Online research in philosophy

v. 1. Plato's Parmenides: history and interpretation from the old academy to later platonism and gnosticism -- Section 1: Plato, from the the old academy to middle platonism -- Section 2: Middle platonic and gnostic texts -- v. 2. Plato's Parmenides: its reception in neoplatonic, Jewish, and Christian texts -- Section 1: Parmenides interpretation from Plotinus to Damascius -- Section 2: The hidden influence of the Parmenides in philo, origen, and later patristic thought.

Let Λ be a singular cardinal of uncountable confinality ψ. Under various assumptions about the sizes of covering families for cardinals below Λ, we prove upper bounds for the covering number cov(Λ, Λ, v⁺, 2). This covering number is closely related to the cofinality of the partial order ([Λ]", ⊆).

Many known tools for proving expressibility bounds for first-order logic are based on one of several locality properties. In this paper we characterize the relationship between those notions of locality. We note that Gaifman's locality theorem gives rise to two notions: one deals with sentences and one with open formulae. We prove that the former implies Hanf's notion of locality, which in turn implies Gaifman's locality for open formulae. Each of these implies the bounded degree property, which is one of the easiest tools for proving expressibility bounds. These results apply beyond the first-order case. We use them to derive expressibility bounds for first-order logic with unary quantifiers and counting. We also characterize the notions of locality on structures of small degree.

The following is my interpretation of the philosophy of Parmenides of Elea , the Greek father of metaphysics. His only work, On Nature , is written in rather obscure verse, and so his thesis can be viewed from a variety of perspectives, of which mine is only one (although a fairly standard one). Parmenides' most important principle, hereafter called "Parmenides' Principle", was that anything rationally conceivable must exist. Nonbeing is not a thing and can neither be thought of nor spoken about in any meaningful or coherent way. Parmenides forbade talking as if there are possible things that nonetheless do not exist. He illustrated this principle by showing us three possible methods of inquiry, of which only one is valid. The following chart summarizes them.

In his great poem, Parmenides uses an argument by elimination to select the correct "way of inquiry" from a pool of two, the ways of is and of is not , joined later by a third, "mixed" way of is and is not . Parmenides' first two ways are soon given modal upgrades - is becomes cannot not be , and is not becomes necessarily is not (B2, 3-6) - and these are no longer contradictories of one another. And is the common view right, that Parmenides rejects the "mixed" way because it is a contradiction? I argue that the modal upgrades are the product of an illicit modal shift. This same shift, built into two Exclusion Arguments, gives Parmenides a novel argument to show that the "mixed" way fails. Given the independent failure of the way of is not , Parmenides' argument by elimination is complete.

Parmenides formulated a formal ontology, to which various additions and alternatives were proposed by Melissus, Gorgias, Leucippus and Democritus. These systems are here interpreted as modifications of a minimal Le?niewskian ontology.