We present a Revised Projectivity Calculus that extends the scope of inclusion and exclusion inferences derivable under the Projectivity Calculus developed by Icard :705–725, 2012). After pointing out the inadequacies of C, we introduce four opposition properties which have been studied by Chow Proceedings of the 18th Amsterdam Colloquium, Springer, Berlin, 2012; Beziau, Georgiorgakis New dimensions of the square of opposition, Philosophia Verlag GmbH, München, 2017) and are more appropriate for the study of exclusion reasoning. Together with the monotonicity properties, (...) the OPs will form the basis of RC instead of the additive/multiplicative properties used in C. We also prove some important results of the OPs and their relation with the MPs. We then introduce a set of projectivity signatures together with the associated operations and conditions for valid inferences, and develop RC by inheriting the key features of C. We then show that under RC, we can derive some inferences that are not derivable under C. We finally discuss some properties of RC and point to possible directions of further studies. (shrink)

In the first half of this paper, we present a fragment of relational syllogisms named RELSYLL consisting of quantified statements with a special set of numerical quantifiers, and introduce a number of concepts that are useful for the later sections, including indirect reduction, quantifier transformations and equivalence of syllogisms. After determining the valid and invalid syllogisms in RELSYLL, we then introduce two Derivation Methods which can be used to derive valid relational syllogisms based on known valid simple syllogisms. We also (...) show that the two Methods are sound and complete for RELSYLL. In the second half of this paper, we discuss ways to extend the Derivation Methods, including the use of more valid syllogisms and the use of existential assumptions. In this way, we are able to derive more relational syllogisms that contain other types of non-classical quantifiers, including “only” and proportional quantifiers. Finally, we state and prove a proposition concerning the relationship between the two Methods. (shrink)