In this paper I address the question of whether the incompleteness theorems imply that “the mind cannot be mechanized,” where this is understood in the specific sense that “the mathematical outputs of the idealized human mind do not coincide with the mathematical outputs of any idealized finite machine.” Gödel argued that his incompleteness theorems implied a weaker, disjunctive conclusion to the effect that either “the mind cannot be mechanized” or “mathematical truth outstrips the idealized human mind.” Others, most notably, Lucas (...) and Penrose, have claimed more—they have claimed that the incompleteness theorems actually imply the first disjunct. I will show that by sharpening the fundamental concepts involved and articulating the background assumptions governing them, one can prove Gödel’s disjunction, one can show that the arguments of Lucas and Penrose fail, and one can see what likely led proponents of the first disjunct astray. (shrink)

The paper begins with an examination of Gödel's views on absolute undecidability and related topics in set theory. These views are sharpened and assessed in light of recent developments. It is argued that a convincing case can be made for axioms that settle many of the questions undecided by the standard axioms and that in a precise sense the program for large cardinals is a complete success “below” CH. It is also argued that there are reasonable scenarios for settling CH (...) and that there is not currently a convincing case to the effect that a given statement is absolutely undecidable. (shrink)

Gödel argued that his incompleteness theorems imply that either “the mind cannot be mechanized” or “there are absolutely undecidable sentences.” In the precursor to this paper I examined the early arguments for the first disjunct. In the present paper I examine the most sophisticated argument for the first disjunct, namely, Penrose’s new argument. It turns out that Penrose’s argument requires a type-free notion of truth and a type-free notion of absolute provability. I show that there is a natural such system, (...) DTK. I prove a series of results which show that Gödel’s disjunction is provable in the system, Penrose’s argument is invalid in the system, there can be no proof or refutation of either disjunct in the system, the independence results are robust in that they persist when one strengthens the principles governing absolute provability, and there are reasons to believe that the situation will not improve under any plausible alteration of the underlying theory of truth. (shrink)

Gödel initiated the program of finding and justifying axioms that effect a significant reduction in incompleteness and he drew a fundamental distinction between intrinsic and extrinsic justifications. Reflection principles are the most promising candidates for new axioms that are intrinsically justified. Taking as our starting point Tait’s work on general reflection principles, we prove a series of limitative results concerning this approach. These results collectively show that general reflection principles are either weak ) or inconsistent. The philosophical significance of these (...) results is discussed. (shrink)

In this paper we investigate strong logics of first and second order that have certain absoluteness properties. We begin with an investigation of first order logic and the strong logics ω-logic and β-logic, isolating two facets of absoluteness, namely, generic invariance and faithfulness. It turns out that absoluteness is relative in the sense that stronger background assumptions secure greater degrees of absoluteness. Our aim is to investigate the hierarchies of strong logics of first and second order that are generically invariant (...) and faithful against the backdrop of the strongest large cardinal hypotheses. We show that there is a close correspondence between the two hierarchies and we characterize the strongest logic in each hierarchy. On the first-order side, this leads to a new presentation of Woodin's Ω-logic. On the second-order side, we compare the strongest logic with full second-order logic and argue that the comparison lends support to Quine's claim that second-order logic is really set theory in sheep's clothing. (shrink)

The independence results in set theory invite the search for new and justified axioms. In Chapter 1 I set the stage by examining three approaches to justifying the axioms of standard set theory and argue that the approach via reflection principles is the most successful. In Chapter 2 I analyse the limitations of ZF and use this analysis to set up a mathematically precise minimal hurdle which any set of new axioms must overcome if it is to effect a significant (...) reduction in incompleteness. In Chapter 3 I examine the standard method of justifying new axioms---reflection principles---and prove a result which shows that no reflection principle can overcome the minimal hurdle and yield a significant reduction in incompleteness. In Chapter 4 I introduce a new approach to justifying new axioms---extension principles---and show that such principles can overcome the minimal hurdle and much more, in particular, such principles imply PD and that the theory of second-order arithmetic cannot be altered by set size forcing. I show that in a sense these principles are inevitable. In Chapter 5 I close with a brief discussion of meta-mathematical justifications stemming from the work of Woodin. These touch on the continuum hypothesis and other questions which are beyond the reach of standard large cardinals. (shrink)

In 1985 the second author showed that if there is a proper class of measurable Woodin cardinals and $V^{B1} $ and $V^{B2} $ are generic extensions of V satisfying CH then $V^{B1} $ and $V^{B2} $ agree on all $\Sigma _1^2 $ -statements. In terms of the strong logic Ω-logic this can be reformulated by saying that under the above large cardinal assumption ZFC + CH is Ω-complete for $\Sigma _1^2 $ Moreover. CH is the unique $\Sigma _1^2 $ -statement (...) with this feature in the sense that any other $\Sigma _1^2 $ -statement with this feature is Ω-equivalent to CH over ZFC. It is natural to look for other strengthenings of ZFC that have an even greater degree of Ω-completeness. For example, one can ask for recursively enumerable axioms A such that relative to large cardinal axioms ZFC + A is Ω-complete for all of third-order arithmetic. Going further, for each specifiable segment $V_\lambda $ of the universe of sets (for example, one might take $V_\lambda $ to be the least level that satisfies there is a proper class of huge cardinals), one can ask for recursively enumerable axioms A such that relative to large cardinal axioms ZFC + A is Ω-complete for the theory of $V_\lambda $ . If such theories exist, extend one another, and are unique in the sense that any other such theory B with the same level of Ω-completeness as A is actually Ω-equivalent to A over ZFC, then this would show that there is a unique Ω-complete picture of the successive fragments of the universe of sets and it would make for a very strong case for axioms complementing large cardinal axioms. In this paper we show that uniqueness must fail. In particular, we show that if there is one such theory that Ω-implies CH then there is another that Ω-implies¬CH. (shrink)

In this paper I examine Feferman’s reasons for maintaining that while the statements of first-order number theory are “completely clear'” and “completely definite,”' many of the statements of analysis and set theory are “inherently vague'” and “indefinite.”' I critique his four central arguments and argue that in the end the entire case rests on the brute intuition that the concept of subsets of natural numbers—along with the richer concepts of set theory—is not “clear enough to secure definiteness.” My response to (...) this final, remaining point will be that the concept of “being clear enough to secure definiteness” is about as clear a case of an inherently vague and indefinite concept as one might find, and as such it can bear little weight in making a case against the definiteness of analysis and set theory. (shrink)

The HOD Dichotomy Theorem states that if there is an extendible cardinal, δ, then either HOD is “close” to V or HOD is “far” from V. The question is whether the future will lead to the first or the second side of the dichotomy. Is HOD “close” to V, or “far” from V? There is a program aimed at establishing the first alternative—the “close” side of the HOD Dichotomy. This is the program of inner model theory. In recent years the (...) third author has provided evidence that there is an ultimate inner model—Ultimate-L—and he has isolated a natural conjecture associated with the model—the Ultimate-L Conjecture. This conjecture implies that that the first alternative holds—HOD is “close” to V. This is the future in which pattern prevails. In this paper we introduce a very different program, one aimed at establishing the second alternative—the “far” side of the HOD Dichotomy. This is the program of large cardinals beyond choice. Kunen famously showed that if AC holds then there cannot be a Reinhardt cardinal. It has remained open whether Reinhardt cardinals are consistent in ZF alone. It turns out that there is an entire hierarchy of choiceless large cardinals of which Reinhardt cardinals are only the beginning, and, surprisingly, this hierarchy appears to be highly ordered and amenable to systematic investigation, as we shall show in this paper. The point is that if these choiceless large cardinals are consistent then the Ultimate-L Conjecture must fail. This is the future where chaos prevails. (shrink)

This volume contains the proceedings of the Logic at Harvard conference in honor of W. Hugh Woodin's 60th birthday, held March 27–29, 2015, at Harvard University. It presents a collection of papers related to the work of Woodin, who has been one of the leading figures in set theory since the early 1980s. The topics cover many of the areas central to Woodin's work, including large cardinals, determinacy, descriptive set theory and the continuum problem, as well as connections between set (...) theory and Banach spaces, recursion theory, and philosophy, each reflecting a period of Woodin's career. Other topics covered are forcing axioms, inner model theory, the partition calculus, and the theory of ultrafilters. This volume should make a suitable introduction to Woodin's work and the concerns which motivate it. The papers should be of interest to graduate students and researchers in both mathematics and philosophy of mathematics, particularly in set theory, foundations and related areas. (shrink)

Conventionalism about mathematics claims that mathematical truths are true by linguistic convention. This is often spelled out by appealing to facts concerning rules of inference and formal systems, but this leads to a problem: since the incompleteness theorems we’ve known that syntactic notions can be expressed using arithmetical sentences. There is serious prima facie tension here: how can mathematics be a matter of convention and syntax a matter of fact given the arithmetization of syntax? This challenge has been pressed in (...) the literature by Hilary Putnam and Peter Koellner. In this paper I sketch a conventionalist theory of mathematics, show that this conventionalist theory can meet the challenge just raised , and clarify the type of mathematical pluralism endorsed by the conventionalist by introducing the notion of a semantic counterpart. The paper’s aim is an improved understanding of conventionalism, pluralism, and the relationship between them. (shrink)

In 5, Tait identifies a set of reflection principles called equation image-reflection principles which Peter Koellner has shown to be consistent relative to the existence of κ, the first ω-Erdős cardinal 1. Tait also defines a set of reflection principles called equation image-reflection principles; however, Koellner has shown that these are inconsistent when m > 2, but identifies restricted versions of them which he proves consistent relative to κ 2. In this paper, we introduce a new large-cardinal property, the (...) α-reflective cardinals. Their definition is motivated by Tait's remarks on parameters of third or higher order in reflection principles. We prove that if κ is ℶκ + α + 1-supercompact and 0 < α < κ then κ is α-reflective. Furthermore we show that α-reflective cardinals relativize to L, and that if κ exists then the set of cardinals λ < κ, such that λ is α-reflective for all α such that 0 < α < λ, is a stationary subset of κ. We show that an ω-reflective cardinal satisfies some restricted versions of equation image-reflection, as well as all the reflection properties proved consistent in 2. (shrink)

An argument going back to Russell shows that the view that propositions are structured is inconsistent in standard type theories. Here, it is shown that such type theories may nevertheless provide entities which can serve as proxies for structured propositions. As an illustration, such proxies are applied to the case of grounding, as standard views of grounding require a degree of propositional structure which suffices for a version of Russell’s argument. While this application solves some of the problems grounding faces, (...) it introduces problematic limitations: it becomes impossible to quantify unrestrictedly over the relata of ground. The proposed proxies may thus not save grounding, but they shed light on what exactly Russell’s argument does and does not show. (shrink)

It is argued that it is not sufficient to consider only the sentences included in the explanans and explanandum when determining whether they constitute an explanation, but these sentences must always be evaluated relative to a knowledge situation. The central criterion on an explanation is that the explanans in a non-trivial way increases the belief value of the explanandum, where the belief value of a sentence is determined from the given knowledge situation. The outlined theory of explanations is applied to (...) some well-known examples and is also compared to other theories of explanation. (shrink)

David Chalmers argues that virtual objects exist in the form of data structures that have causal powers. I argue that there is a large class of virtual objects that are social objects and that do not depend upon data structures for their existence. I also argue that data structures are themselves fundamentally social objects. Thus, virtual objects are fundamentally social objects.

Studies suggest that people's moral intuitions are sensitive to morally irrelevant factors, such as personal force, spatial distance, ethnicity or nationality. Findings of this sort have been used to construct debunking arguments. The most prominent champion of this approach is Joshua Greene, who has attempted to undermine deontology by showing that deontological intuitions are triggered by morally irrelevant factors. This article offers a critical analysis of such empirically informed debunking arguments from moral irrelevance, and of Greene’s effort to undermine deontology. (...) One problem with arguments from moral irrelevance concerns the hierarchy between the targeted case-specific intuitions and the more general intuitions about which factors are morally (ir)relevant. If we assume that general intuitions always take precedence over case-specific intuitions, arguments from moral irrelevance become dialectically useless. By contrast, if we assume that both case-specific and general intuitions should be taken seriously, particularly sweeping debunking projects, such as Greene’s, are unlikely to go through. Another problem concerns the experimental aspect of arguments from moral irrelevance. Empirically informed arguments from moral irrelevance have been presented as examples of how empirical moral psychology can advance moral theory. But arguments from moral irrelevance can also be constructed from the armchair. And basing these arguments on empirical findings about laypeople’s intuitions is even counterproductive. (shrink)

In der 1970 gegründeten Reihe erscheinen Arbeiten, die philosophiehistorische Studien mit einem systematischen Ansatz oder systematische Studien mit philosophiehistorischen Rekonstruktionen verbinden. Neben deutschsprachigen werden auch englischsprachige Monographien veröffentlicht. Gründungsherausgeber sind: Erhard Scheibe (Herausgeber bis 1991), Günther Patzig (bis 1999) und Wolfgang Wieland (bis 2003). Von 1990 bis 2007 wurde die Reihe von Jürgen Mittelstraß mitherausgegeben.

According to certain models of the musical work-performance relationship, musical works persist through time. Dodd and Thomasson argue that perdurantist accounts of musical persistence—according to which musical works persist by having temporal parts at every time they exist—are untenable, and Tillman argues that musical endurantism—according to which persisting works are wholly present at each time they exist—avoids Dodd’s worries. In this paper, I argue that both Dodd’s and Thomasson’s arguments—and Tillman’s response—rely on assumptions linking theories of persistence to common-sense views (...) about musical works and, moreover, that these assumptions are unwarranted. As a result, only an attitude of neutrality towards questions about the nature of musical persistence is warranted. (shrink)

Michael Hand's recent paper, ‘Religious Upbringing Reconsidered', re-opens a debate that was flourishing over a decade ago in this journal and, long before that, in the works of others. In this response I examine Hand's claims that earlier contributions to the debate passed over the central problem and that he can solve that problem. I endeavour to show that several of Hand's arguments, such as those dealing with indoctrination, as well as his claims may be flawed, that the relevance of (...) his inquiries is open to question, but that his hidden and, apparently, unacknowledged agenda should not remain hidden or passed over. (shrink)

The relation-theory of mental acts proposes that a mental act is a kind of relative entity founded upon the mind and directed at the object of perception or thought. While most medieval philosophers recognized that there is something importantly relational about thought, they nevertheless rejected the view that mental acts are wholly relations. Rather, the dominant view was that a mental act is either in whole or part an Aristotelian quality added to the mind upon which such a relation to (...) the object can be founded. In this paper, I examine Durand of St.-Pourçain's defense of the relation-theory of mental acts against two objections raised against it: the first from John Duns Scotus, among others, and the second from an anonymous Thomist and Adam Wodeham. -/- . (shrink)

Volume 52, Issue 12, November 2020, Page 1243-1248.

ABSTRACTThe ability to regulate emotions is essential for psychological well-being. Therefore, it is particularly important to investigate the specific dynamics of emotion regulation. In a new appr...

Are men oppressed as men? The evidence given in support of affirmative responses to this question usually consists in examples of harms, limitations, or requirements masculinity imposes on men: men are expected to pay on dates, men must be breadwinners for their families, men can be drafted for war, and so forth. This article explicates three hypotheses that account for the harms, limitations, and requirements masculinity imposes on men and, drawing on the work of Alison Jaggar, seeks to show that (...) these hypotheses collectively are explanatorily superior to the hypothesis the men are oppressed as men. (shrink)

Populism in politics and policy orientations in law have thrown the jurisdiction of the academy and the disciplines of interpretation into disarray. Critique flounders in abstraction and negativity, law loses itself in particularity. Administering Interpretation brings together philosophers, humanists, and jurists from both continental and Anglophone jurisdictions to reassess the status and trajectory of interpretative theory as applied in the art of law. Tracking the thread of philosophical influences upon the community of legal interpretation, the essays move from the translation (...) and wake of Derrida to the work of Agamben, from deconstruction to oikononmia. Sharing roots in the philological excavation of the political theology of modern law, contributors assess the failure of secularism and the continuing theological borrowings of juridical interpretation. The book brings contemporary critique to bear upon the interpretative apparatuses of exclusion, the law of spectacular sovereignty, and the bodies that lie in its wake. (shrink)

In this paper, we consider what is commonly referred to as Leibniz’s argument for primitive concepts. After presenting and criticizing (in sections 1 and 2) one recent rather straightforward way of interpreting this argument, by Paul Lodge and Stephen Puryear, which takes the argument to be merely about the structure of concepts, we offer an alternative way of looking at the argument. We think it is best seen as being fundamentally about the relation between thought and reality. In order to (...) prepare the ground for our reconstruction (which we present in section 5), we have to introduce his view of ideas or concepts (section 3), as well as some metaphysical principles concerning reality-dependence (section 4). (shrink)

Both Reid and Berkeley reject ‘representationalism’, an epistemological position whereby we (mediately) perceive things in the world indirectly via ideas in our mind, on the grounds of anti-scepticism and common sense. My aim in this paper is to draw out the similarities between Reid and Berkeley's ‘anti-representationalist’ arguments, whilst also identifying the root of their disagreements on certain fundamental metaphysical issues. Reid famously rejects Berkeley's idealism, in which all that exists are ideas and minds, because it undermines the dictates of (...) common sense. Reid also charges Berkeley with not only accepting but furthering the progress of ‘the Way of Ideas’, a longstanding tradition which has drawn philosophy away from true science and common sense. From Berkeley's perspective, Reid is a ‘materialist’; that is, he dogmatically accepts that mind-independent things exist. I argue that these important differences can be explained by both thinkers’ construal of certain ‘philosophical prejudices’. Finally, I conclude that despite these differences, both ought to be characterised as ‘anti-representationalists’ in light of their shared epistemological concerns. (shrink)

Our approach aims at accounting for causal claims in terms of how the physical states of the underlying dynamical system evolve with time. Causal claims assert connections between two sets of physicals states—their truth depends on whether the two sets in question are genuinely connected by time evolution such that physical states from one set evolve with time into the states of the other set. We demonstrate the virtues of our approach by showing how it is able to account for (...) typical causes, causally relevant factors, being ‘the’ cause, and cases of overdetermination and causation by absences. (shrink)

Russell wrote in 1918 in The Philosophy of Logical Atomism: -/- When I speak of a fact ... I mean the kind of thing that makes a proposition true or false. If I say 'It is raining', what I say is true in a certain condition of weather and is false in other conditions of the weather. The condition of weather that makes my state­ment true (or false as the case may be), is what I should call a 'fact'. If (...) I say, 'Socrates is dead', my statement will be true owing to a certain physiological occurrence which happened in Athens long ago. -/- This classic statement of Russell's version of the correspondence theory of truth is defective on one point only: Russell was unwise to use the word 'fact'. Objects which perform the semantic role Russell accords to facts I shall call simply truth-makers. (shrink)

We will be in a better position to evaluate some important skeptical theses if we first investigate two questions about justified suspended judgment. One question is this: when, if ever, does one justified suspension confer justification on another suspension? and the other is this: what is the structure of justified suspension? the goal of this essay is to make headway at answering these questions. After surveying the four main views about the non-normative nature of suspended judgment and offering a taxonomy (...) of the epistemic principles that might govern which suspended judgments are justified, I will isolate five important principles that might govern which suspended judgments are justified. I will call these suspension-to-suspension principles. I will then evaluate these principles by the lights of each of the four views about what suspensions are. I close by drawing some conclusions about the prospects for skepticism, the structure of justified suspended judgment, and the importance of theorizing about justified suspended judgment. (shrink)