. When creatures act in coordination through a gesture, signaling that the gesture means the (...) same to them; it becomes language. Focused attention is one of the ways one has of influencing the clarity and extent of memories. Over the arc of Deep Space Nine's tales, all the characters change. Yet Odo goes from being a one‐dimensional to a multidimensional character by learning languages from those who start out as generalized others, but become specific others as time goes on. (shrink)

On principle, there are no language classifications which are right or wrong, but only classifications which are more or less useful or useless. This statement at the beginning of my paper is intended to indicate the teleological perspective in which I want to view the more general problems involved in the classification of African languages. I shall discuss these within a framework which I derive from the four main components of any language classification, namely:1.the aims and objectives of language classification,2.the (...) basic units to be classified,3.the criteria of classification,4.the models of language classification. (shrink)

The Prologue offers an overview of the Reformation of method from Augustine of Hippo through to the Ramist movement, providing an orientation to the rest of the book. It highlights and explains an important nexus of Realism, exemplarism and illumination fundamental to Ramism. Beginning with Augustine it shows how these themes coalesced into a distinctive Christian philosophy taken up and refined by Franciscans such as Bonaventure of Bagnoregio and John Duns Scotus, as well as by Ramon Lull, the Franciscan-inspired encyclopaedist. (...) It traces the continuation of this methodological impulse into the Platonism and Christian humanism of the Renaissance, highlighting especially the topical and Realist (Scotistic) reform of logic pioneered by Lorenzo Valla and Rudolph Agricola. It concludes by showing the reception of this topical and biblical reform in Philipp Melanchthon and the movement of Reformed scholasticism, revealing Reformed Scotism and Christian philosophy as important preconditions for Ramism. (shrink)

The first English translation of Fichte’s second set of 1804 lectures on the Wissenschaftslehre.

Pocock, J. G. A. (John Greville Agard) 1924- "Quentin Skinner: The History of Politics and the Politics of History" Common Knowledge - Volume 10, Issue 3, Fall 2004, pp. 532-550.

In recent work, Adlam (2022b), Chen & Goldstein (2022), and Meacham (2023) have defended accounts of laws that take laws to be primitive global constraints. A major advantage of these accounts is that they’re able to accommodate the many different kinds of laws that appear in physical theories. In this paper I’ll present these three accounts, highlight their distinguishing features, and note some key differences that might lead one to favor one of these accounts over the others. I’ll conclude by (...) briefly discussing a version of a “constraint” account that I think is especially attractive. (shrink)