Second-order Knowledge

Canadian Journal of Philosophy, 2019

It has recently been argued that a sensitivity theory of knowledge cannot account for intuitively appealing instances of higher-order knowledge. In this paper, we argue that it can once careful attention is paid to the methods or processes by which we typically form higher-order beliefs. We base our argument on what we take to be a well-motivated and commonsensical view on how higher-order knowledge is typically acquired, and we show how higher-order knowledge is possible in a sensitivity theory once this view is adopted. Keywords Higher-order knowledge · Sensitivity theory · Transmission principle for knowledge · Belief-forming methods * Thanks to two anonymous referees for helpful comments.

Ratio, 2005

The paper argues that Jackson's knowledge argument fails to undermine physicalist ontology. First, it is argued that, as this argument stands, it begs the question. Second, it is suggested that, by supplementing the argument (and taking one of its premises for granted), this flaw can be remedied insofar as the argument is taken to be an argument against type-physicalism; however, this flaw cannot be remedied insofar as the argument is taken to be an argument against token-physicalism. The argument cannot be supplemented so as to show that experiences have properties which are illegitimate from a physicalist perspective.

Outline of a logico-philosophical and pedagogical project.

“Second-order Logic” in Anderson, C.A. and Zeleny, M., Eds. Logic, Meaning, and Computation: Essays in Memory of Alonzo Church. Dordrecht: Kluwer, 2001. This expository article focuses on the fundamental differences between second- order logic and first-order logic. It is written entirely in ordinary English without logical symbols. It employs second-order propositions and second-order reasoning in a natural way to illustrate the fact that second-order logic is actually a familiar part of our traditional intuitive logical framework and that it is not an artificial formalism created by specialists for technical purposes. To illustrate some of the main relationships between second-order logic and first-order logic, this paper introduces basic logic, a kind of zero-order logic, which is more rudimentary than first-order and which is transcended by first-order in the same way that first-order is transcended by second-order. The heuristic effectiveness and the historical importance of second-order logic are reviewed in the context of the contemporary debate over the legitimacy of second-order logic. Rejection of second-order logic is viewed as an incipient paradigm shift involving radical repudiation of a part of our scientific tradition that is defended by traditionalists. But it is also viewed as analogous to the reactionary repudiation of symbolic logic by supporters of “Aristotelian” traditional logic. But even if “genuine” logic comes to be regarded as excluding second-order reasoning, which seems less likely today than fifty years ago, its effectiveness as a heuristic instrument will remain and its importance for understanding the history of logic and mathematics will not be diminished. Second-order logic may someday be gone, but it will never be forgotten. Technical formalisms have been avoided entirely in an effort to reach a wide audience, but every effort has been made to limit the inevitable sacrifice of rigor. People who do not know second-order logic cannot understand the modern debate over its legitimacy and they are cut-off from the heuristic advantages of second-order logic. And, what may be worse, they are cut-off from an understanding of the history of logic and thus are constrained to have distorted views of the nature of the subject. As Aristotle first said, we do not understand a discipline until we have seen its development. It is a truism that a person's conceptions of what a discipline is and of what it can become are predicated on their conception of what it has been.

Bulletin of Symbolic Logic, 2001

We discuss the differences between first-order set theory and secondorder logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if second-order logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on informal reasoning. On the other hand, if it is given a weak semantics, it loses its power in expressing concepts categorically. First-order set theory and second-order logic are not radically different: the latter is a major fragment of the former. * I am grateful to Juliette Kennedy for many helpful discussions while developing the ideas of this paper. † Research partially supported by grant 40734 of the Academy of Finland.

My book is about how-possible questions in epistemology. I focus on three such questions, 'How is knowledge of the external world possible?', 'How is knowledge of other minds possible?', and 'How is a priori knowledge possible?'. I explain how questions of this form arise and suggest how they should be answered. The basic idea is that we start by assuming that knowledge of the kind in question is possible but then encounter apparent obstacles to its existence or acquisition. So the issue is: how is knowledge of such-and-such a kind possible given the factors that make it look impossible? Since such questions are obstacle-dependent a satisfying response will need to be an obstacle-removing response, one that shows how the obstacles that led to the question being asked in the first place can be overcome or dissipated.

forthcoming in Nôus

A knowledge-based decision theory faces what has been called the prodigality problem (Greco 2013): given that many propositions are assigned probability 1, agents will be inclined to risk everything when betting on propositions which are known. In order to undo probability 1 assignments in high risk situations, the paper develops a theory which systematically connects higher level goods with higher-order knowledge.

Springer eBooks, 2017

positions and logical systems lore meet in new ways. In this little piece, a programmatic sequel to van Benthem 2011 and a prequel to Baltag et al. 2014, I add some further perspectives and issues to this mix from dynamic--epistemic logics of information and inquiry. My aim is to show that we can have a yet richer agenda of epistemic themes, and a richer view of the interplay of logic and epistemology, when we make epistemic action a major focus on a par with knowledge or belief per se. Modern encounters of logic and epistemology started with epistemic logic (Hintikka 1962). 3 This system reads its key modality Kϕ as truth of the proposition ϕ in the current range of epistemically accessible worlds for the relevant agent. The connection of this notion to knowledge in the philosopher's sense has long been a focus of discussion (cf. Stalnaker 1985, Williamson 2000). It may be viewed as an account of implicit ideal knowledge of ordinary agents, of explicit knowledge of ideal agents, or as a mere statement of the content of an agent's knowledge, without any claim to define knowledge in that way (Holliday 2012B). Be this as it may, among many epistemic logicians, Kϕ has come to be viewed as standing for yet a different notion, equally important in philosophy and the sciences, namely the 'semantic information' that the agent has available. Especially in large open--ended spaces of epistemic options, this seems too demanding to allow for significant knowledge, and current philosophical accounts of knowledge have taken other roads. 1 Rohit Parikh is an inspirational leader at the interface of logic, epistemology, mathematics, com-puter science and game theory whose influence (some might say, grace) has touched so many people. This piece does no justice to the depth of his thinking, but I hope it is in his spirit of free inquiry. 2 I thank Shi Chenwei and, especially, Wesley Holliday for valuable comments on this paper. 3 Hintikka's views on the interface of logic and epistemology are still highly relevant, and they have kept evolving, witness Hintikka 1973 and the 'Socratic epistemology' of Hintikka 2007. Indeed, there has been a veritable wave of creativity in the literature since the 1960s. Many philosophers analyze knowledge as belief, a notion that scans less than the full range of options for semantic information, and then upgrade belief with extra requirements of various kinds, such as tracking the truth (Nozick 1981), robustness under update with new information (Stalnaker 2006), temporal convergence to true belief (Kelly 1996), or defensibility in challenge games (Lehrer 1990). 4 The resulting styles of reasoning about knowledge diverge considerably from classical epistemic logic, in line with philosophical intuitions that have been much discussed in the literature. In particular, there is no auto-matic closure of knowledge under simple logical inferences such as weakening the known proposition, or combining two known propositions to knowing their conjunction. 5 2 Knowledge and logic of relevant alternatives These richer philosophical accounts allow for significant senses in which one can know that ϕ at the present stage of inquiry even when not all possible ¬ϕ-worlds (alternatives, states, scenarios) have been ruled out: knowledge ventures beyond the semantic information. In what follows I will be looking at the influential proposals of Dretske 1970, Dretske 1981, and Lewis 1996 that read knowledge of ϕ as a state of having ruled out all relevant potential counter--examples to the proposition ϕ. This is the view that I will be discussing as the running theme of this paper, since it seems the best suited for making my general points. 6 Recently, logicians have looked in new ways at what makes such relevant alternatives ('RA', for short) theories tick, going beyond the basics of epistemic logic. A trail--blazing study is Holliday 2012B that presents a joint analysis of relevant--alternative and tracking theories of knowledge, and determines the complete valid closure principles governing reasoning about knowledge in these styles. Further analyses, using ideas from dynamic--epistemic logic (van Benthem 2011), are found in Cornelisse 2011, Xu 2010, Shi 2013, Holiday 2012A. I will not go into details of these systems here. The technical discussion of RA theory to follow is in the spirit of these papers, but the general issues I will raise may be all my own. 7 4 Cf. Parikh et al. 2013 for another take on the striking entanglements of knowledge and games. 5 These are not the usual omniscience failures of bounded rationality that are often bandied against epistemic logic. As Dretske points out, they would even occur for "ideally astute logicians". 6 Personally, I am also attracted to accounts of knowledge that also involve belief and truth--tracking, but this second love on the side should not be a serious bias in the following presentation. 7 Analogies with the cited work may still be worth pursuing for their independent interest.

Philosophia Mathematica, 1999

I once heard a story about a museum that claimed to have the skull of Christopher Columbus. In fact, they claimed to have two such skulls, one of Columbus when he was a small boy and one when he was a grown man. Whether there was such a museum or not, the clear moral is that one should not claim too much. The purpose of this paper is to apply the moral to the contrast between first-order logic and second-order logic, as articulated in my Foundations without foundationalism: A case for second-order logic (Shapiro [1991]; see also Shapiro [1985]). Important philosophical issues concerning the nature of logic and logical theory lie in the vicinity. In a review of my book, John Burgess [1993] wrote: ... there is a tendency, signaled by the use of the word 'case' in the subtitle and the phrase 'the competition' as the title for the last chapter, for the author to step into the role of a lawyer or salesman for Second-Order, Inc., and this approach leads to some exaggerated and tendentious formulations. Thus Burgess thinks that in my enthusiastic defense of second-order logic, I claim too much. So do a few other commentators. There is little point to an exegetical study of my own book, to see whether it contains the exaggerated claims in question, but a study of the critical remarks will reveal what should and should not be claimed for second-order logic. The focus in my book, and here, is on second-order languages with standard model-theoretic semantics. In each interpretation, the property or set variables range over the entire powerset of the domain d, the binary relation variables range over the powerset of d 2 , etc. I do not insist on extensionality. If one takes the higher-order variables to range over intensional items, like concepts, then the issue of standard semantics is whether, for each subset s of d, there is a concept whose extension is s, and similarly for relation and function variables. Let AR be the conjunction of the standard Peano axioms, including the