Turing progressions have been often used to measure the proof-theoretic strength of mathematical theories: iterate adding consistency of some weak base theory until you “hit” the target theory. Turing progressions based on n-consistency give rise to a \ proof-theoretic ordinal \ also denoted \. As such, to each theory U we can assign the sequence of corresponding \ ordinals \. We call this sequence a Turing-Taylor expansion or spectrum of a theory. In this paper, we relate Turing-Taylor expansions of sub-theories (...) of Peano Arithmetic to Ignatiev’s universal model for the closed fragment of the polymodal provability logic \. In particular, we observe that each point in the Ignatiev model can be seen as Turing-Taylor expansions of formal mathematical theories. Moreover, each sub-theory of Peano Arithmetic that allows for a Turing-Taylor expansion will define a unique point in Ignatiev’s model. (shrink)

According to Paul Grice, when we address someone, we intend to make ourselves understood, partly by the addressee’s recognition of that intention. Call this set of nested audience-directed intentions an M-intention. The standard Gricean analysis of speaker’s meaning goes as follows: “U meant something by uttering x” is true iff, for some audience A, U uttered x intending: (1) A to produce a particular response r (2) A to think (recognize) that U intends (1) (3) A to fulfill (1) on (...) the basis of his fulfillment of (2). (Grice 1969, p. 92) Call this analysis the M-schema, and to fulfill it, to M-intend. In the Gricean tradition there has been a debate as to the exact form of Mintentions (especially whether clause (3) is needed or should be left out of the analysis), but these are details that I will not address here. The phenomenon of talking to infants challenges the Gricean analysis of addressing others. We talk to infants, but such cases cannot be described as M-intending anything, since the speaker must realize that he will not be able to make himself understood in the required manner. The Gricean mechanism lacks an appropriate addressee. Some Griceans have argued that in cases of no appropriate addressee, the speaker is actually addressing himself (Schiffer 1972, p. 80; Avramides 1989, p. 66). This response might be tempting for soliloquy, but I will show that it is unsuited for the phenomenon of talking to infants. (shrink)

We introduce the logics GLP Λ, a generalization of Japaridze’s polymodal provability logic GLP ω where Λ is any linearly ordered set representing a hierarchy of provability operators of increasing strength. We shall provide a reduction of these logics to GLP ω yielding among other things a finitary proof of the normal form theorem for the variable-free fragment of GLP Λ and the decidability of GLP Λ for recursive orderings Λ. Further, we give a restricted axiomatization of the variable-free fragment (...) of GLP Λ. (shrink)

ABSTRACTThis paper follows the text of Leviticus 25:10 in the Hebrew Bible and in selected works of the exegetical tradition of both Rabbinic Judaism and Western Christianity, in order to provide a lens through which to assess the use of a biblical text which was instrumental during the early modern period in formulating ideas about the Republic and its use in the modern liberal state. The main argument of the paper is that over time the meaning of the text shifted (...) from the socio-economic to the salvific to the political, depending on the context in which it was read. Further, all the authors cited here approached the text as an authoritative normative text, and did not look at the text as a textual artefact. While the move to re-introduce Jewish Sources into the debate in political theory is to be welcomed, it is argued that the results would be improved by balanced reading strategies and by interaction with critical academic biblical scholarship. (shrink)