We analyse the concept of a second-order characterisable structure and divide this concept into two parts—consistency and categoricity—with different strength and nature. We argue that categorical characterisation of mathematical structures in second-order logic is meaningful and possible without assuming that the semantics of second-order logic is defined in set theory. This extends also to the so-called Henkin structures.

Second-order logic has a number of attractive features, in particular the strong expressive resources it offers, and the possibility of articulating categorical mathematical theories (such as arithmetic and analysis). But it also has its costs. Five major charges have been launched against second-order logic: (1) It is not axiomatizable; as opposed to first-order logic, it is inherently incomplete. (2) It also has several semantics, and there is no criterion to choose between them (Putnam, J Symbol (...) Logic 45:464–482, 1980 ). Therefore, it is not clear how this logic should be interpreted. (3) Second-order logic also has strong ontological commitments: (a) it is ontologically committed to classes (Resnik, J Phil 85:75–87, 1988 ), and (b) according to Quine (Philosophy of logic, Prentice-Hall: Englewood Cliffs, 1970 ), it is nothing more than “set theory in sheep’s clothing”. (4) It is also not better than its first-order counterpart, in the following sense: if first-order logic does not characterize adequately mathematical systems, given the existence of non - isomorphic first-order interpretations, second-order logic does not characterize them either, given the existence of different interpretations of second-order theories (Melia, Analysis 55:127–134, 1995 ). (5) Finally, as opposed to what is claimed by defenders of second-order logic [such as Shapiro (J Symbol Logic 50:714–742, 1985 )], this logic does not solve the problem of referential access to mathematical objects (Azzouni, Metaphysical myths, mathematical practice: the logic and epistemology of the exact sciences, Cambridge University Press, Cambridge, 1994 ). In this paper, I argue that the second-order theorist can solve each of these difficulties. As a result, second-order logic provides the benefits of a rich framework without the associated costs. (shrink)

We discuss the differences between first-order set theory and second-order logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if second-order logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on informal reasoning. On the other hand, if it is given a weak semantics, it loses its power in expressing concepts categorically. First-order set theory (...) and second-order logic are not radically different: the latter is a major fragment of the former. (shrink)

We try to answer the question which is the “right” foundation of mathematics, second order logic or set theory. Since the former is usually thought of as a formal language and the latter as a first order theory, we have to rephrase the question. We formulate what we call the second order view and a competing set theory view, and then discuss the merits of both views. On the surface these two views seem to be (...) in manifest conflict with each other. However, our conclusion is that it is very difficult to see any real difference between the two. We analyze a phenomenon we call internal categoricity which extends the familiar categoricity results of second order logic to Henkin models and show that set theory enjoys the same kind of internal categoricity. Thus the existence of non-standard models, which is usually taken as a property of first order set theory, and categoricity, which is usually taken as a property of second order axiomatizations, can coherently coexist when put into their proper context. We also take a fresh look at complete second order axiomatizations and give a hierarchy result for second order characterizable structures. Finally we consider the problem of existence in mathematics from both points of view and find that second order logic depends on what we call large domain assumptions, which come quite close to the meaning of the axioms of set theory. (shrink)

The characterization of second-order type isomorphisms is a purely syntactical problem that we propose to study under the enlightenment of game semantics. We study this question in the case of second-order λμ-calculus, which can be seen as an extension of system F to classical logic, and for which we define a categorical framework: control hyperdoctrines.Our game model of λμ-calculus is based on polymorphic arenas which evolve during the play. We show that type isomorphisms coincide with the (...) “equality” on arenas associated with types. Finally we deduce the equational characterization of type isomorphisms from this equality. We also recover from the same model Roberto Di Cosmo’s characterization of type isomorphisms for system F.This approach leads to a geometrical comprehension on the question of second-order type isomorphisms, which can be easily extended to some other polymorphic calculi including additional programming features. (shrink)

There is little doubt that a second-order axiomatization of Zermelo-Fraenkel set theory plus the axiom of choice (ZFC) is desirable. One advantage of such an axiomatization is that it permits us to express the principles underlying the first-order schemata of separation and replacement. Another is its almost-categoricity: M is a model of second-order ZFC if and only if it is isomorphic to a model of the form Vκ, ∈ ∩ (Vκ × Vκ) , for κ (...) a strongly inaccessible ordinal. (shrink)

This article improves two existing theorems of interest to neologicist philosophers of mathematics. The first is a classification theorem due to Fine for equivalence relations between concepts definable in a well-behaved second-order logic. The improved theorem states that if an equivalence relation E is defined without nonlogical vocabulary, then the bicardinal slice of any equivalence class—those equinumerous elements of the equivalence class with equinumerous complements—can have one of only three profiles. The improvements to Fine’s theorem allow for an (...) analysis of the well-behaved models had by an abstraction principle, and this in turn leads to an improvement of Walsh and Ebels-Duggan’s relative categoricity theorem. (shrink)

Bird’s Ultimate Argument sought to show that Armstrong’s N relationships involving categorical universals can’t entail nomic regularities. In N’s place Bird offered the non-categorical SR relation. Two kinds of objection have been raised: either Bird’s own alternative metaphysics fails in just the same way as Armstrong’s or the target of Bird’s argument may anyway have a way out of the problem. My aim is to reclaim the victory for Bird. I argue that the responses in defence of Armstong’s N relationships (...) fail to acknowledge that Bird was explicitly concerned with Armstrong’s commitment to a categoricalist view of universals. Moreover, Bird’s alternative account does not suffer the same problem since his metaphysics of properties is essentialist. Nevertheless, Bird’s account does need elaborating on to explainwhySR relationships entail their regularities. I offer Schaffer’s Axiomatic Solution as a candidate for this purpose. (shrink)

We show that the categoricity of second-order Peano axioms can be proved from the comprehension axioms. We also show that the categoricity of second-order Zermelo–Fraenkel axioms, given the order type of the ordinals, can be proved from the comprehension axioms. Thus these well-known categoricity results do not need the so-called “full” second-order logic, the Henkin second-order logic is enough. We also address the question of “consistency” of these axiom systems in the (...) second-order sense, that is, the question of existence of models for these systems. In both cases we give a consistency proof, but naturally we have to assume more than the mere comprehension axioms. (shrink)

This paper casts doubt on a recent criticism of Frege's theory of concepts and extensions by showing that it misses one of Frege's most important contributions: the derivation of the infinity of the natural numbers. We show how this result may be incorporated into the conceptual structure of Zermelo- Fraenkel Set Theory. The paper clarifies the bearing of the development of the notion of a real-valued function on Frege's theory of concepts; it concludes with a brief discussion of the claim (...) that the standard interpretation of second-order logic is necessary for the derivation of the Peano Postulates and the proof of their categoricity. (shrink)

The first round about ‘The myth of ASCOT and its rival ASCO2.T: tech-noetic vs. techno-logic’ exposed the hazard in colliding obsolete disciplinary categories under outdated procedures. The orthodox jurisdiction of Ars Electronica and CERN in Collide@CERN, one of the most prominent ongoing programmes of this kind, does not eliminate the risk of missing the target by operating with categories of artists and scientists. Art is one of those disciplines with a long expired warranty, but with decay on its periphery that (...) is turning into fertile forefront territories. Fresh temporary categories are marking and spreading over these uncharted territories and sensibly interconnecting with peripheries of other disciplines. The ex-artist that is reborn in this peripheral transdisciplinary zone can be provisionally categorized as a polyphibian for its features to be carefully studied. Like amphibians, a polyphibian can coherently transcend from one medium to another, in between and beyond the disciplines. In order to research the implications of such ‘categorical’ mutations, a readymade was submitted to the organization of Collide@CERN under the licence of R. Mutt. Namely, the readymade ASCOTT (Apparatus with Super COnducting Thought Transduction) dubbed also ASCO2.T is needed as the second-order upgrade of the existing plan for ASCOT (Apparatus with Super COnducting Toroids) invented by the scientists for CERN. As is expected from the reputation of R. Mutt’s readymades – this submission was ignored and refused in just the same quiet manner as his most notorious one. But if society would not provide salons for the refused artists (Salon des Refuses) right next to the salons under the scrutiny of academically established artists, there would be no mutations and no evolution in art. Such mutations of the artist into a ‘complex and widely distributed system’ at the dispersing fringes of decomposing art was already predicted by R. Ascott. Polyphibians are the species surviving the ripening metamorphosis of disciplines by taking refuge in the unexplored transdisciplinary buffer zone. This is the only zone where a confrontation of ASCOT technology with tech-noetics of ASCO2.T is possible. The refused unknown demands not a Salon but the Interval of Suspended Judgement. In the sequel to the first round of this debate, the Interval of Suspended Judgement will be investigated. (shrink)

This article praises the development of second order cybernetics by von Foerster, Maturana, and Varela as an important step in deepening our understanding of the bio-psychological foundation of the dynamics of information, cognition, and communication. Luhmann's development of the theory into the realm of social communication is seen as a necessary and important move. The triple autopoietic differentiation between biological, psychological, and social-communicative autopoiesis and the introduction of a technical concept of meaning is central. Finally, the paper shows (...) that second order cybernetics lacks explicit and ontological concepts of emotion, meaning, and a concept of signs. C. S. Peirce's theory is introduced for this purpose. It is then shown, through Varela's development of Spencer Brown's ‘Laws of Form’ from a dual to a dynamic triadic categorical structure, that both theories are triadic and second order, and therefore can be fruitfully fused to a Cybersemiotics. (shrink)

We define a new class of Kripke structures for the second-order λ-calculus, and investigate the soundness and completeness of some proof systems for proving inequalities as well as equations. The Kripke structures under consideration are equipped with preorders that correspond to an abstract form of reduction, and they are not necessarily extensional. A novelty of our approach is that we define these structures directly as functors A: → Preor equipped with certain natural transformations corresponding to application and abstraction (...) . We make use of an explicit construction of the exponential of functors in the Cartesian-closed category Preor, and we also define a kind of exponential ΠΦs ε T to take care of type abstraction. However, we strive for simplicity, and we only use very elementary categorical concepts. Consequently, we believe that the models described in this paper are more palatable than abstract categorical models which require much more sophisticated machinery . We obtain soundness and completeness theorems that generalize some results of Mitchell and Moggi to the second-order λ-calculus, and to sets of inequalities. (shrink)

This paper presents a polarized phase semantics, with respect to which the linear fragment of second order polarized linear logic of Laurent [15] is complete. This is done by adding a topological structure to Girard's phase semantics [9]. The topological structure results naturally from the categorical construction developed by Hamano—Scott [12]. The polarity shifting operator ↓ (resp. ↑) is interpreted as an interior (resp. closure) operator in such a manner that positive (resp. negative) formulas correspond to open (resp. (...) closed) facts. By accommodating the exponentials of linear logic, our model is extended to the polarized fragment of the second order linear logic. Strong forms of completeness theorems are given to yield cut-eliminations for the both second order systems. As an application of our semantics, the first order conservativity of linear logic is studied over its polarized fragment of Laurent [16]. Using a counter model construction, the extension of this conservativity is shown to fail into the second order, whose solution is posed as an open problem in [16]. After this negative result, a second order conservativity theorem is proved for an eta expanded fragment of the second order linear logic, which fragment retains a focalized sequent property of [3]. (shrink)

It is often remarked that first-order Peano Arithmetic is non-categorical but deductively well-behaved, while second-order Peano Arithmetic is categorical but deductively ill-behaved. This suggests that, when it comes to axiomatizations of mathematical theories, expressive power and deductive power may be orthogonal, mutually exclusive desiderata. In this paper, I turn to Hintikka’s :69–90, 1989) distinction between descriptive and deductive approaches in the foundations of mathematics to discuss the implications of this observation for the first-order logic versus (...) class='Hi'>second-order logic divide. The descriptive approach is illustrated by Dedekind’s ‘discovery’ of the need for second-order concepts to ensure categoricity in his axiomatization of arithmetic; the deductive approach is illustrated by Frege’s Begriffsschrift project. I argue that, rather than suggesting that any use of logic in the foundations of mathematics is doomed to failure given the impossibility of combining the descriptive approach with the deductive approach, what this apparent predicament in fact indicates is that the first-order versus second-order divide may be too crude to investigate what an adequate axiomatization of arithmetic should look like. I also conclude that, insofar as there are different, equally legitimate projects one may engage in when working on the foundations of mathematics, there is no such thing as the One True Logic for this purpose; different logical systems may be adequate for different projects. (shrink)

Two models of second-order ZFC need not be isomorphic to each other, but at least one is isomorphic to an initial segment of the other. The situation is subtler for impure set theory, but Vann McGee has recently proved a categoricity result for second-order ZFCU plus the axiom that the urelements form a set. Two models of this theory with the same universe of discourse need not be isomorphic to each other, but the pure sets of (...) one are isomorphic to the pure sets of the other. This paper argues that similar results obtain for considerably weaker second-order axiomatizations of impure set theory that are in line with two different conceptions of set, the iterative conception and the limitation of size doctrine. (shrink)

Theoria, Volume 87, Issue 4, Page 986-1000, August 2021.

Due to Gӧdel’s incompleteness results, the categoricity of a sufficiently rich mathematical theory and the semantic completeness of its underlying logic are two mutually exclusive ideals. For first- and second-order logics we obtain one of them with the cost of losing the other. In addition, in both these logics the rules of deduction for their quantifiers are non-categorical. In this paper I examine two recent arguments –Warren (2020), Murzi and Topey (2021)– for the idea that the natural deduction (...) rules for the first-order universal quantifier are categorical, i.e., they uniquely determine its semantic intended meaning. Both of them make use of McGee’s open-endedness requirement and the second one uses in addition Garson’s (2013) local models for defining the validity of these rules. I argue that the success of both these arguments is relative to their semantic or infinitary assumptions, which could be easily discharged if the introduction rule for the universal quantifier is taken to be an infinitary rule, i.e. non-compact. Consequently, I reconsider the use of the ω-rule and I show that the addition of the ω-rule to the standard formalizations of first-order logic is categorical. In addition, I argue that the open-endedness requirement does not make the first-order Peano Arithmetic categorical and I advance an argument for its categoricity based on the inferential conservativity requirement. (shrink)

One of the philosophical uses of Dedekind’s categoricity theorem for Peano Arithmetic is to provide support for semantic realism. To this end, the logical framework in which the proof of the theorem is conducted becomes highly significant. I examine different proposals regarding these logical frameworks and focus on the philosophical benefits of adopting open-ended schemas in contrast to second order logic as the logical medium of the proof. I investigate Pederson and Rossberg’s critique of the ontological advantages of (...) open-ended arithmetic when it comes to establishing the categoricity of Peano Arithmetic and show that the critique is highly problematic. I argue that Pederson and Rossberg’s ontological criterion deliver the bizarre result that certain first order subsystems of Peano Arithmetic have a second order ontology. As a consequence, the application of the ontological criterion proposed by Pederson and Rossberg assigns a certain type of ontology to a theory, and a different, richer, ontology to one of its subtheories. (shrink)

On a widespread naturalist view, the meanings of mathematical terms are determined, and can only be determined, by the way we use mathematical language—in particular, by the basic mathematical principles we’re disposed to accept. But it’s mysterious how this can be so, since, as is well known, minimally strong first-order theories are non-categorical and so are compatible with countless non-isomorphic interpretations. As for second-order theories: though they typically enjoy categoricity results—for instance, Dedekind’s categoricity theorem for second- (...) class='Hi'>order PA and Zermelo’s quasi-categoricity theorem for second-order ZFC—these results require full second-order logic. So appealing to these results seems only to push the problem back, since the principles of second-order logic are themselves non-categorical: those principles are compatible with restricted interpretations of the second-order quantifiers on which Dedekind’s and Zermelo’s results are no longer available. In this paper, we provide a naturalist-friendly, non-revisionary solution to an analogous but seemingly more basic problem—Carnap’s Categoricity Problem for propositional and first-order logic—and show that our solution generalizes, giving us full second-order logic and thereby securing the categoricity or quasi-categoricity of second-order mathematical theories. Briefly, the first-order quantifiers have their intended interpretation, we claim, because we’re disposed to follow the quantifier rules in an open-ended way. As we show, given this open-endedness, the interpretation of the quantifiers must be permutation-invariant and so, by a theorem recently proved by Bonnay and Westerståhl, must be the standard interpretation. Analogously for the second-order case: we prove, by generalizing Bonnay and Westerståhl’s theorem, that the permutation invariance of the interpretation of the second-order quantifiers, guaranteed once again by the open-endedness of our inferential dispositions, suffices to yield full second-order logic. (shrink)

After a short preface, the first of the three sections of this paper is devoted to historical and philosophic aspects of categoricity. The second section is a self-contained exposition, including detailed definitions, of a proof that every mathematical system whose domain is the closure of its set of distinguished individuals under its distinguished functions is categorically characterized by its induction principle together with its true atoms (atomic sentences and negations of atomic sentences). The third section deals with applications especially (...) those involving the distinction between characterizing a system and axiomatizing the truths of a system. (shrink)

Frege’s project has been characterized as an attempt to formulate a complete system of logic adequate to characterize mathematical theories such as arithmetic and set theory. As such, it was seen to fail by Gödel’s incompleteness theorem of 1931. It is argued, however, that this is to impose a later interpretation on the word ‘complete’ it is clear from Dedekind’s writings that at least as good as interpretation of completeness is categoricity. Whereas few interesting first-order mathematical theories are categorical (...) or complete, there are logical extensions of these theories into second-order and by the addition of generalized quantifiers which are categorical. Frege’s project really found success through Gödel’s completeness theorem of 1930 and the subsequent development of first- and higher-order model theory. (shrink)

In proof-theoretic semantics, model-theoretic validity is replaced by proof-theoretic validity. Validity of formulae is defined inductively from a base giving the validity of atoms using inductive clauses derived from proof-theoretic rules. A key aim is to show completeness of the proof rules without any requirement for formal models. Establishing this for propositional intuitionistic logic raises some technical and conceptual issues. We relate Sandqvist’s (complete) base-extension semantics of intuitionistic propositional logic to categorical proof theory in presheaves, reconstructing categorically the soundness and (...) completeness arguments, thereby demonstrating the naturality of Sandqvist’s constructions. This naturality includes Sandqvist’s treatment of disjunction that is based on its second-order or elimination-rule presentation. These constructions embody not just validity, but certain forms of objects of justifications. This analysis is taken a step further by showing that from the perspective of validity, Sandqvist’s semantics can also be viewed as the natural disjunction in a category of sheaves. (shrink)

We propose a criterion to regard a property of a theory (in first or second order logic) as virtuous: the property must have significant mathematical consequences for the theory (or its models). We then rehearse results of Ajtai, Marek, Magidor, H. Friedman and Solovay to argue that for second order logic, ‘categoricity’ has little virtue. For first order logic, categoricity is trivial; but ‘categoricity in power’ has enormous structural consequences for any of the theories satisfying (...) it. The stability hierarchy extends this virtue to other complete theories. The interaction of model theory and traditional mathematics is examined by considering the views of such as Bourbaki, Hrushovski, Kazhdan, and Shelah to flesh out the argument that the main impact of formal methods on mathematics is using formal definability to obtain results in ‘mainstream’ mathematics. Moreover, these methods (e.g., the stability hierarchy) provide an organization for much mathematics which gives specific content to dreams of Bourbaki about the architecture of mathematics. (shrink)

We investigate the reverse-mathematical status of several theorems to the effect that the natural number system is second-order categorical. One of our results is as follows. Define a system to be a triple A,i,f such that A is a set and i∈A and f:A→A. A subset X⊆A is said to be inductive if i∈X and ∀a ∈X). The system A,i,f is said to be inductive if the only inductive subset of A is A itself. Define a Peano system (...) to be an inductive system such that f is one-to-one and i∉the range of f. The standard example of a Peano system is N,0,S where N={0,1,2,…,n,…}=the set of natural numbers and S:N→N is given by S=n+1 for all n∈N. Consider the statement that all Peano systems are isomorphic to N,0,S. We prove that this statement is logically equivalent to WKL0 over RCA0⁎ source. From this and similar equivalences we draw some foundational/philosophical consequences. (shrink)

Structure is central to the realist view of mathematical disciplines with intended interpretations and categoricity is a model-theoretic notion that captures the idea of the determination of structure by theory. By considering the cases of arithmetic and (pure) set theory, I investigate how categoricity results might offer support from within to the realist view. I argue, amongst other things, that second-order quantification is essential to the support the categoricity results provide. I also note how the findings on categoricity (...) relate to a fundamental feature of the realist position. (shrink)

LetMbe a countably infinite ω-categorical structure. Consider Aut(M) as a complete metric space by definingd(g, h) = Ω{2−n:g(xn) ≠h(xn) org−1(xn) ≠h−1(xn)} where {xn:n∈ ω} is an enumeration ofMAn automorphism α ∈ Aut(M) is generic if its conjugacy class is comeagre. J. Truss has shown in [11] that if the set P of all finite partial isomorphisms contains a co-final subset P1closed under conjugacy and having the amalgamation property and the joint embedding property then there is a generic automorphism. In the (...) present paper we give a weaker condition of this kind which is equivalent to the existence of generic automorphisms. Really we give more: a characterization of the existence of generic expansions (defined in an appropriate way) of an ω-categorical structure. We also show that Truss' condition guarantees the existence of a countable structure consisting of automorphisms ofMwhich can be considered as an atomic model of some theory naturally associated toM. We do it in a general context of weak models for second-order quantifiers.The author thanks Ludomir Newelski for pointing out a mistake in the first version of Theorem 1.2 and for interesting discussions. Also, the author is grateful to the referee for very helpful remarks. (shrink)

f The purpose of this paper is to investigate categoricity arguments conducted in second order logic and the philosophical conclusions that can be drawn from them. We provide a way of seeing this result, so to speak, through a first order lens divested of its second order garb. Our purpose is to draw into sharper relief exactly what is involved in this kind of categoricity proof and to highlight the fact that we should be reserved (...) before drawing powerful philosophical conclusions from it. (shrink)

James Walmsley in “Categoricity and Indefinite Extensibility” argues that a realist about some branch of mathematics X (e.g. arithmetic) apparently cannot use the categoricity of an axiomatisation of X to justify her belief that every sentence of the language of X has a truth-value. My discussion note first corrects Walmsley’s formulation of his claim. It then shows that his argument for it hinges on the implausible idea that grasping that there is some model of the axioms amounts to grasping that (...) there is a unique such model. (shrink)

According to the standard definition, a first-order theory is categorical if all its models are isomorphic. The idea behind this definition obviously is that of capturing semantic notions in axiomatic terms: to be categorical is to be, in this respect, successful. Thus, for example, we may want to axiomatically delimit the concept of natural number, as it is given by the pre-theoretic semantic intuitions and reconstructed by the standard model. The well-known results state that this cannot be done within (...) first-order logic, but it can be done within second-order one. Now let us consider the following question: can we axiomatically capture the semantic concept of conjunction? Such question, to be sure, does not make sense within the standard framework: we cannot construe it as asking whether we can form a first-order (or, for that matter, whatever-order) theory with an extralogical binary propositional operator so that its only model (up to isomorphism) maps the operator on the intended binary truth-function. The obvious reason is that the framework of standard logic does not allow for extralogical constants of this type. But of course there is also a deeper reason: an existence of a constant with this semantics is presupposed by the very definition of the framework1. Hence the question about the axiomatic capturability of concunction, if we can make sense of it at all, cannot be asked within the framework of standard logic, we would have to go to a more abstract level. To be able to make sense of the question we would have to think about a propositional ‘proto-language’, with uninterpreted logical constants, and to try to search out axioms which would fix the denotations of the constants as the intended truth-functions. Can we do this? It might seem that the answer to this question is yielded by the completeness theorem for the standard propositional calculus: this theorem states that the axiomatic delimitation of the calculus and the semantic delimitation converge to the same result.. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:First Order Relationality and Its Implications:A Response to David ElsteinRoger T. Ames (bio)David Elstein has asked a series of important questions about Human Becomings that provide me with an opportunity to try to bring the argument of the book into clearer focus. Let me begin by thanking David for his always generous and intelligent reflection on not only my new monograph [End Page 181] but also on Henry (...) Rosemont's and my best efforts to expand upon the idea of "role ethics" as a productive way to read Confucian ethics.David asks: Is relationality truly a part of original Confucian thought? Does relationality rule out any notion of essential human nature, and does it necessarily imply or entail role ethics? Is relationality true or merely a useful way of thinking (if these can be distinguished)? Do we know that it's useful?I think the answer to these questions turns on two very different ways of understanding relationality, namely the doctrines of first and second order relationality. The term 'ontology' as "the science of being per se or in itself" gives us a doctrine of external, second order relations. Ontology was introduced into our philosophical vocabulary as a new term for an old way of thinking at the beginning of the seventeenth century by two German philosophers.1 For the classical Greeks, "the science of being per se" (to on he on) was posited as their "first philosophy" with all of its implications for metaphysics, epistemology, logic, ethics, philosophy of religion, and so on. Ontology is that branch of metaphysics that seeks to classify and explain the things that exist, with its underlying assumption that there are unchanging substances or essences internal to things that are available to us to classify them as this and not that. Ontology privileges "being per se" with its categorical language and its "substance" and "attribute" dualism, giving us a grammar of substances as property-bearers and properties that are borne, respectively.Such ontological thinking animates Plato's pursuit of formal, "real" definitions in his quest for certainty (that is, definitions not of words but of what really is), and underlies Aristotle's taxonomical science of knowing "what is what." For these classical Greek philosophers, only what is real and is thus true can be the proper object of knowledge, giving us a logic of the changeless. Corollary to ontology is causal thinking, formal definitions, and, important for David's question, a doctrine of external, second order relationality in which relations conjoin discrete and self-sufficient things, such as human "beings." As the contemporary philosopher Zhao Tingyang 趙汀陽 avers, this kind of substance ontology defining the real things that constitute the content of an orderly and structured cosmosprovides a "dictionary" kind of explanation of the world, seeking to set up an accurate understanding of the limits of all things. In simple terms, it determines "what is what," and all concepts are footnotes to "being" or "is."2In the Book of Changes, we find a vocabulary making explicit cosmological assumptions that stand in stark contrast to this substance ontology and that brings with it a doctrine of first order, constitutive relations. This alternative "first philosophy" has prompted me to create the neologism "zoetology," with the Greek zoe- "life" and -logia "discourse," as my own new term for an old way of thinking that we might translate into modern Chinese as shengshenglun 生生論: "the art of living." The starting point in this zoetological cosmology, then, is that nothing does anything by [End Page 182] itself; association is a fact. Since the very nature of life is associative and transactional, the vocabulary appealed to in defining Confucian cosmology is irreducibly dyadic and collateral: always multiple, never one.And further, it is the nature of life itself that it seeks to optimize the available conditions for its continuing growth, where such growth has a persistent existential and thus intentional aspect. "Life" is the productive correlation between organism and environment that gives rise to correlative, relational, ecological thinking that many of our best sinologists (Marcel Granet, David Keightley, Tang Junyi, K. C. Chang, A. C. Graham, and many others) would ascribe to the earliest semblances of Chinese... (shrink)

This paper has three parts. First, we establish some of the basic model theoretic facts about, the Tsirelson space of Figiel and Johnson [20]. Second, using the results of the first part, we give some facts about general Banach spaces. Third, we study model‐theoretic dividing lines in some Banach spaces and their theories. In particular, we show: (1) has the non independence property (NIP); (2) every Banach space that is ℵ0‐categorical up to small perturbations embeds c0 or () almost (...) isometrically; consequently the (continuous) first‐order theory of does not characterize, up to almost isometric isomorphism. (shrink)

Some of the features of animal and human categorical perception (CP) for color, pitch and speech are exhibited by neural net simulations of CP with one-dimensional inputs: When a backprop net is trained to discriminate and then categorize a set of stimuli, the second task is accomplished by "warping" the similarity space (compressing within-category distances and expanding between-category distances). This natural side-effect also occurs in humans and animals. Such CP categories, consisting of named, bounded regions of similarity space, may (...) be the ground level out of which higher-order categories are constructed; nets are one possible candidate for the mechanism that learns the sensorimotor invariants that connect arbitrary names (elementary symbols?) to the nonarbitrary shapes of objects. This paper examines how and why such compression/expansion effects occur in neural nets. (shrink)

Their important exegetical and philosophical disagreements notwithstanding, Pauline Kleingeld and Marcus Willaschek, on the one hand, and Alyssa Bernstein, on the other, seem to agree that Kant’s Categorical Imperative transcends the contemporary dichotomy between moral realism and ethical constructivism. My contribution is an attempt to further elaborate on the third, unique, conceptual option that they have identified. I employ the notion of an “enabling condition,” introduced in epistemology and action theory by Jonathan Dancy, in order to show that the (...) Categorical Imperative enables action and is enabled by action. The Categorical Imperative (as distinct from specific categorical imperatives and moral laws) is discussed as a structural, but still normative, coherence and consistence norm of practical reasoning. As such it is two things at the same time, namely, first, the most fundamental normative background condition of practical deliberation, choice, and action. Second, however, the mutual enabler-enablee-relationship between action and Categorical Imperative points in the other – “not-so-Kantian” – direction as well: individual practical deliberation, understood as the mundane phenomenon that penetrates the lives of agents, is the (existence) condition that enables the Categorical Imperative in normatively-structured action. (shrink)

This article offers a new interpretation of Adorno’s new categorical imperative : it suggests that the new imperative is an important element of Adorno’s moral philosophy and at the same time runs counter to some of its essential features. It is suggested that Adorno’s moral philosophy leads to two aporiae, which create an impasse that the new categorical imperative attempts to circumvent. The first aporia results from the tension between Adorno’s acknowledgement that praxis is an essential part of moral philosophy, (...) and his view according to which existing social conditions make it impossible for moral knowledge to be translated into right action. The second aporia results from the tension between the uncompromising sensitivity to suffering that underlies Adorno’s moral thought, and his analysis of the culture industry mechanisms which turn people into happy, satisfied customers—an incompatibility which threatens to pull the rug out from under Adorno’s moral philosophy. My interpretation of the new categorical imperative focuses on two characteristics it inherits from the old, Kantian one—self-evidence and unconditionality—in order to present the new imperative as a response to these two aporiae. (shrink)

Given a π -institution I , a hierarchy of π -institutions I is constructed, for n ≥ 1. We call I the n-th order counterpart of I . The second-order counterpart of a deductive π -institution is a Gentzen π -institution, i.e. a π -institution associated with a structural Gentzen system in a canonical way. So, by analogy, the second order counterpart I of I is also called the “Gentzenization” of I . In the main (...) result of the paper, it is shown that I is strongly Gentzen , i.e. it is deductively equivalent to its Gentzenization via a special deductive equivalence, if and only if it has the deduction-detachment property. (shrink)

In the 1920's Fraenkel and Carnap raised the question of whether or not every finitely axiomatizable semantically complete theory formulated in the theory of types is categorical. Partial answers to this and a related question are presented for theories formulated in second-order logic.

This paper is the second in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the (...) usual set-theoretic semantics so as to shed new light on the relevant strengths and limits of higher-order logic. (shrink)

It is shown that the second-order theory of a Dedekind algebra is categorical if it is finitely axiomatizable. This provides a partial answer to an old and neglected question of Fraenkel and Carnap: whether every finitely axiomatizable semantically complete second-order theory is categorical. It follows that the second-order theory of a Dedekind algebra is finitely axiomatizable iff the algebra is finitely characterizable. It is also shown that the second-order theory of a Dedekind (...) algebra is quasi-finitely axiomatizable iff the algebra is quasi-finitely characterizable. (shrink)

For each positive n , two alternative axiomatizations of the theory of strings over n alphabetic characters are presented. One class of axiomatizations derives from Tarski's system of the Wahrheitsbegriff and uses the n characters and concatenation as primitives. The other class involves using n character-prefixing operators as primitives and derives from Hermes' Semiotik. All underlying logics are second order. It is shown that, for each n, the two theories are definitionally equivalent [or synonymous in the sense of (...) deBouvere]. It is further shown that each member of one class is synonymous with each member of the other class; thus that all of the theories are definitionally equivalent with each other and with Peano arithmetic. Categoricity of Peano arithmetic then implies categoricity of each of the above theories. (shrink)

Recently, the second author, Briseid, and Safarik introduced nonstandard Dialectica, a functional interpretation capable of eliminating instances of familiar principles of nonstandard arithmetic—including overspill, underspill, and generalizations to higher types—from proofs. We show that the properties of this interpretation are mirrored by first-order logic in a constructive sheaf model of nonstandard arithmetic due to Moerdijk, later developed by Palmgren, and draw some new connections between nonstandard principles and principles that are rejected by strict constructivism. Furthermore, we introduce a (...) variant of the Diller–Nahm interpretation with two different kinds of quantifiers, similar to Hernest’s light Dialectica interpretation, and show that one can obtain nonstandard Dialectica by weakening the computational content of the existential quantifiers—a process called herbrandization. We also define a constructive sheaf model mirroring this new functional interpretation, and show that the process of herbrandization has a clear meaning in terms of these sheaf models. (shrink)

This paper proposes the “conceptual pragmatism” of C. I. Lewis as a useful epistemological orientation for studying the relationship between the social and individual registers, in particular as it is set out in Bourdieu’s sociology of practice. Bourdieu’s concepts of habits based upon ‘social schematism’ and of culture as a ‘second nature,’ as well as Lewis’ conception of logical schemas constructed by humans are all formulated in the wake of Kant, and mediating between sensorial experience and the conceptual level (...) is a common issue for Lewis and the French sociologist. This paper discusses Lewis’ controversial idea of “sense meaning” in order to show his commitment to a pragmatic, interactive theory of knowledge and language. Moreover, I emphasize Lewis’ intertwining of conceptual schemas, actions, and reality as a theoretical network for reshaping analysis of the relationship between sociality and individuality according to a pragmatist perspective, and for implementing and improving Lewis’ commitment to observing, describing, and interpreting the effective functioning of conceptual schemas, one that includes an option in favor of an externalist philosophical methodology which, in principle, tallies with the methodological attitude that mostly characterizes the contemporary social sciences. (shrink)

In “Properties and the Interpretation of Second-Order Logic” Bob Hale develops and defends a deflationary conception of properties where a property with particular satisfaction conditions actually exists if and only if it is possible that a predicate with those same satisfaction conditions exists. He argues further that, since our languages are finitary, there are at most countably infinitely many properties and, as a result, the account fails to underwrite the standard semantics for second-order logic. Here a (...) more lenient version of the view is explored, which allows for the possibility of countably infinite predicates understood as the product of linguistic supertasks. This enriched deflationist account of properties—the Infinitary Deflationary Conception of Existence—supports the standard semantics for models with countable first-order domains, and allows one to prove the categoricity of the second-order Peano axioms. (shrink)

There exist two conjectures for constraint satisfaction problems of reducts of finitely bounded homogeneous structures: the first one states that tractability of the CSP of such a structure is, when the structure is a model-complete core, equivalent to its polymorphism clone satisfying a certain nontrivial linear identity modulo outer embeddings. The second conjecture, challenging the approach via model-complete cores by reflections, states that tractability is equivalent to the linear identities satisfied by its polymorphisms clone, together with the natural uniformity (...) on it, being nontrivial. We prove that the identities satisfied in the polymorphism clone of a structure allow for conclusions about the orbit growth of its automorphism group, and apply this to show that the two conjectures are equivalent. We contrast this with a counterexample showing that [Formula: see text]-categoricity alone is insufficient to imply the equivalence of the two conditions above in a model-complete core. Taking another approach, we then show how the Ramsey property of a homogeneous structure can be utilized for obtaining a similar equivalence under different conditions. We then prove that any polymorphism of sufficiently large arity which is totally symmetric modulo outer embeddings of a finitely bounded structure can be turned into a nontrivial system of linear identities, and obtain nontrivial linear identities for all tractable cases of reducts of the rational order, the random graph, and the random poset. Finally, we provide a new and short proof, in the language of monoids, of the theorem stating that every [Formula: see text]-categorical structure is homomorphically equivalent to a model-complete core. (shrink)

In this paper we prove three theorems about first-order theories that are categorical in a higher power. The first theorem asserts that such a theory either is totally categorical or there exist prime and minimal models over arbitrary base sets. The second theorem shows that such theories have a natural notion of dimension that determines the models of the theory up to isomorphism. From this we conclude that $I(T, \aleph_\alpha) = \aleph_0 +|\alpha|$ where ℵ α = the number (...) of formulas modulo T-equivalence provided that T is not totally categorical. The third theorem gives a new characterization of these theories. (shrink)

The papers of this special issue are the outcome of a two-­‐day conference entitled “The Second-­‐Person Standpoint in Law and Morality,” that took place at the University of Vienna in March 2013 and was organized by the ERC Advanced Research Grant “Distortions of Normativity.” -/- The aim of the conference was to explore and discuss Stephen Darwall’s innovative and influential second-­‐personal account of foundational moral concepts such as „obligation“, „responsibility“, and „rights“, as developed in his book The (...) class='Hi'>Second-­‐Person Standpoint: Morality, Respect, and Accountability (Harvard University Press 2006) and further elaborated in Morality, Authority and Law: Essays in Second-­‐Personal Ethics I and Honor, History, and Relationships: Essays in Second-­‐Personal Ethics II (both Oxford University Press 2013). -/- With the second-­‐person standpoint Darwall refers to the unique conceptual normative space that practical deliberators and agents occupy when they address claims and demands to one another (and to themselves). The very first sentence of Darwall’s examination of the second-­‐personal conceptual paradigm summarizes the gist of the argument succinctly when he claims that “the second-­‐person standpoint [is] the perspective that you and I take up when we make and acknowledge claims on one another’s conduct and will.” (Darwall 2006, 3) The Second-­‐Person Standpoint reminds us that this perspective has been ignored for much too long and that it better take centre stage in any philosophical analysis of moral phenomena, in order to yield a satisfying account of morality as a social institution. The negative part of Darwall’s strategy is to show that neither a purely first-­‐personal approach (represented by Kant and contemporary Kantians), nor a third-­‐personal state-­‐of-­‐affairs-­‐perspective (represented by most varieties of contemporary consequentialism) are capable of accounting for the categorical bindingness characteristic of moral obligation. The latter feat can only be accomplished, and this is the positive part of Darwall’s argument, when those second-­‐ personal normative “felicity conditions” and conceptual presuppositions are acknowledged and spelled out that are already presupposed in every instance of issuing (putatively valid) claims and demands. It is especially second-­‐personal competence and second-­‐personal authority that are the bedrock of these normative conceptual presuppositions, without which engaging in any meaningful address would be impossible. Kantians and utilitarians alike have neglected this critical dimension of the normative landscape. -/- In addition to working out an original conception of moral obligation, the first eight chapters of The Second-­‐Person Standpoint articulate this fundamental insight with respect to a variety of traditional projects in ethical theory such as developing accounts of moral responsibility, rights, dignity, and autonomy. In this context, special emphasis is to be awarded, on the one hand, to Darwall’s refreshing second-­‐personal interpretation of Strawson’s influential account of reactive attitudes and moral responsibility and, on the other, to his historically well-­‐informed reconstruction of Samuel Pufendorf’s often neglected version of an enlightened theistic voluntarism concerning moral authority. Darwall dedicates the second part of The Second-­‐Person Standpoint to the urgent question: how should one respond to the sceptical challenge that expresses utter indifference to the second-­‐person standpoint, including all its multifarious normative presuppositions and implications? What commits us to all this? It is at this point that Darwall, firstly, refines his criticisms of the Kantian, first-­‐personal, paradigm of normativity and emphasizes that only if one already incorporates the second-­‐personal conceptual apparatus into a Kantian analysis of moral obligation is the latter going to yield a convincing account. Secondly, and this certainly is one of the highlights of Darwall’s theory, the Second-­‐Person Standpoint employs themes from Fichte’s philosophy of right in order to strengthen the case for the inescapability of taking up the second-­‐person standpoint of moral obligation. In his contribution for this special issue Darwall further develops his diagnosis that Fichte’s thought offers in many respects a more promising, since more second-­‐personal, foundation of morality than, for example, Kant’s. -/- By now, the impact of Darwall’s second-­‐person standpoint theory has far transcended the confines of contemporary debates on moral obligation. Darwall has put to use the second-­‐personal apparatus to critical engagements with Joseph Raz’s theory of legal authority and Derek Parfit’s convergence arguments for his recent Triple Theory of moral wrongness. The constant theme that unifies all these diverse applications remains the one so impressively presented in The Second-­‐Person Standpoint: without paying attention to the “interdefinable” and “irreducible” circle of (four) foundational second-­‐ personal concepts (valid demand, practical authority, second-­‐personal reason, and accountability), neither superior epistemic status (Raz) nor the identification of optimific states of affairs (Parfit) are potent enough sources to generate anything close to the authority relationships that underlie the idea involved in obligating ourselves and one another. Given all of the above, it comes as no surprise that Darwall reserves his strongest sympathies for a specific ethical theory, namely contractualism. Our commitment to equal basic second-­‐personal authority, that Darwall arrives at through his Fichtean rectification of the Kantian project, leads him to the endorsement of a contractualist paradigm in the spirit of broadly Rawls and Scanlon. -/- . (shrink)

I develop a new theory of properties by considering two central arguments in the debate whether properties are dispositional or categorical. The first claims that objects must possess categorical properties in order to be distinct from empty space. The second argument, however, points out several untoward consequences of positing categorical properties. I explore these arguments and argue that despite appearances, their conclusions need not be in conflict with one another. In particular, we can view the second argument (...) as supporting only the claim that there is not a plurality of categorical properties, and not the stronger claim that there are no categorical properties whatsoever. I then develop a new account of properties which capitalizes on this insight. (shrink)

Second-order axiomatizations of certain important mathematical theories—such as arithmetic and real analysis—can be shown to be categorical. Categoricity implies semantic completeness, and semantic completeness in turn implies determinacy of truth-value. Second-order axiomatizations are thus appealing to realists as they sometimes seem to offer support for the realist thesis that mathematical statements have determinate truth-values. The status of second-order logic is a controversial issue, however. Worries about ontological commitment have been influential in the debate. Recently, (...) Vann McGee has argued that one can get some of the technical advantages of second-order axiomatizations—categoricity, in particular—while walking free of worries about ontological commitment. In so arguing he appeals to the notion of an open-ended schema—a schema that holds no matter how the language of the relevant theory is extended. Contra McGee, we argue that second-order quantification and open-ended schemas are on a par when it comes to ontological commitment. (shrink)

A major challenge in the philosophy of mathematics is to explain how mathematical language can pick out unique structures and acquire determinate content. In recent work, Button and Walsh have introduced a view they call ‘internalism’, according to which mathematical content is explained by internal categoricity results formulated and proven in second-order logic. In this paper, we critically examine the internalist response to the challenge and discuss the philosophical significance of internal categoricity results. Surprisingly, as we argue, while (...) internalism arguably explains how we pick out unique mathematical structures, this does not suffice to account for the determinacy of mathematical discourse. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Mental Disorder and Intentional OrderRichard Gipps (bio)Bengt Brülde and Filip Radovic inform the reader that they will assume "there is such a thing as a general category of disorder, of which mental and somatic disorders can be regarded as subcategories" (2006, 100). With this assumption in place, they take up a fascinating discussion of what warrants our categorizations of certain disorders as mental as opposed to physical. The answers (...) they then canvass include the possibilities that it is the mentality of the internal causes, or of the symptomatic effects, of these disorders which renders them mental, or that there is no single answer, or that there is no valid distinction to be drawn.In what follows I wish to question the initial assumption, and then offer an understanding of the nature of mentality that I hope will lead to a different way of answering the question, what is mental about mental disorder? It worries me that the views I develop here may share too little by way of common framework assumptions with the views developed in Brülde and Radovic's paper. I hope that this does not render dialogue impossible, or make the thoughts that were inspired by their thoughtful paper appear just too off field to be seriously entertained.The Concept of DisorderBrülde and Radovic assume that "mental disorders" and "physical disorders" are subtypes of a general category of disorder. (In what follows I describe this as the assumption.) They also distinguish between two questions: (1) What is mental about mental disorder? (in what follows I describe this as the question) and (2) What does it take for a condition to be a disorder? This latter question is, they acknowledge, certainly important, but is set aside for the time being.Mental disorder, physical disorder, and we might perhaps also add social disorder, are then subcategories of disorder. They are perhaps not natural kinds, but nevertheless they are subcategories within a general category. With their assumption in place it makes good sense to reject accounts of the mentality of mental disorders that seem to simply presuppose the concepts of mind and body when they address the question.There is of course a trivial sense in which everyone must agree that mental disorders and physical disorders are to be considered subcategories of disorder. There is what could be described as a logical sense of category and subcategory, in which any noun is said to designate a category and any adjectival qualification of it to introduce a subcategory. However, we can also distinguish what I call a "semantic" sense, in which talk of categories and subcategories is more restricted. Here we rely on there being a unitary sense to the putative category if we are to talk meaningfully of subcategories of this single category. [End Page 117]The philosophy of the individuation of senses is not within my expertise, but I am relying on an intuitive notion spelled out by the following. First, that a good dictionary would elaborate these senses by providing different definitions of the putative category rather than a single definition followed by examples. Second, that when we do not have to do with categories in the semantic sense, we would find ourselves more in want of true understanding of what was being said, and not so much in want of mere further knowledge, if we are offered a description that included the noun (disorder) without the qualifier (mental). Third, that we are less likely to be dealing with subcategories within a general semantic category if we easily can turn the adjective–noun pair (e.g., mental disorder) around to provide two nouns in a short phrase (e.g., disorder of the mind), or if there are verbal and adjectival forms of the second noun that can also be used in conjunction with the first noun (e.g., disordered mind).An example serves to make things clearer. Consider a chemist who has a reaction in her laboratory. What is referred to may be an emotional reaction (to her sudden scientific breakthrough), or a chemical reaction. In this case, I suggest, we do not have to do with a semantic category of "reaction," which... (shrink)