The main objective of this PhD Thesis is to present a method of obtaining strong normalization via natural ordinal, which is applicable to natural deduction systems and typed lambda calculus. The method includes (a) the definition of a numerical assignment that associates each derivation (or lambda term) to a natural number and (b) the proof that this assignment decreases with reductions of maximal formulas (or redex). Besides, because the numerical assignment used coincide with the length of a specific sequence of (...) reduction - the worst reduction sequence - it is the lowest upper bound on the length of reduction sequences. The main commitment of the introduced method is that it is constructive and elementary, produced only through analyzing structural and combinatorial properties of derivations and lambda terms, without appeal to any sophisticated mathematical tool. Together with the exposition of the method, it is presented a comparative study of some articles in the literature that also get strong normalization by means we can identify with the natural ordinal methods. Among them we highlight Howard[1968], which performs an ordinal analysis of Godel’s Dialectica interpretation for intuitionistic first order arithmetic. We reveal a fact about this article not noted by the author himself: a syntactic proof of strong normalization theorem for the system of typified lambda calculus λ⊃ is a consequence of its results. This would be the first strong normalization proof in the literature. (written in Portuguese). (shrink)

Cantor’s proof that the powerset of the set of all natural numbers is uncountable yields a version of Richard’s paradox when restricted to the full definable universe, that is, to the universe containing all objects that can be defined not just in one formal language but by means of the full expressive power of natural language: this universe seems to be countable on one account and uncountable on another. We argue that the claim that definitional contexts impose restrictions on the (...) scope of quantifiers reveals a natural way out. (shrink)

Inductively defined structures are ubiquitous in mathematics; their specification is unambiguous and their properties are powerful. All fields of mathematical logic feature these structures prominently: the formula of a language, the set of theorems, the natural numbers, the primitive recursive functions, the constructive number classes and segments of the cumulative hierarchy of sets. -/- This dissertation gives a mathematical characterization of a species of inductively defined structures, called accessible domains, which include all of the above examples except the set (...) of theorems. The concept of an accessible domain comes from Wilfried Sieg's analysis of proof-theoretic practices, starting with his dissertation. In particular, he noticed the special epistemological character of elements of an accessible domain: they can always be uniquely identified with their build-up. Generally, the unique build-up of elements justifies the principles of induction and recursion. -/- I use category theory to give an abstract characterization of accessible domains. I claim that accessible domains are all instances of initial algebras for endofunctors. Grounded in the historical roots of Sieg's discussions, this dissertation shows how the properties of initial algebras for endofunctors and accessible domains coincide in a satisfying and natural way. -/- Filling out this characterization, I show how important examples of accessible domains fit into this broad characterization. I first characterize some accessible domains by relatively simple functors (e.g. finite, polynomial, those that preserve certain colimits) where we can see how iterating the functor can produce the accessible domain. Then I describe accessible domains that result from more involved specifications (e.g. ordinals and segments of the cumulative hierarchies associated with CZF, IZF, and ZF) by relying heavily on algebraic set theory. I end with a discussion of some of the methodological features of category theory in particular that helped characterize accessible domains. (shrink)

It is well known that the following features hold of AR + T under the strong Kleene scheme, regardless of the way the language is Gödel numbered: 1. There exist sentences that are neither paradoxical nor grounded. 2. There are 2ℵ0 fixed points. 3. In the minimal fixed point the weakly definable sets (i.e., sets definable as {n∣ A(n) is true in the minimal fixed point where A(x) is a formula of AR + T) are precisely the Π1 1 sets. (...) 4. In the minimal fixed point the totally defined sets (sets weakly defined by formulae all of whose instances are true or false) are precisely the ▵1 1 sets. 5. The closure ordinal for Kripke's construction of the minimal fixed point is ωCK 1. In contrast, we show that under the weak Kleene scheme, depending on the way the Gödel numbering is chosen: 1. There may or may not exist nonparadoxical, ungrounded sentences. 2. The number of fixed points may be any positive finite number, ℵ0, or 2ℵ0 . 3. In the minimal fixed point, the sets that are weakly definable may range from a subclass of the sets 1-1 reducible to the truth set of AR to the Π1 1 sets, including intermediate cases. 4. Similarly, the totally definable sets in the minimal fixed point range from precisely the arithmetical sets up to precisely the ▵1 1 sets. 5. The closure ordinal for the construction of the minimal fixed point may be ω, ωCK 1, or any successor limit ordinal in between. In addition we suggest how one may supplement AR + T with a function symbol interpreted by a certain primitive recursive function so that, irrespective of the choice of the Godel numbering, the resulting language based on the weak Kleene scheme has the five features noted above for the strong Kleene language. (shrink)

In his 1927 Analysis of Matter and elsewhere, Russell argued that we can successfully infer the structure of the external world from that of our explanatory schemes. While nothing guarantees that the intrinsic qualities of experiences are shared by their objects, he held that the relations tying together those relata perforce mirror relations that actually obtain (these being expressible in the formal idiom of the Principia Mathematica). This claim was subsequently criticized by the Cambridge mathematician Max Newman as true but (...) trivial, insofar as from a closed body of observations (or “Ramsey sentence”) one can always generate other equally-satisfactory networks of relations, provided they respect the original set’s cardinality. Since any model thus generated will be empirically adequate, “[t]he defence is therefore driven back from the fairly safe fictitious-real classification to the much less tenable ‘trivial’ and ‘important’” (Newman 1928). Given the definitional rigour afforded by the initial appeal to isomorphism (via one-to-one correspondences in extension), the received assessment, shared by Russell himself, is that retreating to a pragmatic adjudication would betoken a fatal blow. However, I suggest that reliance on “importance” can be avoided if we incorporate an impersonal criterion of diachronic precedence. When collecting observations, an ordinality emerges alongside the cardinality which gives that underived structure an irrevocable epistemological privilege. Hence, I argue that, all other things being equal, any construct parasitic on an antecedent theory ought to be regarded as inferior/dispensable, since it was generated by an algorithm lacking the world-involving pedigree of its host structure. (shrink)

The purpose of this paper is to communicate - as a proposal - a general method of assigning a 'truthmaker' to any 1st order sentence in each of its models. The respective construct is derived from the standard model theoretic (recursive) satisfaction definition for 1st order languages and is a conservative extension thereof. The heuristics of the proposal (which has been somewhat idiosyncratic from the current point of view) and some more technical detail of the construction may be found in (...) my article on part I of Spinoza's 'ethica, ordine geometrico demonstrata', which is the context within which I elaborated the assignment. But this context need not be repeated here, the presentation of the truthmaker assignment will be comprehensible to anybody with solid basic knowledge in 1st order model theory, anything used is standard and no advanced techniques are required. (shrink)

We aim to compile some means for a rational reconstruction of a named part of the start-over of Baruch (Benedictus) de Spinoza's metaphysics in 'de deo' (which is 'pars prima' of the 'ethica, ordine geometrico demonstrata' ) in terms of 1st order model theory. In so far, as our approach will be judged successful, it may, besides providing some help in understanding Spinoza, also contribute to the discussion of some or other philosophical evergreen, e.g. 'ontological commitment'. For this text we (...) assume the reader familiar with 'de deo' as well as with some basic concepts and results of 1st order model theory. Before we start reconstruction, we will first revisit shortly the concept of 'attributum' (definitio IV) in it's setting in 'de deo' , next scan for formalizable aspects of 'in suo genere finita' ('de deo', definitio II), subsequently list the model theoretic constructs we will make use of. Then we begin reconstruction by stating "coordinative definitions" for the notions of 'attribute (of a substance)', 'modus (as conceived via an attribute)' and 'substance (as conceived via an attribute)', reasoning shortly for each of them. The "coordinative definitions", we will arrive at, must not be understood as literal translations of Spinoza's concepts - of course, there can't be such a thing as a literal translation - they are meant as formal analoga of these concepts, mapping some logical structure. But even with this caveat they may seem strange to the reader at this stage of discussion. Additional justification for them then should be found in our endeavour, to map some argumentation of Spinoza's proofs of some of his propositions from this starting point. -/- [ Version 1 of this paper originated from 2020-03-14. This is version 2 from 2021-09-12 with some readability amendments. ]. (shrink)

It is discerned what light can bring the recent historical reconstructions of maxwellian optics and electromagnetism unification on the following philosophical/methodological questions. I. Why should one believe that Nature is ultimately simple and that unified theories are more likely to be true? II. What does it mean to say that a theory is unified? III. Why theory unification should be an epistemic virtue? To answer the questions posed genesis and development of Maxwellian electrodynamics are elucidated. It is enunciated that the (...) Maxwellian Revolution is a far more complicated phenomenon than it may be seen in the light of Kuhnian and Lakatosian epistemological models. Correspondingly it is maintained that maxwellian electrodynamics was elaborated in the course of the old pre-maxwellian programmes’ reconciliation: the electrodynamics of Ampére-Weber, the wave theory of Young-Fresnel and Faraday’s programme. To compare the different theoretical schemes springing from the different language games James Maxwell had constructed a peculiar neutral language. Initially it had encompassed the incompressible fluid models; eventually – the vortices ones. The three programmes’ encounter engendered the construction of the hybrid theory at first with an irregular set of theoretical schemes. However, step by step, on revealing and gradual eliminating the contradictions between the programmes involved, the hybrid set is “put into order” (Maxwell’s term). A hierarchy of theoretical schemes starting from ingenious crossbreeds (the displacement current) and up to usual hybrids is set up. After the displacement current construction the interpenetration of the pre-maxwellian programmes begins that marks the commencement of theoretical schemes of optics, electricity and magnetism real unification. Maxwell’s programme surpassed that of Ampére-Weber because it did absorb the ideas of the Ampére-Weber programme, as well as the presuppositions of the programmes of Young-Fresnel and Faraday properly co-ordinating them with each other. But the opposite statement is not true. The Ampére-Weber programme did not assimilate the propositions of the Maxwellian programme. Maxwell’s victory over his rivals became possible because the gist of Maxwell’s unification strategy was formed by Kantian epistemology looked in the light of William Whewell and such representatives of Scottish Enlightenment as Thomas Reid and Sir William Hamilton. Maxwell did put forward as basic synthetic principles the ideas that radically differed from that of Ampére-Weber approach by their open, flexible and contra-ontological, genuinly epistemological, Kantian character. For Maxwell, ether was not the ultimate building block of physical reality, from which all the charges and fields should be constructed. “Action at a distance”, “incompressible fluid”, “molecular vortices”, etc. were contrived analogies for Maxwell, capable only to direct the researcher at the “right” mathematical relations. Key words: J.C. Maxwell, unification of optics and electromagnetism, I. Kant, T. Reid, W. Hamilton . (shrink)

A possible world is a junky world if and only if each thing in it is a proper part. The possibility of junky worlds contradicts the principle of general fusion. Bohn (2009) argues for the possibility of junky worlds, Watson (2010) suggests that Bohn‘s arguments are flawed. This paper shows that the arguments of both authors leave much to be desired. First, relying on the classical results of Cantor, Zermelo, Fraenkel, and von Neumann, this paper proves the possibility of junky (...) worlds for certain weak set theories. Second, the paradox of Burali-Forti shows that according to the Zermelo-Fraenkel set theory ZF, junky worlds are possible. Finally, it is shown that set theories are not the only sources for designing plausible models of junky worlds: Topology (and possibly other "algebraic" mathematical theories) may be used to construct models of junky worlds. In sum, junkyness is a relatively widespread feature among possible worlds. (shrink)

Georg Cantor was the genuine discoverer of the Mathematical Infinity, and whatever he claimed, suggested, or even surmised should be taken seriously -- albeit not necessary at its face value. Because alongside his exquisite in beauty ordinal construction and his fundamental powerset description of the continuum, Cantor has also left to us his obsessive presumption that the universe of sets should be subjected to laws similar to those governing the set of natural numbers, including the universal principles of cardinal comparability (...) and well-ordering -- and implying an ordinal re-creation of the continuum. During the last hundred years, the mainstream set-theoretical research -- all insights and adjustments due to Kurt G\"odel's revolutionary insights and discoveries notwithstanding -- has compliantly centered its efforts on ad hoc axiomatizations of Cantor's intuitive transfinite design. We demonstrate here that the ontological and epistemic sustainability} of this design has been irremediably compromised by the underlying peremptory, Reductionist mindset of the XIXth century's ideology of science. (shrink)

The concept of expression has always held an important place in Fernando Gil’s thought. The author’s reflection on the idea of expression is well reflected among his writings – either in his early essays or throughout his entire work –, attaining a special focus in the lectures that he gave at Johns Hopkins University during the years of 2002, 2003 and 2004. Actually, these lectures represent the strongest testimony of the relevancy given by Fernando Gil to this concept. However, despite (...) being a concept which crosses all his work, clearly having a central place in his thought, Fernando Gil never got to write a systematic theory on the concept of expression. Thus, closely following the author’s works, the main purpose of this study is to suggest a personal vision of what could have been Fernando Gil’s theory on expression. Combining some of the main ideas of Johns Hopkins lectures to some fundamental notions of Fernando Gil’s thought, we will try to show that Fernando Gil’s theory on expression is a theory on the ordination of thought, on cognitive construction and, above all, a theory on the elevation of thought to what shall be considered its higher modality, the aesthetical thought. In fact, our major hypothesis is that Fernando Gil’s theory on expression is not only a cognitive theory but also an aesthetical one. As such, its major guidelines shall be: to explain the adequacy of expression to its objects; to show how expression evolves according to the cognitive development of the subject; and, finally, to explain and to justify the elevation of thought to universality, harmony and beauty. (shrink)

Higher-order logic, with its type-theoretic apparatus known as the simple theory of types (STT), has increasingly come to be employed in theorizing about properties, relations, and states of affairs—or ‘intensional entities’ for short. This paper argues against this employment of STT and offers an alternative: ordinal type theory (OTT). Very roughly, STT and OTT can be regarded as complementary simplifications of the ‘ramified theory of types’ outlined in the Introduction to Principia Mathematica (on a realist reading). While STT, understood as (...) a theory of intensional entities, retains the Fregean division of properties and relations into a multiplicity of categories according to their adicities and ‘input types’ and discards the division of intensional entities into different ‘orders’, OTT takes the opposite approach: it retains the hierarchy of orders (though with some modifications) and discards the categorisation of properties and relations according to their adicities and input types. In contrast to STT, this latter approach avoids intensional counterparts of the Epimenides and related paradoxes. Fundamental intensional entities lie at the base of the proposed hierarchy and are also given a prominent part to play in the individuation of non-fundamental intensional entities. (shrink)

It is widely held that under ordinal utility, utility differences are ill-defined. Allegedly, for these to be well-defined (without turning to choice under risk or the like), one should adopt as a new kind of primitive quaternary relations, instead of the traditional binary relations underlying ordinal utility functions. Correlatively, it is also widely held that the key structural properties of quaternary relations are entirely arbitrary from an ordinal point of view. These properties would be, in a nutshell, the hallmark of (...) cardinal utility. While much is obviously true in these two tenets, this note explains why, as stated, they should be abandoned. Any ordinal utility function induces a rich quaternary relation. There is such a thing as ordinal utility differences. Furthermore, this induced quaternary relation respects, apart from completeness, the most standard structural properties of quaternary relations. These properties are, from an ordinal point of view, anything but arbitrary; from a quaternary perspective only completeness should be considered the hallmark—if any—of cardinal utility. These facts are explained to be especially relevant to the critical appreciation of the ordinalist methodology. (shrink)

This paper offers a critique of an account of explanatory integration that claims that explanations of cognitive capacities by functional analyses and mechanistic explanations can be seamlessly integrated. It is shown that achieving such explanatory integration requires that the terms designating cognitive capacities in the two forms of explanation are stable but that experimental practice in the mind-brain sciences currently is not directed at achieving such stability. A positive proposal for changing experimental practice so as to promote such stability is (...) put forward and its implications for explanatory integration are briefly considered. (shrink)

There are multiple formal characterizations of the natural numbers available. Despite being inter-derivable, they plausibly codify different possible applications of the naturals – doing basic arithmetic, counting, and ordering – as well as different philosophical conceptions of those numbers: structuralist, cardinal, and ordinal. Nevertheless, some influential philosophers of mathematics have argued for a non-egalitarian attitude according to which one of those characterizations is more “legitmate” in virtue of being “more basic” or “more fundamental”. This paper addresses two related issues. First, (...) we review some of these non-egalitarian arguments, lay out a laundry list of different, legitimate, notions of relative priority, and suggest that these arguments plausibly employ different such notions. Secondly, we argue that given a metaphysical-cum-epistemological gloss suggested by Frege’s foundationalist epistomology, the ordinals are plausibly more basic than the cardinals. This is just one orientation to relative priority one could take, however. Ultimately, we subscribe to an egalitarian attitude towards these formal characterizations: they are, in some sense, equally “legitimate”. (shrink)

Among Dunn’s many important contributions to relevance logic was his work on the system RM (R-mingle). Although RM is an interesting system in its own right, it is widely considered to be too strong. In this chapter, I revisit a closely related system, RM0 (sometimes known as ‘constructive mingle’), which includes the mingle axiom while not degenerating in the way that RM itself does. My main interest will be in examining this logic from two related semantical perspectives. First, I (...) give a purely operational bisemilattice semantics for it by adapting previous work of Humberstone. Second, I examine a more conventional algebraic semantics for it and discuss how this relates to the operational semantics. A novel operational semantics for J (intuitionistic logic) as well as its conventional Heyting algebraic semantics emerge as special cases of the corresponding semantics for RM0. The results of this chapter suggest that RM0 is a more interesting logic than has been appreciated and that Humberstone’s operational semantic framework similarly deserves more attention than it has received. (shrink)

K denotes both the knowledge predicate satisfied by every currently known theorem and the finite set of all currently known theorems. The set K is time-dependent, publicly available, and contains theorems both from formal and constructive mathematics. Any theorem of any mathematician from past or present forever belongs to K. Mathematical statements with known constructive proofs exist in K separately and form the set K_c⊆K. We assume that mathematical sets are atemporal entities. They exist formally in ZFC theory (...) although their properties can be time-dependent (when they depend on K) or informal. The true statement "K is non-empty" is outside K forever because K is not a formal set. Paul Cohen proved in 1963 that the equality 2^{\aleph_0}=\aleph_1 is independent of ZFC. Before 1963, the statement "There is a known constructively defined integer n⩾1 such that 2^{\aleph_0}=\aleph_n" was outside K. Since 1963, this statement is outside K forever. The true statement "There exists a set X⊆{1,...,49} such that card(X)=6 and X never occurred as the winning six numbers in the Polish Lotto lottery" refers to the current non-mathematical knowledge and is outside K forever. In Martin-Löf's terminology, every currently known theorem is actually true whereas every theorem (known or unknown) is potentially true. Actual truth is knowledge dependent and tensed. Potential truth is knowledge independent and tenseless. Algorithms always terminate. We explain the distinction between algorithms whose existence is provable in ZFC and constructively defined algorithms which are currently known. By using this distinction, we obtain non-trivially true statements on decidable sets X⊆N that belong to constructive and informal mathematics and refer to the current mathematical knowledge on X. For every such statement, its truth concerning the current mathematical knowledge is implied by a true statement of the form: (the conjunction of i conditions of the form α∈K)∧(the conjunction of j conditions of the form β∉K)∧(the conjunction of k conditions of the form γ∈K_c)∧(the conjunction of l conditions of the form δ∉K_c), where i,j,k,l∈N and α,β,γ,δ are mathematical statements. In constructive mathematics and the traditional Brouwerian intuitionism, the truth of a mathematical statement means that we know a constructive proof. Therefore, the truth of a mathematical statement depends on time, where the statement is formally stated in the classical mathematics without the predicate K. In this article, mathematical statements on decidable sets X⊆N refer to time because they refer to the current mathematical knowledge on X. They cannot be formally stated in the classical mathematics without the predicate K and their logical values may change in time. For a set X⊆N whose infiniteness is false or unproven, we define which elements of X are classified as known. No known set X⊆N satisfies Conditions (1)-(4) and is widely known in number theory or naturally defined, where this term has only informal meaning. (1) A known algorithm with no input returns an integer n satisfying card(X)<ω ⇒ X⊆(-∞,n]. (2) A known algorithm for every k∈N decides whether or not k∈X. (3) For every known algorithm A with no input, the statement "A returns the logical value of the statement card(X)=ω" is outside K. (4) There are many elements of X and it is conjectured, though so far unproven, that X is infinite. (5) X is naturally defined. The infiniteness of X is false or unproven. X has the simplest definition among known sets Y⊆N with the same set of known elements. No set X⊆N will satisfy Conditions (1)-(4) forever, if for every algorithm with no input, at some future day, a computer will be able to execute this algorithm in 1 second or less. The physical limits of computation disprove the assumption of the above statement. We prove that the set X={n∈N: the interval [-1,n] contains more than 29.5+(11!/(3n+1))∙sin(n) primes of the form k!+1} satisfies Conditions (1)-(5) except the requirement that X is naturally defined. We present a table that shows satisfiable conjunctions of the form #(Condition1)∧(Condition 2)∧#(Condition 3)∧(Condition 4)∧#(Condition 5), where # denotes the negation ¬ or the absence of any symbol. We prove that there exists a naturally defined set C⊆N which satisfies the following conditions (6)-(11). (6) A known and simple algorithm for every k∈N decides whether or not k∈C. (7) For every known algorithm A with no input, the statement "A returns the logical value of the statement card(C)=ω" is outside K. (8) For every known algorithm A with no input, the statement "A returns the logical value of the statement card(N\C)=ω" is outside K. (9) It is conjectured, though so far unproven, that C is infinite. (10) For every known algorithm A with no input, the statement "A returns an integer n satisfying card(C)<ω ⇒ C⊆(-∞,n]" is outside K. (11) For every known algorithm A with no input, the statement "A returns an integer m satisfying card(N\C)<ω ⇒ N\C⊆(-∞,m]" is outside K. The set C={k∈N: 2^{2^k}+1 is composite} is naturally defined and satisfies conditions (6)-(11). (shrink)

What’s the difference between those psychological posits that are ‘me” and those that are not? Distinguishing between these psychological kinds is important in many domains, but an account of what the distinction consists in is challenging. I argue for Psychological Constructionism: those psychological posits that correspond to the kinds within folk psychology are personal, and those that don’t, aren’t. I suggest that only constructionism can answer a fundamental challenge in characterizing the personal level – the plurality problem. The things that (...) plausibly qualify as personal are motley. Other attempts at accounting for the personal level either cannot accommodate this plurality, or cannot explain what unifies the personal. Given arguments others have given for a pluralistic conception of folk psychology, constructionism explains and predicts this plurality in a systematic and unified way, thereby solving the plurality problem. (shrink)

How Moore’s open question argument works, insofar as it does, remains a matter of controversy. My purpose here is to construct an open question argument based on a novel interpretation of how Moore’s argument might work. In order to sidestep exegetical questions, I do not claim here to be offering Moore’s own argument. Rather, I offer a reconstruction making use of important elements of Moore’s methodology and assumptions that could be reasonable within a Moorean viewpoint. The crucial role within the (...) argument is played by what I call the real thought move. I show that the reconstructed argument is more defensible from some standard objections than the common construction. I finish by drawing attention to a neglected objective that would make sense within Moore’s viewpoint, showing that it fits with a major commitment of Bonjour’s moderate rationalism, and show how the package might be important for the non-naturalist today. (shrink)

Transfinite ordinal numbers enter mathematical practice mainly via the method of definition by transfinite recursion. Outside of axiomatic set theory, there is a significant mathematical tradition in works recasting proofs by transfinite recursion in other terms, mostly with the intention of eliminating the ordinals from the proofs. Leaving aside the different motivations which lead each specific case, we investigate the mathematics of this action of proof transforming and we address the problem of formalising the philosophical notion of elimination which (...) characterises this move. (shrink)

Inspired by Rudolf Carnap's Der Logische Aufbau Der Welt, David J. Chalmers argues that the world can be constructed from a few basic elements. He develops a scrutability thesis saying that all truths about the world can be derived from basic truths and ideal reasoning. This thesis leads to many philosophical consequences: a broadly Fregean approach to meaning, an internalist approach to the contents of thought, and a reply to W. V. Quine's arguments against the analytic and the a priori. (...) Chalmers also uses scrutability to analyze the unity of science, to defend a conceptual approach to metaphysics, and to mount a structuralist response to skepticism. Based on the 2010 John Locke lectures, Constructing the World opens up debate on central philosophical issues involving language, consciousness, knowledge, and reality. This major work by a leading philosopher will appeal to philosophers in all areas. This entry contains uncorrected proofs of front matter, chapter 1, and first excursus. (shrink)

The aim of this paper is to bring recent work on metaphysical grounding to bear on the phenomenon of social construction. It is argued that grounding can be used to analyze social construction and that the grounding framework is helpful for articulating various claims and commitments of social constructionists, especially about social identities, e.g., gender and race. The paper also responds to a number of objections that have been leveled against the application of grounding to social construction from Elizabeth Barnes, (...) Mari Mikkola, and Jessica Wilson. (shrink)

This paper looks at the question of what it means for a psychological test to have construct validity. I approach this topic by way of an analysis of recent debates about the measurement of implicit social cognition. After showing that there is little theoretical agreement about implicit social cognition, and that the predictive validity of implicit tests appears to be low, I turn to a debate about their construct validity. I show that there are two questions at stake: First, what (...) level of detail and precision does a construct have to possess such that a test can in principle be valid relative to it? And second, what kind of evidence needs to be in place such that a test can be regarded as validated relative to a given construct? I argue that construct validity is not an all-or-nothing affair. It can come in degrees, because both our constructs and our knowledge of the explanatory relation between constructs and data can vary in accuracy and level of detail, and a test can fail to measure all of the features associated with a construct. I conclude by arguing in favor of greater philosophical attention to processes of construct development. (shrink)

Constructive empiricism implies that if van Fraassen does not believe that scientific theories and his positive philosophical theories, including his contextual theory of explanation, are empirically adequate, he cannot accept them, and hence he cannot use them for scientific and philosophical purposes. Moreover, his epistemic colleagues, who embrace epistemic reciprocalism, would not believe that his positive philosophical theories are empirically adequate. This epistemic disadvantage comes with practical disadvantages in a social world.

Review of "The Construction of Logical Space", by Augustin Rayo. Oxford: OxfordUniversity Press, 2013. Pp. xix+220. H/b$35.00.

Georg Cantor's absolute infinity, the paradoxical Burali-Forti class Ω of all ordinals, is a monstrous non-entity for which being called a "class" is an undeserved dignity. This must be the ultimate vexation for mathematical philosophers who hold on to some residual sense of realism in set theory. By careful use of Ω, we can rescue Georg Cantor's 1899 "proof" sketch of the Well-Ordering Theorem––being generous, considering his declining health. We take the contrapositive of Cantor's suggestion and add Zermelo's choice (...) function. This results in a concise and uncomplicated proof of the Well-Ordering Theorem. (shrink)

Niche construction theory highlights the active role of organisms in modifying their environment. A subset of these modifications is the developmental niche, which concerns ecological, epistemic, social and symbolic legacies inherited by organisms as resources that scaffold their developmental processes. Since in this theory development is a situated process that takes place in a culturally structured environment, we may reasonably ask if implicit cultural biases may, in some cases, be responsible for maladaptive developmental niches. In this paper we wish to (...) argue for an affirmative answer. In order to do so, we first propose to conceptualize implicit bias as embodied perceptual habits, and then proceed to show that these habits are at least partially responsible for maladaptive developmental niches of children with Down syndrome and autism. With this framework we thus hope to bring together two fields of research that haven’t been explicitly connected: implicit bias and niche construction theory. Linking these two theories may bring benefits both to implicit bias researchers, who can extend this concept to characterize other sets of processes as biased, as well as to niche construction theorists, who will have a useful theoretical tool to diagnose maladaptive features of niches brought about by sociocultural biases. (shrink)

Niche construction is a concept that captures a wide array of biological phenomena, from the environmental effects of metabolism to the creation of complex structures such as termite mounds and beaver dams. A central point in niche construction theory is that organisms do not just passively undergo developmental, ecological, or evolutionary processes, but are also active participants in them Evolution: From molecules to men, Cambridge University Press, Cambridge, 1983; Laland KN, Odling-Smee J, Feldman MW, In: KN Laland and T Uller (...) Evolutionary causation: Biological and philosophical reflections, MIT Press, Cambridge, MA, 2019). In this paper, we distinguish between two fundamentally different ways in which organisms are active participants: as agents and as contributors. Roughly, organisms act as agents when niche constructing effects are a result of a goal-directed behavior over which the organisms have some degree of control. Organisms act as contributors when the niche constructing effects do not arise from a goal to perform the constructive activity. As illustrative examples we discuss plants altering leaf-morphology to optimize light exposure as reported by Sultan and bacteria creating novel niches through excreting energy-rich metabolites. The difference between agential and contributional niche construction is important for understanding the different ways organisms can actively participate in development, ecology, and evolution. Additionally, this distinction can increase our understanding of how the capacity of agency is distributed across the tree of life and how agency influences developmental and evolutionary processes. (shrink)

It is an integral part of the phenomenology of mature perceptual experience that it seems to present to us an experience-independent world. I shall call this feature 'perceptual intentionality'. In this paper, I argue that perceptual intentionality is constructed by the structure of more basic sensory features, features that are not intentional themselves. This theory can explain why the same sensory feature can figure both in presentational and non-presentational experiences. There is a fundamental difference between the intentionality of sensory experiences (...) and the intentionality of thoughts: unlike the former, the latter is not constructed. (shrink)

Part 1 of this exchange consists in a critique by Smith of Searle’s The Construction of Social Reality focusing on Searle’s use of the formula ‘X counts as Y in context C’. Smith argues that this formula works well for social objects such as dollar bills and presidents where the corresponding X terms (pieces of paper, human beings) are easy to identify. In cases such as debts and prices and money in a bank's computers, however, the formula fails, because these (...) are cases of what he calls ‘free-standing Y terms’, since there is here no X which can count as the corresponding Y. In his response in Part 2, Searle argues that Smith’s critique rests on three misunderstandings: 1. in wrongly presupposing that Searle is trying to analyze the nature of what he calls “social objects”, rather than of social facts; 2. in thinking that the counts as formula is intended as a definition, rather than as a mere mnemonic; and 3. in neglecting the naturalism of Searle’s account. (shrink)

In this article, I explain why stabilizing constructs is important to the success of the Research Domain Criteria Project and identify one measure for facilitating such stability.

Kant's reasoning in his special metaphysics of nature is often opaque, and the character of his a priori foundation for Newtonian science is the subject of some controversy. Recent literature on the Metaphysical Foundations of Natural Science has fallen well short of consensus on the aims and reasoning in the work. Various of the doctrines and even the character of the reasoning in the Metaphysical Foundations have been taken to present insuperable obstacles to accepting Kant's claim to ground Newtonian science. (...) Gordon Brittan and Gerd Buchdahl, amongst others, have argued that Kant's stated aims in this case are not to be taken at face value, and that prior ontological commitments play a hidden but central role in Kant's special metaphysics. ;Michael Friedman has shown how Kant's stated aims can be taken seriously with his ingenious reconstruction of the Metaphysical Foundations as a demonstration of the a priori basis for our thinking bodies to be in true motion and in absolute space. However, Friedman does not address the issue of matter theory--despite the importance of the issue to Kant. I argue that a strict reading of both the stated aims and doctrines of the Metaphysical Foundations is possible, since much of Kant's reasoning about the empirical concept of matter can be explained by his views on how the construction of empirical concepts is possible. ;Kant's quasi-mathematical constructions are pivotal in Friedman's interpretation. Constructibility is Kant's criterion of acceptability for the concepts of natural science. Yet Kant notoriously fails to construct the dynamical concept of matter, and accepts this failure with an equally notorious complacency. I argue that Kant's criteria of empirical concept construction, apart from any prior ontological commitments, are enough to generate his views on matter. Kant's failure to construct the requisite concept of matter can be ascribed to a missing law of nature, a law of the relation of forces the discovery of which Kant thought imminent. I conclude that matter theory is central to the Metaphysical Foundations, but that this does not undermine Kant's stated aim of giving the a priori ground of Newtonian science. (shrink)

Memory perspectives on past events allegedly take one of two shapes. In field memories, we recall episodes from a first-person point of view, while in observer memories, we look at a past scene from a third-person perspective. But this mere visuospatial dichotomy faces several practical and conceptual challenges. First, this binary distinction is not exhaustive. Second, this characterization insufficiently accounts for the phenomenology of observer memories. Third, the focus on the visual aspect of memory perspective neglects emotional, agential, and self-related (...) social aspects. Fourth, the focus on the time of recall neglects the fact that visual, emotional, agential, and social aspects of perspective can also be dissociated in the original experience. In this chapter, we move away from the standard visual dichotomy. Instead, we propose that memory perspective is better understood along four lines: visual, agential, emotional, and social. Drawing on empirical research, we argue that these dimensions predict a disposition to recall a past event based on the present situation of the memorizer. This account supports seeing episodic memory as a natural kind, supported by scenario construction mechanisms and minimal memory traces. By remapping the classic distinction between field and observer perspectives along four dimensions, our proposal provides explanatory advantages and secures practical gains by enabling testable hypotheses. (shrink)

It is argued that, contrary to prevailing opinion, Bas van Fraassen nowhere uses the argument from underdetermination in his argument for constructive empiricism. It is explained that van Fraassen’s use of the notion of empirical equivalence in The Scientific Image has been widely misunderstood. A reconstruction of the main arguments for constructive empiricism is offered, showing how the passages that have been taken to be part of an appeal to the argument from underdetermination should actually be interpreted.

A construction principle for natural deduction systems for arbitrary, finitely-many-valued first order logics is exhibited. These systems are systematically obtained from sequent calculi, which in turn can be automatically extracted from the truth tables of the logics under consideration. Soundness and cut-free completeness of these sequent calculi translate into soundness, completeness, and normal-form theorems for natural deduction systems.

In Chapter I, I discuss Buchdahl’s view that the possibility of empirical lawlikeness could not have been established in the Principles of the Critique given the differences between transcendental, metaphysical and empirical lawlikeness, and the connection between the faculty of Reason and empirical lawlikeness. I then discuss the general conditions for empirical hypotheses according to Kant, which include the justification of the method by which an empirical hypothesis is obtained and the establishment of the general and specific constructability of the (...) empirical concept. -/- *In Chapter II, I discuss the nature of the general construction of concepts which is treated in the Schematism of the Critique, surveying the views of Pippin, Allison, Bennett and Butts in an effort both to make sense of a difficult part of the Critique and to demonstrate that the Schematism is indeed where Kant demonstrates how the construction of empirical concepts in general is possible. -/- *In Chapter III, I discuss Brittan’s and Butts’ views on the nature of the specific construction of empirical concepts, defending Butts’ interpretation as compatible with Buchdahl’s view that gaps exist between kinds of lawlikeness for Kant and, because of its connection with an interpretation of how metaphysical lawlikeness figures in the Metaphysical Foundations of Natural Science, insofar as it helps us to establish the possibility of empirical lawlikeness and natural science. (shrink)

ABSTRACT (ENG) One of the concerns of Greek philosophy centred on the question of how a manifold and ordered universe arose out of the primitive state of things. From the mythical accounts dating around the seventh century B.C. to the cosmologies of the Classical period in Ancient Greece, many theories have been proposed in order to answer to this question. How these theories differ in positing a “something” that pre-existed the ordered cosmos has been widely discussed. However, scholars have rarely (...) made explicit how they differ in style of thought. In the span of four centuries the first deductive arguments of the Eleatic philosophers culminated in the emergence of logical proof and a form of explanation of natural phenomena, which consisted of searching for the simplest and fewest premises and deducting implications. In this paper it will be discussed how, at distinct stages of its development, the deductive thinking informed the solutions proposed to solve the chaos-order problem, that of how an ordered universe has been possible. -/- ABSTRACT (ITA) Una delle più importanti questioni della filosofia greca è stata quella di comprendere come sia stato possibile un universo ordinato a partire da uno stato primordiale. Dalle teogonie del VII secolo a.C. fino alle cosmologie dei filosofi dell’età classica, sono state proposte diverse teorie per dare risposta a questa domanda. Come esse differiscano nel postulare l’esistenza di un “qualcosa” di primordiale che preesisteva all’ordine del cosmo è stato molto discusso. Pochi studiosi, però, le hanno esaminate sullo sfondo della lenta evoluzione del pensiero deduttivo, culminata nella dimostrazione in geometria e in una forma di spiegazione dei fenomeni che consisteva nel cercare semplici premesse e inferire conclusioni. In questo articolo si mostrerà come il lento affermarsi della spiegazione razionale prima, dell’argomento deduttivo e della dimostrazione in geometria poi, abbiano dato forma alle diverse risposte al problema caos-ordine, e in particolare alla domanda su come sia sorto un universo ordinato. (shrink)

There is a widely held view on measurement inferences, that goes back to Stevens’s ([1946]) theory of measurement scales and ‘permissible statistics’. This view defends the following prohibition: you should not make inferences from averages taken with ordinal scales (versus quantitative scales: interval or ratio). This prohibition is general—it applies to all ordinal scales—and it is sometimes endorsed without qualification. Adhering to it dramatically limits the research that the social and biomedical sciences can conduct. I provide a Bayesian analysis of (...) this inferential problem, determining when measurements from ordinal scales can be used to confirm hypotheses about relative group averages. The prohibition, I conclude, cannot be upheld, even in a qualified sense. The beliefs needed to make average comparisons are less demanding than those appropriate for quantitative scales. I illustrate with the alleged paradigm ordinal scale, Mohs’ scale of mineral hardness, arguing that the literature has mischaracterized it. (shrink)

In this paper, I argue that Constructive Empiricism (CE) is ambiguous between two interpretations: CE as a normative epistemology of science and CE as a descriptive philosophy of science. When they present CE, constructive empiricists write as if CE is supposed to be more than a normative epistemology of science and that it is meant to be responsible to actual scientific practices. However, when they respond to objections, constructive empiricists fall back on a strictly normative interpretation of (...) CE. This ambiguity seems to make CE immune to objections in a rather ad hoc fashion. (shrink)

My main purpose here is to provide an account of context selection in utterance understanding in terms of the role played by schemata and goals in top-down processing. The general idea is that information is organized hierarchically, with items iteratively organized in chunks—here called “schemata”—at multiple levels, so that the activation of any items spreads to schemata that are the most accessible due to previous experience. The activation of a schema, in turn, activates its other components, so as to predict (...) a likely context for the original item. Since each input activates its own schemata, conflicting schemata compete with (and inhibit) each other, while multiple activations of a schema raise its likelihood to win the competition. There is therefore a double movement—with bottom-up activation of schemata enabling top-down prediction of other contextual components—triggered by multiple sources. Another claim of the paper is that goals are represented by schemata placed at the highest-levels of the executive hierarchy, in accordance with Fuster’s model of the brain as a hierarchically organized perception action cycle. This account can be considered, in part at least, a development of ideas contained in Relevance Theory, though it may imply that some other claims of the theory are in need of revision. Therefore, a secondary purpose of the paper is a contribution to the analysis of that theory. (shrink)

This article explores some open questions related to the problem of verification of theories in the context of empirical sciences by contrasting three epistemological frameworks. Each of these epistemological frameworks is based on a corresponding central metaphor, namely: (a) Neo-empiricism and the gambling metaphor; (b) Popperian falsificationism and the scientific tribunal metaphor; (c) Cognitive constructivism and the object as eigen-solution metaphor. Each of one of these epistemological frameworks has also historically co-evolved with a certain statistical theory and method for testing (...) scientific hypotheses, respectively: (a) Decision theoretic Bayesian statistics and Bayes factors; (b) Frequentist statistics and p-values; (c) Constructive Bayesian statistics and e-values. This article examines with special care the Zero Probability Paradox (ZPP), related to the verification of sharp or precise hypotheses. Finally, this article makes some remarks on Lakatos’ view of mathematics as a quasi-empirical science. (shrink)

A naturalistic theory of intentionality is proposed that differs from previous evolutionary and tracking theories. Full-blown intentionality is constructed through a series of evolvable refinements. A first, minimal version of intentionality originates from a conjectured internal process that estimates an organism’s own fitness and that continually modifies the organism. This process produces the directedness of intentionality. The internal estimator can be parsed into intentional components that point to components of the process that produces fitness. It is argued that such intentional (...) components can point to mistaken or non-existing entities. Different Fregian senses of the same reference correspond to different components that have different roles in the estimator. Intentional components that point to intentional components in other organisms produce directedness towards semi-abstract entities. Finally, adding a general, population-wide means of communication enables intentional components that point to fully abstract entities. Intentionality thus naturalized has all of its expected properties: being directed; potentially making errors; possibly pointing to non-existent, abstract, or rigid entities; capable of pointing many-to-one and one-to-many; distinguishing sense and reference; having perspective and grain; and having determinate content. Several examples, such as ‘swampman’ and ‘brain-in-a-vat’, illustrate how the theory can be applied. (shrink)

Martin Heidegger devotes extensive discussion to medieval philosophers, particularly to their treatment of Truth and Being. On both these topics, Heidegger accuses them of forgetting the question of Being and of being responsible for subjugating truth to the modern crusade for certainty: ‘truth is denied its own mode of being’ and is subordinated ‘to an intellect that judges correctly’. Though there are some studies that discuss Heidegger’s debt to and criticism of medieval thought, particularly that of Thomas Aquinas, there is (...) no constructive reply to his assertions. As a result, Heidegger’s critique had an unprecedented effect on the credibility of medieval philosophy, whereby great portions of the philosophical community dismiss it altogether as an illegitimate Onto‐Theology. It is the aim of this study to offer a constructive reply that will fundamentally grapple with these allegations. By constructive reply we mean not only a reply that avoids the problems Heidegger raises regarding existence, essence and truth, but more importantly, one that uses Heidegger’s criticism in order to present a more insightful account of these notions. The present study is composed of two parts where the second serves as a sort of addendum. The first part, the core of this study, is an attempt to develop an understanding of the distinction between essence and existence that, on the one hand, accords with Heidegger’s criticism while on the other hand advances our understanding of how we think and understand reality. After presenting Heidegger’s depiction of Aquinas’s distinction between essence and existence (esse) as a real distinction, the study will present several views propounded by scholars of Aquinas regarding the status of this distinction. It will be argued that it is not clear whether the distinction is real, formal or conceptual, and that different types of distinction are applied in different places, particularly in regard to the phantasm that Aquinas considers essential to the human act of thinking. The second part diverges from the first part and focuses on Heidegger’s criticism of Aquinas’s conception of truth as adequation, i.e., what it is that grounds the possibility of truth as adequation. This divergence is necessary in order to present a full metaphysical response to Heidegger’s criticism. Since the aim of the present study is to argue that Aquinas’s philosophical system can contend with Heidegger’s criticism, a partial reply would greatly diminish its effectiveness. (shrink)

An ideal constructor produces geometry from scratch, modelled through the bottom-up assembly of a graph-like lattice within a space that is defined, bootstrap-wise, by that lattice. Construction becomes the problem of assembling a homogeneous lattice in three-dimensional space; that becomes the problem of resolving geometrical frustration in quasicrystalline structure; achieved by reconceiving the lattice as a dynamical system. The resulting construction is presented as the introductory model sufficient to motivate the formal argument that it is a fundamental structure; based on (...) which, it is proposed that where mathematics’ numbers conventionally correspond to dimensionless points on the stateless number line, numbers more fundamentally correspond to an ordering of discrete objects constructed within the stateful number lattice. A second observation is that this fundamental lattice structure is helically configured with fractal character, which, as it relates to the geometry underlying spacetime, has relevance to questions in physics, particularly those involving wave-particle duality. (shrink)

This paper considers logics which are formally dual to intuitionistic logic in order to investigate a co-constructive logic for proofs and refutations. This is philosophically motivated by a set of problems regarding the nature of constructive truth, and its relation to falsity. It is well known both that intuitionism can not deal constructively with negative information, and that defining falsity by means of intuitionistic negation leads, under widely-held assumptions, to a justification of bivalence. For example, we do not (...) want to equate falsity with the non-existence of a proof since this would render a statement such as “pi is transcendental” false prior to 1882. In addition, the intuitionist account of negation as shorthand for the derivation of absurdity is inadequate, particularly outside of purely mathematical contexts. To deal with these issues, I investigate the dual of intuitionistic logic, co-intuitionistic logic, as a logic of refutation, alongside intuitionistic logic of proofs. Direct proof and refutation are dual to each other, and are constructive, whilst there also exist syntactic, weak, negations within both logics. In this respect, the logic of refutation is weakly paraconsistent in the sense that it allows for statements for which, neither they, nor their negation, are refuted. I provide a proof theory for the co-constructive logic, a formal dualizing map between the logics, and a Kripke-style semantics. This is given an intuitive philosophical rendering in a re-interpretation of Kolmogorov’s logic of problems. (shrink)

Economies are complicated systems encompassing micro behaviors, interaction patterns, and global regularities. Whether partial or general in scope, studies of economic systems must consider how to handle difficult real-world aspects such as asymmetric information, imperfect competition, strategic interaction, collective learning, and the possibility of multiple equilibria. Recent advances in analytical and computational tools are permitting new approaches to the quantitative study of these aspects. One such approach is Agent-based Computational Economics (ACE), the computational study of economic processes modeled as dynamic (...) systems of interacting agents. This chapter explores the potential advantages and disadvantages of ACE for the study of economic systems. General points are concretely illustrated using an ACE model of a two-sector decentralized market economy. Six issues are highlighted: Constructive understanding of production, pricing, and trade processes; the essential primacy of survival; strategic rivalry and market power; behavioral uncertainty and learning; the role of conventions and organizations; and the complex interactions among structural attributes, institutional arrangements, and behavioral dispositions. (shrink)

Recent debates in metaphysics have highlighted the significance of type theories, such as Simple Type Theory (STT), for our philosophical analysis. In this chapter, I present the salient features of a constructive type theory in the style of Martin-Löf, termed CTT. My principal aim is to convey the flavour of this rich, flexible and sophisticated theory and compare it with STT. I especially focus on the forms of quantification which are available in CTT. A further aim is to argue (...) that a comparison between a plurality of theories is beneficial to the philosophical analysis. We may, for example, discover helpful features of one theory that we may want to import into another context, thus enriching our repertoire of formal tools. Or, through comparison with a less well-known theory, we may gain a better understanding of the characteristics of the theories we are familiar with. As argued in this chapter, CTT has much to offer in all of these respects. (shrink)

The paper presents a new approach to the history of analytic philosophy. Instead of exploring different kinds of analysis (Michael Beaney), or to marry analytic philosophy to the analytic / synthetic distinction (Scott Soames), we turn attention to the fact that it was rooted in two different types of logical constructing. The discrepancy between the two concepts of logical constructing produced much unclarity in our understanding of analytic philosophy.

You and I can differ in what we say, or believe, even though the things we say, or believe, are logically equivalent. Discussing what is said, or believed, requires notions of content which are finer-grained than sets of (metaphysically or logically) possible worlds. In this paper, I develop the approach to fine-grained content in terms of a space of possible and impossible worlds. I give a method for constructing ersatz worlds based on theory of substantial facts. I show how this (...) theory overcomes an objection to actualist constructions of ersatz worlds and argue that it naturally gives rise to useful notions of fine-grained content. (shrink)

An increasing number of philosophers argue that indeterminacy is metaphysical (or worldly) in the sense that indeterminacy has its source in the world itself (rather than how the world is represented or known). The standard arguments for metaphysical indeterminacy are centered around the sorites paradox. In this essay, I present a novel argument for metaphysical indeterminacy. I argue that metaphysical indeterminacy follows from the existence of constitutive social construction; there is indeterminacy in the social world because there is indeterminacy in (...) how the social world is constructed. (shrink)