First-Order Frege Theory is Undecidable

The system whose only predicate is identity, whose only nonlogical vocabulary is the abstraction operator, and whose axioms are all first-order instances of Frege's Axiom V is shown to be undecidable.     

Frege,Boolos, and Logical Objects

In this paper, the authors discuss Frege's theory of logical objects (extensions, numbers, truth-values) and the recent attempts to rehabilitate it. We show that the eta relation George Boolos deployed on Frege's behalf is similar, if not identical, to the encoding mode of predication that underlies the theory of abstract objects. Whereas Boolos accepted unrestricted Comprehension for Properties and used the eta relation to assert the existence of logical objects under certain highly restricted conditions, the theory of abstract objects uses unrestricted Comprehension for Logical Objects and banishes encoding (eta) formulas from Comprehension for Properties. The relative mathematical and philosophical strengths of the two theories are discussed. Along the way, new results in the theory of abstract objects are described, involving: (a) the theory of extensions, (b) the theory of directions and shapes, and (c) the theory of truth values.     

Frege Puzzles and Mental Files

This paper proposes a novel conception of mental files, aimed at addressing Frege puzzles. Classical Frege puzzles involve ignorance and discovery of identity. These may be addressed by accounting for a more basic way for identity to figure in thought—the treatment of beliefs by the believer as being about the same thing. This manifests itself in rational inferences that presuppose the identity of what the beliefs are about. Mental files help to provide a functional characterization of a mind capable of this presupposition, but more must be said to show how it may be rational. I argue that this can be done by drawing out the way in which mental files interact with a thinker's motivational states and so come to have normative functional properties. I show how this theory works better than some other treatments of mental files.     

The Middle Wittgenstein’s Critique of Frege

ABSTRACT

This article aims to analyse Wittgenstein’s 1929–1932 notes concerning Frege’s critique of what is referred to as old formalism in the philosophy of mathematics. Wittgenstein disagreed with Frege’s critique and, in his notes, outlined his own assessment of formalism. First of all, he approvingly foregrounded its mathematics-game comparison and insistence that rules precede the meanings of expressions. In this article, I recount Frege’s critique of formalism and address Wittgenstein’s assessment of it to show that his remarks are not so much a critique of Frege as rather a defence of the formalist anti-metaphysical investment.     

Concept Cartesianism,Concept Pragmatism,and Frege Cases

This paper concerns the dialectal role of Frege Cases in the debate between Concept Cartesians and Concept Pragmatists. I take as a starting point Christopher Peacocke’s argument that, unlike Cartesianism, his ‘Fregean’ Pragmatism can account for facts about the rationality and epistemic status of certain judgments. I argue that since this argument presupposes that the rationality of thoughts turn on their content, it is thus question-begging against Cartesians, who claim that issues about rationality turn on the form, not the content, of thoughts. I then consider Jerry Fodor’s argument that ‘modes of presentation’ are not identical with Fregean senses, and argue that explanatory considerations should leads us to reject his ‘syntactic’ treatment of Frege cases. Rejecting the Cartesian treatment of Frege cases, however, is not tantamount to accepting Peacocke’s claim that reasons and rationality are central to the individuation of concepts. For I argue that we can steer a middle course between Fodor’s Cartesianism and Peacocke’s Pragmatism, and adopt a form of Pragmatism that is constrained by Fregean considerations, but at the same time denies that concepts are constitutively tied to reasons and rationality.

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In this paper, the author derives the Dedekind–Peano axioms for number theory from a consistent and general metaphysical theory of abstract objects. The derivation makes no appeal to primitive mathematical notions, implicit definitions, or a principle of infinity. The theorems proved constitute an important subset of the numbered propositions found in Frege"s Grundgesetze. The proofs of the theorems reconstruct Frege"s derivations, with the exception of the claim that every number has a successor, which is derived from a modal axiom that (philosophical) logicians implicitly accept. In the final section of the paper, there is a brief philosophical discussion of how the present theory relates to the work of other philosophers attempting to reconstruct Frege"s conception of numbers and logical objects.     

Age-related differences in idea generation and selection for propositional language

Conceptual preparation mechanisms such as novel idea generation and selection from amongst competing alternatives are critical for language production and may contribute to age-related language deficits. This study investigated whether older adults show diminished idea generation and selection abilities, compared to younger adults. Twenty younger (18–35 years) and 20 older (60–80 years) adults completed two novel experimental tasks, an idea generation task and a selection task. Older participants were slower than younger participants overall on both tasks. Importantly, this difference was more pronounced for task conditions with greater demands on generation and selection. Older adults were also significantly reduced on a semantic, but not phonemic, word fluency task. Overall, the older group showed evidence of age-related decline specific to idea generation and selection ability. This has implications for the message formulation stage of propositional language decline in normal aging.     

The unity of the Church according to Calvin and its meaning for the churches in Indonesia

Since the founding of the Council of Churches in Indonesia (DGI), which later changed its name to the Communion of Churches in Indonesia (PGI), a recurring question has been asked about what kind of unity the DGI/PGI intends to achieve. This issue was discussed by the General Assemblies, from the first in 1950 until the 10th in 1984, with various ideas being put forward. Some underscored the unity of structural organisations, whereas others emphasised our spiritual unity in Christ. It was only at the 10th General Assembly held in 1984 at Ambon that Indonesian churches discovered the answer they had been seeking all that time. The unity they sought was a shared understanding of Christian doctrine, mutual recognition and acceptance, and co-operation in working together to carry out their task and calling in Indonesia. In this respect, I believe that the view of Indonesian church unity conforms to Calvin’s concept which does not stress organisational unity but rather spiritual oneness, oneness in Christ, and oneness in the basic principles of the faith, in recognising each other as having the true preaching of the Word and administration of the sacraments. It is this kind of unity which has been expressed in the Five Documents of Church Unity (LDKG), later called Documents on Church Unity (DKG). The churches of Indonesia can learn much from the view of church reformer John Calvin.     

Young infants' perception of unity and form in occlusion displays

Young infants have been reported to perceive the unity of a center-occluded object when the visible ends of the object are aligned and undergo common motion but not when the edges of the object are misaligned (Johnson & Aslin, 1996). Using a recognition-based paradigm, the authors investigated the possibility that past research failed to provide sufficiently sensitive assessments of infants' perception of the unity of misaligned edges in partial occlusion displays. Positive evidence was obtained in 4-month-olds for veridical perception of the motion and location of a hidden region but not its orientation, whereas 7-month-olds, in contrast to the younger infants, appeared to respond to the orientation of the hidden region. Overall, the results suggest that habituation designs tapping recognition processes may be particularly efficacious in revealing infants' perceptual organization. In addition, the findings provide corroborative evidence for the importance of both motion and orientation in young infants' object segregation and for the difficulty in achieving percepts of the global form of a partly occluded object.     

Motion and edge sensitivity in perception of object unity

Although much evidence indicates that young infants perceive unitary objects by analyzing patterns of motion, infants' abilities to perceive object unity by analyzing Gestalt properties and by integrating distinct views of an object over time are in dispute. To address these controversies, four experiments investigated adults' and infants' perception of the unity of a center-occluded, moving rod with misaligned visible edges. Both alignment information and depth information affected adults' and infants' perception of object unity in similar ways, and infants perceived object unity by integrating information about object features over time. However, infants perceived a moving, misaligned, three-dimensional object as indeterminate in its connectedness, whereas adults perceived it as connected behind the occluder. These findings indicate that the effectiveness of common motion in specifying unified surfaces across an occluder is reduced by misalignment of edges. Alignment information enhances perception of object unity either by serving directly as information for unity or by optimizing the detectability of motion-carried information for unity. In addition, young infants are able to retain information about edge orientation over short intervals in determining connectedness via a process of spatiotemporal integration.     

Socrates,the primary question,and the unity of virtue

For Socrates, the virtues are a kind of knowledge, and the virtues form a unity. Sometimes, Socrates suggests that the virtues are all ‘one and the same’ thing. Other times, he suggests they are ‘parts of a single whole.’ I argue that (i) the ‘what is x?’ question is sophisticated, it gives rise to two distinct kinds of investigations into virtue, a conceptual investigation into the ousia and a psychological investigation into the dunamis, (ii) Plato recognized the difference between definitional accounts of the ousia and a psychological accounts of the dunamis, and (iii) the distinction between these two investigations can effectively resolve various interpretive puzzles regarding the unity of the virtues. It is argued that the virtues are ‘one and the same’ psychologically, while they are ‘parts of a single whole’ conceptually.     

Perception of object unity in chicks: A comparison with the human infant

Newly hatched chicks (Gallus gallus) were imprinted on a display consisting of two rod pieces that moved above and below a central occluder. On test trials, the chicks approached a complete rod in preference to two rod pieces. This finding, supported by those from control coditions, suggests that chicks, soon after hatching, perceive object unity. The results are compared with those from human infants.     

Propositions as Structured Cognitive Event-Types

According to act theories, propositions are structured cognitive act-types. Act theories appear to make propositions inherently representational and truth-evaluable, and to provide solutions to familiar problems with alternative theories, including Frege’s and Russell’s problems, and the third-realm and unity problems. Act theories have critical problems of their own, though: acts as opposed to their objects are not truth evaluable, not structured in the right way, not expressed by sentences, and not the objects of propositional attitudes. I show how identifying propositions with other cognitive event-types, namely thoughts, has the perceived virtues of act theories without the defects.     

Diversity in unity: practical unity and personal boundaries

In the spirit of the discussion in Daniel Kolak’s I Am You: The Metaphysical Foundation for Global Ethics, I consider the way in which divisions that we usually think of as borders between distinct people occur within a single life. Starting with the dispute between constructionist and non-constructionist views of persons, I argue for a view that places the unity of persons in the dynamic generated by simultaneously taking a constructionist and non-constructionist view of oneself. In order to unify ourselves as agents we need to treat past and future selves as others, but to motivate this endeavor we need to think of ourselves as temporally extended agents, and so identify with past and future selves. Understanding this dynamic illuminates the structure of our agency and the unity of the self.     

Models,Models, and Models

Michael Dummett famously maintained that analytic philosophy was simply philosophy that followed Frege in treating the philosophy of language as the basis for all other philosophy (1978, 441). But one important insight to emerge from computer science is how difficult it is to animate the linguistic artifacts that the analysis of thought produces. Yet, modeling the effects of thought requires a new skill that goes beyond analysis: procedural literacy. Some of the most promising research in philosophy makes use of a variety of modeling techniques that go beyond basic logic and elementary probability theory. What unifies this approach is a focus on what Alan Perlis called procedural literacy. This essay argues that the future spoils in philosophical research will disproportionally go to those who are procedurally literate.