We have previously established that [Formula: see text]-comprehension is equivalent to the statement that every dilator has a well-founded Bachmann–Howard fixed point, over [Formula: see text]. In this paper, we show that the base theory can be lowered to [Formula: see text]. We also show that the minimal Bachmann–Howard fixed point of a dilator [Formula: see text] can be represented by a notation system [Formula: see text], which is computable relative to [Formula: see text]. The statement that [Formula: see text] (...) is well founded for any dilator [Formula: see text] will still be equivalent to [Formula: see text]-comprehension. Thus, the latter is split into the computable transformation [Formula: see text] and a statement about the preservation of well-foundedness, over a system of computable mathematics. (shrink)

Sortal concepts are at the center of certain logical discussions and have played a significant role in solutions to particular problems in philosophy. Apart from logic and philosophy, the study of sortal concepts has found its place in specific fields of psychology, such as the theory of infant cognitive development and the theory of human perception. In this monograph, different formal logics for sortal concepts and sortal-related logical notions are characterized. Most of these logics are intensional in nature and possess, (...) in addition, a bidimensional character. That is, they simultaneously represent two different logical dimensions. In most cases, the dimensions are those of time and natural necessity, and, in other cases, those of time and epistemic necessity. Another feature of the logics in question concerns second-order quantification over sortal concepts, a logical notion that is also represented in the logics. Some of the logics adopt a constant domain interpretation, others a varying domain interpretation of such quantification. Two of the above bidimensional logics are philosophically grounded on predication sortalism, that is, on the philosophical view that predication necessarily requires sortal concepts. Another bidimensional logic constitutes a logic for complex sortal predicates. These three sorts of logics are among the important novelties of this work since logics with similar features have not been developed up to now, and they might be instrumental for the solution of philosophically significant problems regarding sortal predicates. The book assumes a modern variant of conceptualism as a philosophical background. For this reason, the approach to sortal predicates is in terms of sortal concepts. Concepts, in general, are here understood as intersubjective realizable cognitive capacities. The proper features of sortal concepts are determined by an analysis of the main features of sortal predicates. Posterior to this analysis, the sortal-related logical notions represented in the above logics are discussed. There is also a discussion on the extent to which the set-theoretic formal semantic systems of the book capture different aspects of the conceptualist approach to sortals. These different semantic frameworks are also related to realist and nominalist approaches to sortal predicates, and possible modifications to them are considered that might represent those alternative approaches. (shrink)

This paper investigates the responses of GPT-4, a state-of-the-art AI language model, to ten prominent philosophical paradoxes, and evaluates its capacity to reason and make decisions in complex and uncertain situations. In addition to analyzing GPT-4's solutions to the paradoxes, this paper assesses the model's Theory of Mind (ToM) capabilities by testing its understanding of mental states, intentions, and beliefs in scenarios ranging from classic ToM tests to complex, real-world simulations. Through these tests, we gain insight into AI's potential for (...) social reasoning and its capacity for more sophisticated forms of human-AI interaction. The paper also explores the limitations and biases of AI-generated reasoning and its implications for our comprehension of complex philosophical problems. (shrink)

In previous work, the author has shown that $\Pi ^1_1$ -induction along $\mathbb N$ is equivalent to a suitable formalization of the statement that every normal function on the ordinals has a fixed point. More precisely, this was proved for a representation of normal functions in terms of Girard’s dilators, which are particularly uniform transformations of well orders. The present paper works on the next type level and considers uniform transformations of dilators, which are called 2-ptykes. We show that $\Pi (...) ^1_2$ -induction along $\mathbb N$ is equivalent to the existence of fixed points for all 2-ptykes that satisfy a certain normality condition. Beyond this specific result, the paper paves the way for the analysis of further $\Pi ^1_4$ -statements in terms of well ordering principles. (shrink)

Michael Rathjen and the present author have shown that ‐bar induction is equivalent to (a suitable formalization of) the statement that every normal function has a derivative, provably in. In this note we show that the base theory can be weakened to. Our argument makes crucial use of a normal function f with and. We shall also exhibit a normal function g with and.

We shall distinguish between sortal predication and standard predication. The former kind of predication necessarily involves sortal concepts but the latter, as it is customarily viewed, does not. It is generally thought that the only essential occurrence of a concept in a standard predication is the concept being predicated. In this paper, we shall put forward an alternative view. We shall propose to understand standard predication as a cognitive act essentially requiring sortal concepts. We shall call this view conceptual predication (...) sortalism and ground it on the basis of epistemic-semantic reasons. Concepts are understood as intersubjectively realizable capacities or abilities that fulfill particular cognitive roles, such as those of classification, categorization, individuation, and referring. (shrink)

Let Con↾x denote the finite consistency statement “there are no proofs of contradiction in T with ≤x symbols.” For a large class of natural theories T, Pudlák has shown that the lengths of the shortest proofs of Con↾n in the theory T itself are bounded by a polynomial in n. At the same time he conjectures that T does not have polynomial proofs of the finite consistency statements Con)↾n. In contrast, we show that Peano arithmetic has polynomial proofs of Con)↾n, (...) where Con∗ is the slow consistency statement for Peano arithmetic, introduced by S.-D. Friedman, Rathjen, and Weiermann. We also obtain a new proof of the result that the usual consistency statement Con is equivalent to ε0 iterations of slow consistency. Our argument is proof-theoretic, whereas previous investigations of slow consistency relied on nonstandard models of arithmetic. (shrink)

There is widespread and growing concern among U.S. consumers about the treatment of farmed animals, and consumers are consequently paying attention to food product labels that indicate humane production practices. However, labels vary in their standards for animal welfare, and prior research suggests that consumers are confused by welfare-related labels: many shoppers cannot differentiate between labels that indicate changes in the way animals are raised and those that do not. We administered a survey to 1,000 American grocery shoppers to better (...) understand the extent to which consumers purchase and pay more for food with certain labels based on an assumption of welfare improvement. Results showed that 86% of shoppers reported purchasing at least one product with the following labels in the last year: “cage or crate-free”, “free-range”, “pasture-raised”, “natural”, “organic”, “no hormone”, “no antibiotic”, “no rBST”, “humane”, “vegetarian-fed”, “grass-fed”, “farm-raised”. Of those who purchased one of the aforementioned labels, 89% did so because they thought the label indicated higher-welfare production practices, and 79% consciously paid more for the product with the label because they thought that the label indicated better-than-standard animal welfare. However, many of these labels lack uniform standards for the production practices they represent, and some labels represent production practices that do not influence animal welfare, thus the degree of the animal welfare impact of a given label is highly variable. These results indicate that labels need to clearly and accurately specify their animal welfare benefits to improve the consumers’ ability to purchase products that align with their expectations. (shrink)

In the context of reverse mathematics, effective transfinite recursion refers to a principle that allows us to construct sequences of sets by recursion along arbitrary well orders, provided that each set is ‐definable relative to the previous stages of the recursion. It is known that this principle is provable in. In the present note, we argue that a common formulation of effective transfinite recursion is too restrictive. We then propose a more liberal formulation, which appears very natural and is still (...) provable in. (shrink)

Professors Hildt (2023), Shepherd (2023), and Zilio and Lavazza (2023) jointly considered the ethical and philosophical implications of acknowledging non-human (e.g., machine) consciousness. Althou...

This paper explores the perversion of reality in the context of advanced technologies, such as AI, VR, and AR, through the lens of Jean Baudrillard’s theory of hyperreality and the precession of simulacra. By examining the transformative effects of these technologies on our perception of reality, with a particular focus on the usage of digital avatars, the paper highlights the blurred distinction between the real and the simulated, where the copy becomes more ‘real’ than the original. Drawing on Baudrillard’s concept (...) of hyperreality, the paper delves into the perversion of reality as individuals seek refuge in virtual pleasure paradises and embrace artificial pleasures through their digital avatars, disconnecting from genuine human experiences. The convergence of AI, VR, and AR technologies amplifies this hyperreal condition, where digital avatars mimic or surpass the depth of human relationships, challenging our understanding of what is real. In line with Baudrillard’s theory, the paper explores the objectification and commodification of reality within digital spaces, specifically examining the digital avatars’ role in the erosion of genuine human connections. It explores the implications of these avatars in terms of consent, exploitation, and loss of authenticity, echoing Baudrillard’s concerns about the distortion of reality in contemporary society. Recognizing the implications of these technologies, the paper calls for a critical reflection on their transformative power. It emphasizes the need for a nuanced understanding of the hyperreal condition and ethical responsibility in engaging with AI, VR, and AR, particularly in relation to the usage of digital avatars. By resisting the seductive allure of digital escapism and preserving genuine human connections, we can navigate the perversion of reality and cultivate empathy, compassion, and meaningful interactions that transcend the simulated experiences offered by technology. (shrink)

A. Tarski [22] proposed the study of infinitary consequence operations as the central topic of mathematical logic. He considered monotonicity to be a property of all such operations. In this paper, we weaken the monotonicity requirement and consider more general operations, inference operations. These operations describe the nonmonotonic logics both humans and machines seem to be using when infering defeasible information from incomplete knowledge. We single out a number of interesting families of inference operations. This study of infinitary inference operations (...) is inspired by the results of [12] on nonmonotonic inference relations, and relies on some of the definitions found there. (shrink)

In this text, a variety of modal logics at the sentential, first-order, and second-order levels are developed with clarity, precision and philosophical insight.

With the past and future tense propositional operators in its syntax, a formal logical system for sortal quantifiers, sortal identity and (second order) quantification over sortal concepts is formulated. A completeness proof for the system is constructed and its absolute consistency proved. The completeness proof is given relative to a notion of logical validity provided by an intensional semantic system, which assumes an approach to sortals from a modern form of conceptualism.

In a recent paper by M. Rathjen and the present author it has been shown that the statement “every normal function has a derivative” is equivalent to $\Pi ^1_1$ -bar induction. The equivalence was proved over $\mathbf {ACA_0}$, for a suitable representation of normal functions in terms of dilators. In the present paper, we show that the statement “every normal function has at least one fixed point” is equivalent to $\Pi ^1_1$ -induction along the natural numbers.

A. Tarski [21] proposed the study of infinitary consequence operations as the central topic of mathematical logic. He considered monotonicity to be a property of all such operations. In this paper, we weaken the monotonicity requirement and consider more general operations, inference operations. These operations describe the nonmonotonic logics both humans and machines seem to be using when infering dofeasible information from incomplete knowledge. We single out a number of interesting families of inference operations. This study of infinitary inference operations (...) is inspired by the results of [11] on finitary nonmonotonic operations, but this paper is self-contained. (shrink)

We show that arithmetical transfinite recursion is equivalent to a suitable formalization of the following: For every ordinal \alpha there exists an ordinal /beta such that 1 + \beta \cdot (\beta + \alpha) admits an almost order preserving collapse into \beta. Arithmetical comprehension is equivalent to a statement of the same form, with \beta \cdot \alpha at the place of \beta \cdot (\beta + \alpha). We will also characterise the principles that any set is contained in a countable coded ω-model (...) of arithmetical transfinite recursion and arithmetical comprehension, respectively. (shrink)

An intensional semantic system for languages containing, in their logical syntax, sortal quantifiers, sortal identities, (second-order) quantifiers over sortals and the necessity operator is constructed. This semantics provides non-standard assignments to predicate expressions, which diverge in kind from the entities assigned to sortal terms by the same semantic system. The nature of the entities assigned to predicate expressions shows, at the same time, that there is an internal semantic connection between those expressions and sortal terms. A formal logical system is (...) formulated that is proved to be absolutely consistent, sound and complete with respect to the intensional semantic system. (shrink)

When a proposition is cumulatively entailed by a finite setA of premisses, there exists, trivially, a finite subsetB ofA such thatB B entails for all finite subsetsB that are entailed byA. This property is no longer valid whenA is taken to be an arbitrary infinite set, even when the considered inference operation is supposed to be compact. This leads to a refinement of the classical definition of compactness. We call supracompact the inference operations that satisfy the non-finitary analogue of the (...) above property. We show that for any arbitrary cumulative operationC, there exists a supracompact cumulative operationK(C) that is smaller thenC and agrees withC on finite sets. Moreover,K(C) inherits most of the properties thatC may enjoy, like monotonicity, distributivity or disjunctive rationality. The main part of the paper concerns distributive supracompact operations. These operations satisfy a simple functional equation, and there exists a representation theorem that provides a semantic characterization for this family of operations. We examine finally the case of rational operations and show that they can be represented by a specific kind of model particularly easy to handle. (shrink)

Processing the motivational relevance of a visual scene and reacting accordingly is crucial for survival. Previous work suggests the emotional content of naturalistic scenes affects response speed, such that unpleasant content slows responses whereas pleasant content accelerates responses. It is unclear whether these effects reflect motor-cognitive processes, such as attentional orienting, or vary with the function/outcome of the motor response itself. Four experiments manipulated participants’ ability to terminate the picture and, thereby, the response’s function and motivational value. Attentive orienting was (...) manipulated via picture repetition, which diminishes orienting. A total of N = 81 participants completed versions of a go/no-go task, discriminating between distorted versus intact pictures drawn from six content categories varying in positive, negative, or neutral valence. While all participants responded faster with repetition, only participants without offset control exhibited slower responses to unpleasant and accelerated responses to pleasant content. Emotional engagement, measured by the late positive potential, was not modulated by attentional orienting, suggesting that the interaction between repetition and offset control is not due to altered emotional engagement. Together, results suggest that response time changes as a function of emotional content and sensitivity to attention orienting depends on the motivational function of the motor response. (shrink)

A formal logical system for sortal quantifiers, sortal identity and (second order) quantification over sortal concepts is formulated. The absolute consistency of the system is proved. A completeness proof for the system is also constructed. This proof is relative to a concept of logical validity provided by a semantics, which assumes as its philosophical background an approach to sortals from a modern form of conceptualism.

A. Tarski [22] proposed the study of infinitary consequence operations as the central topic of mathematical logic. He considered monotonicity to be a property of all such operations. In this paper, we weaken the monotonicity requirement and consider more general operations, inference operations. These operations describe the nonmonotonic logics both humans and machines seem to be using when infering defeasible information from incomplete knowledge. We single out a number of interesting families of inference operations. This study of infinitary inference operations (...) is inspired by the results of [12] on nonmonotonic inference relations, and relies on some of the definitions found there. (shrink)

In this commentary, we consider how Balcetis’s proposals may interface with the study of motivation and emotion in lifespan developmental psychology, pointing to open questions regarding the distance perception of long-term chronic goals as well as age-related shifts from informational to emotional goals.

We consider a formal language whose logical syntax involves both modal and tense propositional operators, as well as sortal quantifiers, sortal identities and (second order) quantifiers over sortals. We construct an intensional semantics for the language and characterize a formal logical system which we prove to be sound and complete with respect to the semantics. Conceptualism is the philosophical background of the semantic system.