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)

Kearns (2021) reconstructs Berkeley’s (1713) Master Argument as a formally valid argument against the Materialist Thesis, with the key premise the Distinct Conceivability Thesis, namely the thesis that truths about sensible objects having or lacking thinkable qualities are (distinctly) conceivable and as its conclusion that all sensible objects are conceived. It will be shown that Distinct Conceivability Thesis entails the Reduction Thesis, which states that de dicto propositional (ordinary or distinct) conceivability reduces to de re propositional (ordinary or distinct) conceivability. (...) While Kearns (2021) set out to avoid the confusion of a de re claim with a de dicto claim, which Prior (1955) accused Berkeley of, Kearns can be accused of making a similar conceptual confusion. (shrink)

The subject of the first section is Carnapian modal logic. One of the things I will do there is to prove that certain description principles, viz. the ''self-predication principles'', i.e. the principles according to which a descriptive term satisfies its own descriptive condition, are theorems and that others are not. The second section will be devoted to Carnapian modal arithmetic. I will prove that, if the arithmetical theory contains the standard weak principle of induction, modal truth collapses to truth. Then (...) I will propose a different formulation of Carnapian modal arithmetic and establish that it is free of collapse. Noteworthy is that one can retain the standard strong principle of induction. I will occupy myself in the third section with Carnapian epistemic logic and arithmetic. Here too it is claimed that the standard weak principle of induction is invalid and that the alternative principle is valid. In the fourth and last section I will get back to the self-predication principles and I will point to some of the consequences if one adds them to Carnapian Epistemic arithmetic. The interaction of self-predication principles and the strong principle of induction results in a collapse of de re knowability. (shrink)

Many philosophical discussions hinge on the concept of knowability. For example, there is a blooming literature on the so-called paradox of knowability. How to understand this notion, however? In this paper, we examine several approaches to the notion: the naive approach to take knowability as the possibility to know, the counterfactual approach endorsed by Edgington (1985) and Schlöder (2019) , approaches based on the notion of a capacity or ability to know (Fara 2010, Humphreys 2011), and finally, approaches that make (...) use of the resources of dynamic epistemic logic (van Benthem 2004, Holliday 2017). (shrink)

Semantic externalism in the style of McDowell and Evans faces a puzzle formulated by Pryor: to explain that a sentence such as 'Jack exists' is only a posteriori knowable, despite being logically entailed by the seemingly logical truth 'Jack is self-identical', and hence being itself a logical truth and therefore a priori knowable. Free logics can dissolve the puzzle. Moreover, Pryor has argued that the existentially hedged 'If Jack exists, then Jack is self-identical', when properly formalised, is a logical truth (...) in a system of neutral free logic and therefore a priori knowable, while it does not entail that Jack exists. The latter holds also for negative free logic. In response, Yeakel has argued that on any system of neutral free logic existence hedges will either entail some unwanted existence claims or they will not entail some wanted existence claims. The dilemma also holds for any non-positive free logic. It will be shown that the extension of one of the systems of neutral free logic with a truth operator escapes Yeakel's dilemma, whereas no other non-positive free logic when extended with the truth operator does the same (or it breaks quantifier exchangeability). (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.

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)

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)

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)

Marton ( 2019 ) 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 ( 1985 ) reformulated antirealist theory of truth can deal with his modal-epistemic collapse argument at no cost. (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.

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)

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)

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)

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)

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)

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.

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)

The central topic of this article is (the possibility of) de re knowledge about natural numbers and its relation with names for numbers. It is held by several prominent philosophers that (Peano) numerals are eligible for existential quantification in epistemic contexts (‘canonical’), whereas other names for natural numbers are not. In other words, (Peano) 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)

Both Lowe and Tsai have presented their own versions of the theory that both indicative and subjunctive conditionals are strict conditionals. We critically discuss both versions and we find each version wanting.

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)

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)