In a recent paper Horsten embarked on a journey along the limits of the domain of the unknowable. Rather than knowability simpliciter, he considered a priori knowability, and by the latter he meant absolute provability, i.e. provability that is not relativized to a formal system. He presented an argument for the conclusion that it is not absolutely provable that there is a natural number of which it is true but absolutely unprovable that it has a certain property. The argument depends (...) on a description principle. I will argue that the latter principle implies the knowability of all arithmetical truths. Therefore, Horsten's argument is either sound but its conclusion is trivial, or his argument is unsound. (shrink)

The focus of this article is the question whether the notion of being in a position to know is closed under modus ponens. The question is answered negatively.

Famously, the Church–Fitch paradox of knowability is a deductive argument from the thesis that all truths are knowable to the conclusion that all truths are known. In this argument, knowability is analyzed in terms of having the possibility to know. Several philosophers have objected to this analysis, because it turns knowability into a nonfactive notion. In addition, they claim that, if the knowability thesis is reformulated with the help of factive concepts of knowability, then omniscience can be avoided. In this (...) article we will look closer at two proposals along these lines :557–568, 1985; Fuhrmann in Synthese 191:1627–1648, 2014a), because there are formal models available for each. It will be argued that, even though the problem of omniscience can be averted, the problem of possible or potential omniscience cannot: there is an accessible state at which all truths are known. Furthermore, it will be argued that possible or potential omniscience is a price that is too high to pay. Others who have proposed to solve the paradox with the help of a factive concept of knowability should take note :53–73, 2010; Spencer in Mind 126:466–497, 2017). (shrink)

Carnap's theory of descriptions was restricted in two ways. First, the descriptive conditions had to be non-modal. Second, only primitive predicates or the identity predicate could be used to predicate something of the descriptum . The motivating reasons for these two restrictions that can be found in the literature will be critically discussed. Both restrictions can be relaxed, but Carnap's theory can still be blamed for not dealing adequately with improper descriptions.

Within the context of the Quine–Putnam indispensability argument, one discussion about the status of mathematics is concerned with the ‘Enhanced Indispensability Argument’, which makes explicit in what way mathematics is supposed to be indispensable in science, namely explanatory. If there are genuine mathematical explanations of empirical phenomena, an argument for mathematical platonism could be extracted by using inference to the best explanation. The best explanation of the primeness of the life cycles of Periodical Cicadas is genuinely mathematical, according to Baker (...) :223–238, 2005; Br J Philos Sci 60:611–633, 2009). Furthermore, the result is then also used to strengthen the platonist position :779–793, 2017a). We pick up the circularity problem brought up by Leng Mathematical reasoning, heuristics and the development of mathematics, King’s College Publications, London, pp 167–189, 2005) and Bangu :13–20, 2008). We will argue that Baker’s attempt to solve this problem fails, if Hume’s Principle is analytic. We will also provide the opponent of the Enhanced Indispensability Argument with the so-called ‘interpretability strategy’, which can be used to come up with alternative explanations in case Hume’s Principle is non-analytic. (shrink)

From Leibniz to Krauss philosophers and scientists have raised the question as to why there is something rather than nothing. Why-questions request a type of explanation and this is often thought to include a deductive component. With classical logic in the background only trivial answers are forthcoming. With free logics in the background, be they of the negative, positive or neutral variety, only question-begging answers are to be expected. The same conclusion is reached for the modal version of the Question, (...) namely ‘Why is there something contingent rather than nothing contingent?’. The categorial version of the Question, namely ‘Why is there something concrete rather than nothing concrete?’, is also discussed. The conclusion is reached that deductive explanations are question-begging, whether one works with classical logic or positive or negative free logic. I also look skeptically at the prospects of giving causal-counterfactual or probabilistic answers to the Question, although the discussion of the options is less comprehensive and the conclusions are more tentative. The meta-question, viz. ‘Should we not stop asking the Question’, is accordingly tentatively answered affirmatively. (shrink)

The topic of this article is the closure of a priori knowability under a priori knowable material implication: if a material conditional is a priori knowable and if the antecedent is a priori knowable, then the consequent is a priori knowable as well. This principle is arguably correct under certain conditions, but there is at least one counterexample when completely unrestricted. To deal with this, Anderson proposes to restrict the closure principle to necessary truths and Horsten suggests to restrict it (...) to formulas that belong to less expressive languages. In this article it is argued that Horsten’s restriction strategy fails, because one can deduce that knowable ignorance entails necessary ignorance from the closure principle and some modest background assumptions, even if the expressive resources do not go beyond those needed to formulate the closure principle itself. It is also argued that it is hard to find a justification for Anderson’s restricted closure principle, because one cannot deduce it even if one assumes very strong modal and epistemic background principles. In addition, there is an independently plausible alternative closure principle that avoids all the problems without the need for restriction. (shrink)

The subject of this article is Modal-Epistemic Arithmetic (MEA), a theory introduced by Horsten to interpret Epistemic Arithmetic (EA), which in turn was introduced by Shapiro to interpret Heyting Arithmetic. I will show how to interpret MEA in EA such that one can prove that the interpretation of EA is MEA is faithful. Moreover, I will show that one can get rid of a particular Platonist assumption. Then I will discuss models for MEA in light of the problems of logical (...) omniscience and logical competence. Awareness models, impossible worlds models and syntactical models have been introduced to deal with the first problem. Certain conditions on the accessibility relations are needed to deal with the second problem. I go on to argue that those models are subject to the problem of quantifying in, for which I will provide a solution. (shrink)

The central question of this article is how to combine counterfactual theories of knowledge with the notion of actuality. It is argued that the straightforward combination of these two elements leads to problems, viz. the problem of easy knowledge and the problem of missing knowledge. In other words, there is overgeneration of knowledge and there is undergeneration of knowledge. The combination of these problems cannot be solved by appealing to methods by which beliefs are formed. An alternative solution is put (...) forward. The key is to rethink the closeness relation that is at the heart of counterfactual theories of knowledge. (shrink)

Jonathan Lowe has argued that a particular variation on C.I. Lewis' notion of strict implication avoids the paradoxes of strict implication. We show that Lowe's notion of implication does not achieve this aim, and offer a general argument to demonstrate that no other variation on Lewis' notion of constantly strict implication describes the logical behaviour of natural-language conditionals in a satisfactory way.

The central topic of this article is de re knowledge about natural numbers and its relation with names for numbers. It is held by several prominent philosophers that numerals are eligible for existential quantification in epistemic contexts, whereas other names for natural numbers are not. In other words, numerals are intimately linked with de re knowledge about natural numbers, whereas the other names for natural numbers are not. In this article I am looking for an explanation of this phenomenon. It (...) is argued that the standard induction scheme plays a key role. (shrink)

Marton argues that that it follows from the standard antirealist theory of truth, which states that truth and possible knowledge are equivalent, that knowing possibilities is equivalent to the possibility of knowing, whereas these notions should be distinct. Moreover, he argues that the usual strategies of dealing with the Church–Fitch paradox of knowability are either not able to deal with his modal-epistemic collapse result or they only do so at a high price. Against this, I argue that Marton’s paper does (...) not present any seriously novel challenge to anti-realism not already found in the Church–Fitch result. Furthermore, Edgington reformulated antirealist theory of truth can deal with his modal-epistemic collapse argument at no cost. (shrink)

This article is about the ontological dispute between finitists, who claim that only finitely many numbers exist, and infinitists, who claim that infinitely many numbers exist. Van Bendegem set out to solve the 'general problem' for finitism: how can one recast finite fragments of classical mathematics in finitist terms? To solve this problem Van Bendegem comes up with a new brand of finitism, namely so-called 'apophatic finitism'. In this article it will be argued that apophatic finitism is unable to represent (...) the negative ontological commitments of infinitism or, in other words, that which does not exist according to infinitism. However, there is a brand of infinitism, so-called 'apophatic infinitism', that is able to represent both the positive and the negative ontological commitments of apophatic finitism. Unfortunately, apophatic finitism cannot adopt that way without losing the ability to represent the positive ontological commitments of infinitism. (shrink)

Antirealists who hold the knowability thesis, namely that all truths are knowable, have been put on the defensive by the Church-Fitch paradox of knowability. Rejecting the non-factivity of the concept of knowability used in that paradox, Edgington has adopted a factive notion of knowability, according to which only actual truths are knowable. She has used this new notion to reformulate the knowability thesis. The result has been argued to be immune against the Church-Fitch paradox, but it has encountered several other (...) triviality objections. Schlöder in a forthcoming paper defends the general approach taken by Edgington, but amends it to save it in turn from the triviality objections. In this paper I will argue, first, that Schlöder's justification for the factivity of his version of the concept of knowability is vulnerable to criticism, but I will also offer an improved justification that is in the same spirit as his. To the extent that some philosophers are right about our intuitive concept of knowability being a factive one, it is important to explore factive concepts of knowability that are made formally precise. I will subsequently argue that Schlöder's version of the knowability thesis overgenerates knowledge or, in other words, it leads to attributions of knowledge where there is ignorance. This fits a general pattern for the research programme initiated by Edgington. This paper also contains preliminary investigations into the internal and logical structure of lines of inquiries, which raise interesting research questions. (shrink)

In the first chapter I have introduced Carnapian intensional logic again st the background of Frege s and Quine s puzzles. The main body of the d issertation consists of two parts. In the first part I discussed Carnapi an modal logic and arithmetic with descriptions. In the second chapter, I have described three Carnapian theories, CCL, CFL, and CNL. All three theories have three things in common. F irst, they are formulated in languages containing description terms. Sec ond, they (...) contain a system of modal logic. Third, they do not contain th e unrestricted classical substitution principle, but they do contain the classical substitution principle restricted to non-modal formulas and t he Carnapian substitution principle, which says that two terms can be s ubstituted salva veritate if they are necessarily coreferential. There a re two major differences between the three theories. First, CCL and CFL allow universal instantiation with description ter ms, whereas CNL does not. Moreover, the quantificational theo ry of the CCL is classical, whereas the quantificational theo ry of CFL is a free logic. Another difference is t hat CCL and CFL contain different description principles. Most import antly, the description principle of CCL ensures that even imp roper descriptions have a denotation, whereas the description principle of CFL does not guarantee this. CNL does not have a description prin ciple. In the third chapter, I have studied collapse arguments for CCL, CFL, and CNL. A collapse argument is an argum ent for the following statement: if p is true, then it is nec essarily true. A crucial role in the proofs of these collapse results wa s played by so-called self-predication principles, which say that unde r certain conditions the predicate that expresses the descriptive condition can be combined by the description term formed ou t of that predicate with the result being a true sentence. In this chapt er I have discussed a collapse argument for the extension of CCL with a self-predication principle, I have given a collapse argument for a similarly extended CFL, and most importantly, I have gi ven a collapse argument for the extension of CNL with a self- predication principle. Finally, I have argued that the relevant self-pre dication principles are unsound under a Carnapian interpretation. In the fourth chapter, I have studied the extension of Peano Arithmetic with a Carnapian modal logic C, which is a dummy l etter standing for either CCL or CFL. One can prov e that the principle of the necessity of identity is a theorem of CPA. This implies that one gets a collapse result for CPA. The standard principle of weak induction was crucial for the proof. O ne can also prove that, if one assumes a particular self-predication pri nciple, and if one assumes the principle of strong induction or, equivalently, the least-number principle, then one gets a partial collap se of de re modal truths in de dicto modal tr uths. I have argued that, if the box operator is interpreted as a metaph ysical necessity operator, then Platonists would not be inimical to the collapse result. But if CPA is extended with a physical theor y, then there is a threat that physical truths become physical necessiti es. It was shown that, under a Carnapian interpretation, the standard pr inciple of weak induction is unsound, and that it can be replaced by a C arnapian principle of weak induction that is sound. The probl em of logical and mathematical omniscience prevents ordinary Carnapian i ntensional logic from being taken seriously as a logic adequate for desc ribing the principles of demonstrability. Yet many of the proof-theoreti c results of the first part carry over to the part on Carnapian epistemi c arithmetic with descriptions, since proof-theoretic results are indepe ndent of the informal reading of the operators. In the fifth chapter, I looked at extensions of arithmetic with a modal logic in which the box operator is interpreted as a demonstrability oper ator. A first extension in that sense is Shapiro s Epistemic Arithmetic. Shapiro himself offered the problem of mathematic al omniscience as a reason why it is difficult to find a model theory fo r EA.Horsten attempted to provide a model theory via the deto ur of Modal-Epistemic Arithmetic. The attention of the reade r was drawn to an incoherence in the model theory of. Two al ternative solutions were presented and, after a short discussion of the problem of de re demonstrability one of those alternatives wa s chosen. The discussion of the problem of de re demonstrabil ity made it clear that it would be interesting to study the epistemic pr operties of notation systems. Horsten himself provided a framework for t his, viz. Carnapian Epistemic Arithmetic, and he started a systematic study of the epistemic properti es of notation systems within that framework. However, he did not provid e non-trivial but adequate models. To make a start with solving the prob lem of finding good models for CEA, I introduced Carnapian Mo dal-Epistemic Arithmetic In constructing CMEA I incorporated the lesson about the principle of weak induction learnt in the fourth chapter. In the sixth chapter, I gave a critical assessment of an argument concerning the limits of de re demonst rability about the natural numbers. The conclusion of the Description Ar gument is that it is undemonstrable that there is a natural number that has a certain property but of which it is undemonstrable that it has tha t property. A crucial step in the Description Argument involved a self-p redication principle. Making good use of one of the results obtained in the third chapter, I proved a collapse result for the background theory against which the Description Argument was formulated. I concluded that either the either the Description Argument is sound but its conclusion i s trivial, o r the Description Argument is unsound, or it is a cheapshot. As an appendix I included an article co-authored by prof. dr. Leon Horst en and me. The topic of the article is indirectly related to some other topics investigated in my dissertation. Also, it backs up one of the addition al theses I might be asked to publicly defend during my doctoral exam. T he topic of the appendix is the set of the so-called paradoxes of stric t implication. Jonathan Lowe has argued that a particular variation on C.I. Lewis notion of strict implication avoids the paradoxes of strict implication. Pace Lowe, it is argued that Lowe s notion of implication d oes not achieve this aim. Moreover, a general argument is offered to the effect that no other variation on Lewis notion of constantly strict imp lication describes the logical behaviour of natural language conditional s in a satisfactory way. (shrink)

Het meest succesvolle denken over de natuur vind je in de natuurwetenschappen. Filosofie wordt wel eens omschreven als denken over denken. In het handboek Over wetenschappelijk denken behandelen we het denken over het wetenschappelijk denken. Dat maakt van dit boek zowel een algemene inleiding in de wijsbegeerte als meer in het bijzonder een inleiding tot de wetenschapsfilosofie. -/- Eerst gaan we in dit handboek dieper in op de natuurfilosofische revolutie in het antieke Griekenland. De mythische verklaringen van natuurfenomenen zoals regenbogen (...) moesten toen plaats maken voor natuurfilosofische verklaringen die beroep doen op zuiver fysische oorzaken, zoals gekleurde wolken. Vervolgens komt de natuurwetenschappelijke revolutie aan bod. Het filosofisch denken over de natuur en het Aristotelische wereldbeeld worden op hun beurt verdrongen door de moderne natuurwetenschap en astronomie van Copernicus, Kepler, Galilei, Newton en Lavoisier. -/- Zowel de oude Grieken als moderne filosofen en natuurwetenschappers zoals Bacon en Newton dachten niet alleen over de natuur na, maar ook over de manier waarop het best over de natuur wordt nagedacht. Wetenschappelijke redeneervormen zoals deductie, inductie en abductie hebben daarin een prominente plaats. Elk van deze redeneervormen en ook logica en waarschijnlijkheid bekijken we van naderbij. De observatie van een enkele zwarte zwaan (cover) volstaat om te deduceren dat niet alle zwanen wit zijn, ook als we door talloze vroegere observaties van witte zwanen inductief hadden afgeleid dat het waarschijnlijk is dat alle zwanen wit zijn. Zo komen we tot reflectie op wat wetenschap nu eigenlijk is en hoe we wetenschap kunnen onderscheiden van pseudowetenschap. (shrink)

I will examine three claims made by Ackerman and Kripke. First, they claim that not any arithmetical terms is eligible for universal instantiation and existential generalisation in doxastic or epistemic contexts. Second, Ackerman claims that Peano numerals are eligible for universal instantiation and existential generalisation in doxastic or epistemic contexts. Kripke's position is a bit more subtle. Third, they claim that the successor relation and the smaller-than relation must be effectively calculable. These three claims will be examined from the framework (...) of modal-epistemic arithmetic, i.e. arithmetic extended with certain modal, epistemic and modal-epistemic principles. I will present theorems that give support to the claims made by Ackerman and Kripke. (shrink)

Rosenkranz has recently proposed a logic for propositional, non-factive, all-things-considered justification, which is based on a logic for the notion of being in a position to know, 309–338 2018). Starting from three quite weak assumptions in addition to some of the core principles that are already accepted by Rosenkranz, I prove that, if one has positive introspective and modally robust knowledge of the axioms of minimal arithmetic, then one is in a position to know that a sentence is not provable (...) in minimal arithmetic or that the negation of that sentence is not provable in minimal arithmetic. This serves as the formal background for an example that calls into question the correctness of Rosenkranz’s logic of justification. (shrink)

Fregean theories of descriptions as terms have to deal with improper descriptions. To save bivalence various proposals have been made that involve assigning referents to improper descriptions. While bivalence is indeed saved, there is a price to be paid. Instantiations of the same general scheme, viz. the one and only individual that is F and G is G, are not only allowed but even required to have different truth values.

The focus of the article is the self-predication principle, according to which the/a such-and-such is such-and-such. We consider contemporary approaches (Frege, Russell, Meinong) to the self-predication principle, as well as fourteenth-century approaches (Burley, Ockham, Buridan). In crucial ways, the Ockham-Buridan view prefigures Russell’s view, and Burley’s view shows a striking resemblance to Meinong’s view. In short the Russell-Ockham-Buridan view holds: no existence, no truth. The Burley-Meinong view holds, in short: intelligibility suffices for truth. Both views approach self-predication in a uniform (...) way. We were unable to find a medieval philosopher who, like Frege, approaches self-predication in a non-uniform way. We do not want to dispute that there are also considerable differences between the contemporary and the fourteenth-century approach to self-predication. Importantly, fourteenth-century accounts of self-predication rely strongly on the identity theory of predication, and this theory is not endorsed by contemporary philosophers of language. Nevertheless, some basic tenets seem to us to be clearly shared between fourteenth-century and contemporary thinkers. (shrink)

Halbach has argued that Tarski biconditionals are not ontologically conservative over classical logic, but his argument is undermined by the fact that he cannot include a theory of arithmetic, which functions as a theory of syntax. This article is an improvement on Halbach's argument. By adding the Tarski biconditionals to inclusive negative free logic and the universal closure of minimal arithmetic, which is by itself an ontologically neutral combination, one can prove that at least one thing exists. The result can (...) then be strengthened to the conclusion that infinitely many things exist. Those things are not just all Gödel codes of sentences but rather all natural numbers. Against this background inclusive negative free logic collapses into noninclusive free logic, which collapses into classical logic. The consequences for ontological deflationism with respect to truth are discussed. (shrink)