We prove that a combinatorial consequence of the negation of the PCF conjecture for intervals, involving free subsets relative to set mappings, is not implied by even the strongest known large cardinal axiom.

In this paper we show the embedding of Hybrid Probabilistic Logic Programs into the rather general framework of Residuated Logic Programs, where the main results of (definite) logic programming are validly extrapolated, namely the extension of the immediate consequences operator of van Emden and Kowalski. The importance of this result is that for the first time a framework encompassing several quite distinct logic programming semantics is described, namely Generalized Annotated Logic Programs, Fuzzy Logic Programming, Hybrid Probabilistic Logic Programs, and Possibilistic (...) Logic Programming. Moreover, the embedding provides a more general semantical structure paving the way for defining paraconsistent probabilistic reasoning with a logic programming semantics. (shrink)

We consider simplified representation theorems in pcf-theory and, in particular, we prove that if ${\aleph_{\omega}^{\aleph_{0}} > \aleph_{\omega_{1}}\cdot2^{\aleph_{0}}}$ then there are cofinally many sequences of regular cardinals such that ${\aleph_{\omega_{1}+1}}$ is represented by these sequences modulo the ideal of finite subsets, using a topological approach to the pcf-structure.

Counterfactuals are conjectures about what would have happened, had an alternative event occurred. It provides lessons for the future by virtue of contemplating alternatives; it permits thought debugging; it supports a justification why different alternatives would have been worse or not better. Typical expressions are: “If only I were taller …”, “I could have been a winner …”, “I would have passed, were it not for …”, “Even if... the same would follow”. Counterfactuals have been well studied in Linguistics, Philosophy, (...) Physics, Ethics, Psychology, Anthropology, and Computation, but not much within Critical Thinking. The purpose of this study is to illustrate counterfactual thinking, through logic program abduction and updating, and inspired by Pearl’s structural theory of counterfactuals, with an original application to morality, a common concern for critical thinking. In summary, we show counterfactual reasoning to be quite useful for critical thinking, namely about moral issues. (shrink)