This paper gives a definition of self-reference on the basis of the dependence relation given by Leitgeb (J Philos Logic 34(2):155–192, 2005), and the dependence digraph by Beringer and Schindler (Reference graphs and semantic paradox, 2015. https://www.academia.edu/19234872/reference_graphs_and_semantic_paradox). Unlike the usual discussion about self-reference of paradoxes centering around Yablo’s paradox and its variants, I focus on the paradoxes of finitary characteristic, which are given again by use of Leitgeb’s dependence relation. They are called ‘locally finite paradoxes’, satisfying that any sentence in (...) these paradoxes can depend on finitely many sentences. I prove that all locally finite paradoxes are self-referential in the sense that there is a directed cycle in their dependence digraphs. This paper also studies the ‘circularity dependence’ of paradoxes, which was introduced by Hsiung (Logic J IGPL 22(1):24–38, 2014). I prove that the locally finite paradoxes have circularity dependence in the sense that they are paradoxical only in the digraph containing a proper cycle. The proofs of the two results are based directly on König’s infinity lemma. In contrast, this paper also shows that Yablo’s paradox and its ∀∃\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\forall \exists $$\end{document}-unwinding variant are non-self-referential, and neither McGee’s paradox nor the ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}-cycle liar has circularity dependence. (shrink)

The unwinding that Cook, 767–774 2004) proposed is a simple but powerful method of generating new paradoxes from known ones. This paper extends Cook’s unwinding to a larger class of paradoxes and studies further the basic properties of the unwinding. The unwinding we study is a procedure, by which when inputting a Boolean modal net together with a definable digraph, we get a set of sentences in which we have a ‘counterpart’ for each sentence of the Boolean modal net and (...) each point of the digraph. What is more, whenever a sentence of the Boolean modal net says another sentence is necessary, then the counterpart of the first sentence at a point correspondingly says the counterparts of the second one at all accessible points of that point are all true. The output of the procedure is called ‘the unwinding of a Boolean modal net on a definable digraph’. We prove that the unwinding procedure preserves paradoxicality: a Boolean modal net is paradoxical on a definable digraph, iff the unwinding of it on this digraph is also paradoxical. Besides, the dependence digraph for the unwinding of a Boolean modal net on a definable digraph is proved to be isomorphic to the unwinding of the dependence digraph for the Boolean modal net on the previous definable digraph. So the unwinding of a Boolean modal net on a digraph is self-referential, iff the Boolean modal net is self-referential and the digraph is cyclic. Thus, on the one hand, the unwinding of any Boolean modal net on an acyclic digraph is non-self-referential. In particular, the unwinding of any Boolean modal net on \ is non-self-referential. On the other hand, if a Boolean modal net is paradoxical on a locally finite digraph, the unwinding of it on that digraph must be self-referential. Hence, starting from a Boolean modal paradox, the unwinding can output a non-self-referential paradox only if the digraph is not locally finite. (shrink)

This paper gives a definition of self-reference on the basis of the dependence relation given by Leitgeb (2005), and the dependence digraph by Beringer & Schindler (2015). Unlike the usual discussion about self-reference of paradoxes centering around Yablo's paradox and its variants, I focus on the paradoxes of finitary characteristic, which are given again by use of Leitgeb's dependence relation. They are called 'locally finite paradoxes', satisfying that any sentence in these paradoxes can depend on finitely many sentences. I prove (...) that all locally finite paradoxes are self-referential in the sense that there is a directed cycle in their dependence digraphs. This paper also studies the 'circularity dependence' of paradoxes, which was introduced by Hsiung (2014). I prove that the locally finite paradoxes have circularity dependence in the sense that they are paradoxical only in the digraph containing a proper cycle. The proofs of the two results are based directly on König's infinity lemma. In contrast, this paper also shows that Yablo's paradox and its nested variant are non-self-referential, and neither McGee's paradox nor the omega-cycle liar paradox has circularity dependence. (shrink)

According to the revision theory of truth, the binary sequences generated by the paradoxical sentences in revision sequence are always unstable. In this paper, we work backwards, trying to reconstruct the paradoxical sentences from some of their binary sequences. We give a general procedure of constructing paradoxes with specific binary sequences through some typical examples. Particularly, we construct what Herzberger called “unstable statements with unpredictably complicated variations in truth value.” Besides, we also construct those paradoxes with infinitely many finite primary (...) periods but without any infinite primary period, those with an infinite critical point but without any finite primary period, and so on. This is the first formal appearance of these paradoxes. Our construction demonstrates that the binary sequences generated by a paradoxical sentence are something like genes from which we can even rebuild the original sentence itself. (shrink)

I lay out the framework for my theory of sensory imagination in “Imagining as a guide to possibility.” Sensory imagining involves mental imagery , and crucially, in describing the content of imagining, I distinguish between qualitative content and assigned content. Qualitative content derives from the mental image itself; for visual imaginings, it is what is “pictured.” For example, visually imagine the Philadelphia Eagles defeating the Pittsburgh Steelers to win their first Super Bowl. You picture the greenness of the field and (...) the football’s brown oblong shape. Some of what you imagine isn’t explicitly pictured, however. That it is Sunday, that it is the Super Bowl: these facts are assigned. (shrink)

It is proved that Yablo’s paradox and the Liar paradox are equiparadoxical, in the sense that their paradoxicality is based upon exactly the same circularity condition—for any frame ${\mathcal{K}}$ , the following are equivalent: (1) Yablo’s sequence leads to a paradox in ${\mathcal{K}}$ ; (2) the Liar sentence leads to a paradox in ${\mathcal{K}}$ ; (3) ${\mathcal{K}}$ contains odd cycles. This result does not conflict with Yablo’s claim that his sequence is non-self-referential. Rather, it gives Yablo’s paradox a new significance: (...) his construction contributes a method by which we can eliminate the self-reference of a paradox without changing its circularity condition. (shrink)

According to the revision theory of truth, the paradoxical sentences have certain revision periods in their valuations with respect to the stages of revision sequences. We find that the revision periods play a key role in characterizing the degrees of paradoxicality for Boolean paradoxes. We prove that a Boolean paradox is paradoxical in a digraph, iff this digraph contains a closed walk whose height is not any revision period of this paradox. And for any finitely many numbers greater than 1, (...) if any of them is not divisible by any other, we can construct a Boolean paradox whose primary revision periods are just these numbers. Consequently, the degrees of Boolean paradoxes form an unbounded dense lattice. The area of Boolean paradoxes is proved to be rich in mathematical structures and properties. (shrink)

Cook regards Sorenson’s so-called ‘the no-no paradox’ as only a kind of ‘meta-paradox’ or ‘quasi-paradox’ because the symmetry principle that Sorenson imposes on the paradox is meta-theoretic. He rebuilds this paradox at the object-language level by replacing the symmetry principle with some ‘background principles governing the truth predicate’. He thus argues that the no-no paradox is a ‘new type of paradox’ in that its paradoxicality depends on these principles. This paper shows that any theory is inconsistent with the T-schema instances (...) for the no-no sentences, plus the T-schema instance for a Curry sentence associated with the symmetry of the no-no sentences. It turns out that the no-no paradox still depends on the problematic instances of the T-schema in a way that the liar paradox does. What distinguishes the no-no paradox is the T-schema instance for the above Curry sentence, which encodes Sorensen’s symmetry principle at the object-language level. (shrink)

In various theories of truth, people have set forth many definitions to clarify in what sense a set of sentences is paradoxical. But what, exactly, is _a_ paradox per se? It has not yet been realized that there is a gap between ‘being paradoxical’ and ‘being a paradox’. This paper proposes that a paradox is a minimally paradoxical set meeting some closure property. Along this line of thought, we give five tentative definitions based upon the folk notion of paradoxicality implied (...) in Tarski’s undefinability theorem of truth and the dependence relation proposed by Leitgeb (J Philos Log 34(2):155–192, 2005). A definition of ‘being a paradox’ is acceptable if its extension must be sufficiently comprehensive to include all known paradoxes on the one hand, and its standard must also be high enough to exclude those that are evidently not paradoxes on the other hand. It turns out that only the last attempt is acceptable: a set of sentences is a paradox if it is paradoxical and self-dependent, and at the same time, it and any of its paradoxical and self-dependent subsets bisimulate each other. In this way, we articulate in what sense a set of sentences is a paradox, and so establish a connection between the notion of a paradox and the notion of paradoxicality. (shrink)

Studies of occupational sex segregation rely on the sociocultural model to explain why some occupations are numerically dominated by women and others by men. This model argues that occupational sex segregation is driven by norms about gender-appropriate work, which are frequently conceptualized as gender-typed skills: work-related tasks, abilities, and knowledge domains that society views as either feminine or masculine. The sociocultural model thus explains the primary patterns of occupational sex segregation, which conform to these norms: Requirements for feminine skills increase (...) with women’s representation in the occupation. However, the model does not adequately explain cases of segregation that deviate from these norms or investigate the ways in which feminine and masculine skills co-occur in occupations. The present study fills these gaps by evaluating two previously untested explanations for deviations from the sociocultural model. The findings show that requirements for physical strength increase with women’s representation in professional occupations because physical strength skills co-occur with substantially higher requirements for feminine skills that involve helping and caring for others. These results indicate that the sociocultural model, and more generally explanations for how gender norms drive occupational sex segregation, can be improved by examining patterns of gender-typed skill co-occurrence. (shrink)

Tarski's theorem essentially says that the Liar paradox is paradoxical in the minimal reflexive frame. We generalise this result to the Liar-like paradox $\lambda^\alpha$ for all ordinal $\alpha\geq 1$. The main result is that for any positive integer $n = 2^i(2j+1)$, the paradox $\lambda^n$ is paradoxical in a frame iff this frame contains at least a cycle the depth of which is not divisible by $2^{i+1}$; and for any ordinal $\alpha \geq \omega$, the paradox $\lambda^\alpha$ is paradoxical in a frame (...) iff this frame contains at least an infinite walk which has an arbitrarily large depth. We thus get that $\lambda^n$ has a degree of paradoxicality no more than $\lambda^m$ iff the multiplicity of 2 in the (unique) prime factorisation of $n$ is no more than that in the prime factorisation of $m$; and all tranfinite $\lambda^\alpha$ has the same degree of paradoxcality but has a higher degree of paradoxicality than any $\lambda^n$. (shrink)

This paper aims to explore the relationship between the necessity predicate and the truth predicate by comparing two possible-world interpretations. The first interpretation, proposed by Halbach et al. (J Philos Log 32(2):179–223, 2003), is for the necessity predicate, and the second, proposed by Hsiung (Stud Log 91(2):239–271, 2009), is for the truth predicate. To achieve this goal, we examine the connections and differences between paradoxical sentences that involve either the necessity predicate or the truth predicate. A primary connection is (...) established through two translations that change only one of the predicates to the other while keeping everything else unchanged. We prove that in bijective frames, a set of sentences that contains one of the two semantic predicates has the same paradoxicality as the corresponding set of sentences that contains the other predicate obtained through translation. However, there are substantial differences as well. First, the necessity predicate and the truth predicate, under the two interpretations, cannot be defined by each other. Moreover, for sentences that involve only the truth predicate, their paradoxicality is preserved under the homomorphisms of frames. For sentences containing the necessity predicate, their paradoxicality is preserved under bounded morphisms, but none of these sentences can have their paradoxicality preserved under the extension of frames. Finally, we also show that paradoxical sentences involving the necessity predicate and those involving the truth predicate differ significantly in terms of mirror symmetry, circularity dependence, and frame compactness. (shrink)

This article is a lecture note I wrote for my philosophy of mathematics course. Its main task is to explain the main ideas of Gupta's acceptable model proposed in his paper [J. Philos. Logic 11(1), 1–60, 1982]. I aim to provide detailed information on a result established by Gupta. On the one hand, I hope this explanation can be helpful for those who are learning Gupta's acceptable model, and on the other hand, I also hope to provide a guide for (...) beginners to interpret literature. (shrink)

We can classify the (truth-theoretic) paradoxes according to their degrees of paradoxicality. Roughly speaking, two paradoxes have the same degrees of paradoxicality, if they lead to a contradiction under the same conditions, and one paradox has a (non-strictly) lower degree of paradoxicality than another, if whenever the former leads to a contradiction under a condition, the latter does so under the same condition. In this paper, we outline some results and questions around the degrees of paradoxicality and summarize recent progress.

By introducing the intensional mappings and their properties, we establish a new semantical approach of characterizing intermediate logics. First prove that this new approach provides a general method of characterizing and comparing logics without changing the semantical interpretation of implication connective. Then show that it is adequate to characterize all Kripke_complete intermediate logics by showing that each of these logics is sound and complete with respect to its (unique) ‘weakest characterization property’ of intensional mappings. In particular, we show that classical (...) logic has the weakest characterization property , which is the strongest among all possible weakest characterization properties of intermediate logics. Finally, it follows from this result that a translation is an embedding of classical logic into intuitionistic logic, iff. its semantical counterpart has the property. (shrink)

Substantial average gender differences in childhood play behaviour and occupational interests have been well-documented. Recent research shows that childhood gender-related play behaviour longitudinally predicts gender-related occupational interests in adolescence. The first aim of the present study was to extend this recent finding by examining whether university students’ recalled childhood gender-related play behaviour predicts their current gender-related occupational interests. The second aim of the present study was to investigate whether gender-related socio-cognitive processes mediate the relation between childhood play behaviour and subsequent (...) occupational interests. University students completed scales assessing recalled childhood gender-related play behaviour, gender-related occupational interests, gender typicality, gender contentedness, agentic goal endorsement, communal goal endorsement, and gender-related occupational stereotype flexibility. In the present study, recalled childhood gender-related play behaviour predicted gender-related occupational interests in both men and women. In men, gender typicality and gender contentedness mediated the play-interests link. In women, gender typicality and communal goal endorsement mediated the play-interests link. The present study provides further evidence that childhood gender-related play behaviour is related to subsequent gender-related occupational interests. Although the current study has a correlational design, one interpretation of the current findings is that childhood play may influence socio-cognitive processes, such as gender compatibility and goal endorsement, which may in turn shape occupational interests. (shrink)

Chinese culture is a lively and active organism. If we are to get to the real image of Chinese culture, we should take as a starting point the first passage of the Chung-yung [Doctrine of the Mean], which expounds its basic principles: "That which Heaven has decreed is called Nature, to follow that Nature is called tao [way] and to cultivate the tao is called instruction." This passage uses three phrases and one breath; these three phrases have a unified nature (...) which cannot be separated and which constitute one long sentence. Because of this, the words as a whole lead people to think of a richly vital and great tree which has the will of Heaven as its roots, human nature as its shoots, the tao as its trunk, and cultural instruction as its branches, leaves, and blossoms. As for the fruit, in terms of the individual, it is the achievement of a completely beautiful character, and in terms of the masses of the people, it is the ordering of the country and the pacification of the realm, the progression from the era of peace under the ancient kings to the age of the Grand Unity [ta-t'ung]. (shrink)

Algorithmically driven online dating platforms today promise the ability to sort through relevant data and identify one’s ideal amorous matches effectively. Yet the appeal of technological and scientific solutions to the messy problem of finding partners is hardly new. This introduction to the Focus section “It’s a Match!” argues that the history of amorous matching has long been part and parcel of the history of science, in particular the social sciences. Taking matching as an “applied science of social harmony,” the (...) authors argue that concern over more reliable techniques for determining the suitability of partners has formed an essential part of both the maintenance of social order and the shaping of subjectivities, enabling discourses of informed choice and the rational management of the passions, while also reinforcing and subverting structures of age, gender, race, and sexuality. (shrink)

: It is generally accepted that skeptical scenarios must be possible to raise legitimate skeptical doubt. I argue that if the possibility in question is supposed to be genuine metaphysical possibility, the skeptic's reasoning does not straightforwardly succeed. I first motivate the metaphysical possibility requirement on skeptical scenarios : it's a plausible position that several authors accept and that a family of prominent views—sensitivity, safety, relevant alternatives—are committed to. I argue that plausible constraints in modal epistemology show that justification for (...) believing that certain global skeptical scenarios are metaphysically possible rests on some justified beliefs about the external world, and that this undermines the skeptical argument. While there may still be local skeptical challenges, skeptics cannot appeal to the metaphysical possibility of skeptical scenarios to generate global external world skepticism. (shrink)

The DIALOG between Husserlian Phenomnnology and Analytic Philosophy is severely hampered by the fact that much of the secondary literrature on phenomenology fails to pay attention to certain subtile semantical distinctions which are basic for a clear understanding of epistemological issures. Some European Phenomenologists even take pride in their neglect of what they consider to be shallow scholastic quibbling. I hope to remedy this short-coming by outlining in this paper what I believe to be the keypoints of Husserl's theory of (...) perception and theory of meaning. My apprach will be developmental because contrary to popular belief phenomenology has not been revealed in ont instantaneous intuition; on the contrary Husserl's views developed and shifted constantly throughout his life. (shrink)

Imagination is celebrated as our vehicle for escape from the mundane here and now. It transports us to distant lands of magic and make-believe, and provides us with diversions during boring meetings or long bus rides. Yet the focus on imagination as a means of escape from the real world minimizes the fact that imagination seems also to furnish us with knowledge about it. Imagination seems an essential component in our endeavor to learn about the world in which we live--whether (...) we're planning for the future, aiming to understand other people, or figuring out whether two puzzle pieces fit together. But how can the same mental power that allows us to escape the world as it currently is also inform us about the world as it currently is? Ten original essays grapple with this neglected question; in doing so, they present a diverse array of positions ranging from cautious optimism to deep-seated pessimism. Blending perspectives from philosophy of mind, cognitive science, epistemology, aesthetics, and ethics, Knowledge Through Imagination sheds new light on the epistemic role of imagination. (shrink)