Pure Type Systems, PTSs, were introduced as a generalization of the type systems of Barendregt's lambda cube and were designed to provide a foundation for actual proof assistants which will verify proofs. Systems of illative combinatory logic or lambda calculus, ICLs, were introduced by Curry and Church as a foundation for logic and mathematics. In an earlier paper we considered two changes to the rules of the PTSs which made these rules more like ICL rules. This led to four kinds (...) of PTSs. Most importantly PTSs are about statements of the form M:A, where M is a term and A a type. In ICLs there are no explicit types and the statements are terms. In this paper we show that for each of the four forms of PTS there is an equivalent form of ICL, sometimes if certain conditions hold. (shrink)

If T has only countably many complete types, yet has a type of infinite multiplicity then there is a c.c.c. forcing notion Q such that, in any Q-generic extension of the universe, there are non-isomorphic models M 1 and M 2 of T that can be forced isomorphic by a c.c.c. forcing. We give examples showing that the hypothesis on the number of complete types is necessary and what happens if `c.c.c.' is replaced by other cardinal-preserving adjectives. We also give (...) an example showing that membership in a pseudo-elementary class can be altered by very simple cardinal-preserving forcings. (shrink)

Listening to someone from some distance in a crowded room you may experience the following phenomenon: when looking at them speak, you may both hear and see where the source of the sounds is; but when your eyes are turned elsewhere, you may no longer be able to detect exactly where the voice must be coming from. With your eyes again fixed on the speaker, and the movement of her lips a clear sense of the source of the sound will (...) return. This ‘ventriloquist’ effect reflects the ways in which visual cognition can dominate auditory perception. And this phenomenological observation is one what you can verify or disconfirm in your own case just by the slightest reflection on what it is like for you to listen to someone with or without visual contact with them. (shrink)

This volume offers the first-ever complete English translation of Leibniz’s Dissertatio De Arte Combinatoria together with a critical edition of the original Latin text on fa...

Our research is based on a rather large "library" of various works by M. Drahomanov, which contains his views on religion. Among them: Paradise and Progress, From the History of Relations Between Church and State in Western Europe, Faith and Public Affairs, Fight for Spiritual Power and Freedom of Conscience in the 16th - 17th Centuries,, "Church and State in the Roman Empire", "The Status and Tasks of the Science of Ancient History," "Evangelical Faith in Old England," "Populism and Popular (...) Progress in Austrian Rus, Austrian-Russian Remembrance," "Pious The Legend of the Bulgarians "," The Issues of Religious Freedom in Russia, "" On the Brotherhood of the Baptist or the Baptist in Ukraine, "" The Foreword, " Shevchenko, Ukrainianophiles and Socialism "," Wonderful thoughts about the Ukrainian national affair "," Zazdri gods "," Slavic variants of one Gospel legend "," Resurrection of Christ ", etc. (shrink)