R. S. Woolhouse; IV*—Leibniz's Reaction to Cartesian Interaction, Proceedings of the Aristotelian Society, Volume 86, Issue 1, 1 June 1986, Pages 69–82, https:/.

We generalize two well-known model-theoretic characterization theorems from propositional modal logic to first-order modal logic. We first study FML-definable frames and give a version of the Goldblatt–Thomason theorem for this logic. The advantage of this result, compared with the original Goldblatt–Thomason theorem, is that it does not need the condition of ultrafilter reflection and uses only closure under bounded morphic images, generated subframes and disjoint unions. We then investigate Lindström type theorems for first-order modal logic. We show that FML has (...) the maximal expressive power among the logics extending FML which satisfy compactness, bisimulation invariance and the Tarski union property. (shrink)

An algebraic semantics, based on factor algebras, for one-way and two-way implicative verbs is proposed. Implicative verbs denote elements of filters or of ideals generated by identity functions in factor algebras. This semantics explains in particular the problem of implicational equivalence raised by two-way implicative verbs, and shows that the negation necessary to establish the implicativity of these verbs is the negation which preserves the presuppositions of sentences with implicative verbs. In addition, it follows from the proposed semantics that any (...) two implicative verbs denoting in the same algebra but belonging to different categories, are semantically related. (shrink)

The article studies two related issues. First, it introduces the notion of the contraposition of quantifiers which is a “dual” notion of symmetry and has similar relations to co-intersectivity as symmetry has to intersectivity. Second, it shows how symmetry and contraposition can be generalised to higher order type quantifiers, while preserving their relations with other notions from generalized quantifiers theory.

Available on its own, or as part of the 9-volume reissue of the classic Political Thinkers series.

The problems and the history of contacts between distant continents in bygone ages and long before the age of fast and easy travel, have always fascinated both professional scholars and the interested public. Was ancient history really nothing but the history of co-existing and isolated geographic, cultural and political “islands?” Already at school we learned too much about migrations of peoples, economic contacts, influences on art styles, conquests, and the rise, expansion and fall of empires to believe that. The (highly (...) improbable) theory that certain archaeological finds in America suggest, or even prove, Mediterranean influence (e.g., the arrival of Phoenician ships), or the alleged Viking discovery of America centuries before Columbus, or Thor Heyerdahl's adventurous journey to prove South American influence on remote Pacific islands did not fail to cause widespread interest and even excitement (although many scholars still feel that Heyerdahl's epic adventure, thrilling as it is, failed to prove what it set out to prove). The island of Madagascar is pretty close to the African continent, but its language exhibits more points of contact with Polynesian than with African tongues. And the blow-bellows used by Malagache smiths are similar to those used in Malaysia and unlike those known to African metalworkers. Clearly the Polynesians, those great ancient mariners, sailed further than originally seemed likely. Musicologists studying the cantillation of the Hebrew Bible in the liturgy of the ancient Jewish communities on the Malabar coast in India discovered to their surprise that it was similar to the cantillation not of the Babylonian but of the Yemenite Jews! This surprise was, of course, no surprise to those who knew anything about trade-winds and shipping routes between South Arabia and the Indian coast. Historians of religion, even more than general historians, studying the spread and expansion of religious ideas and movements have realised long ago that the beginning of all wisdom is a basic knowledge of economic geography. (shrink)

Amongst the people who pester Peisthetaerus with unwanted help and advice in the latter part of the Birds is Meton, famous astronomer and mathematician, who produces and demonstrates with instruments a method of laying out the plan of the new town. Peisthetaerus makes no attempt to follow him and quickly bundles him out again without much ceremony. Commentators and readers with few exceptions treat him in a similar way. ʹΕπίτηδες δανόητα, δόλου νοηταίνε, παίζε—such are the comments of the scholiast, and (...) editors are mostly content with that. Van Leeuwen says, ‘Metonis haec verba intellegere velle, id est operam dare ut suo ioco frustretur cbtnicus.’ The passage is of course highly comical; to make it didactic and attribute to Aristophanes a serious excursion into geometry and town-planning would be perverse and pedantic; but in Aristophanes more than in most comic writers there is commonly a grain or two of truth among the chaff. These grains, though of little or no importance for the appreciation of the play as comedy, may still have some possible value in other ways, and should be carefully sifted. (shrink)

It is shown that the notion of the partition of a set can be used to describe in a uniform way the meaning of the expression the same, in its basic uses in transitive and ditransitive sentences. Some formal properties of the function denoted by the same, which follow from such a description are indicated. These properties indicate similarities and differences between functions denoted by the same and generalised quantifiers.