Let µ be a regular cardinal. In this paper I prove two (forcing) existence results concerning structures governed by two parameters, the cardinal µ and an ordinal ρ less than µ+++. The results improve on theorems from [M*2] where the second parameter was always the cardinal µ++.
κ-M-proper forcing, introduced in [K00] when κ = ω1, is a very powerful new technique for generic stepping up, subsuming all previous generic steppings up using auxiliary functions. A general framework for using κ-M-proper forcing is set out, and a couple of examples of such forcings, adding κ−-thin-very tall scattered spaces and long chains in P(κ) modulo <κ−, are given. These objects are not currently obtainable by the previously known techniques.
Local connectedness functions for (κ, 1)-simplified morasses, localisations of the coupling function c studied in [M96, §1], are defined and their elementary properties discussed. Several different, useful, canonical ways of arriving at the functions are examined. This analysis is then used to give explicit formulae for generalisations of the local distance functions which were defined recursively in [K00], leading to simple proofs of the principal properties of those functions. It is then extended to the properties of local connectedness functions in (...) the context of κ-M-proper forcing for sucessor κ. The functions are shown to enjoy substantial strengthenings of the properties (particularly the ∆-properties) hitherto proved for both c and for Todorcevic’s ρ-functions in the special case κ = ω1. A couple of examples of the use of local connectedness functions in consort with κ-M-proper forcing are then given. (shrink)
is a (κ, 1)-simplified morass if θα | α < κ is an increasing sequence of ordinals less than κ, θκ = κ+, and each Fαβ is a collection of maps from θα to θβ such that the following properties hold.
We prove consistent, assuming there is a supercompact cardinal, that there is a singular strong limit cardinal µ, of cofinality ω, such that every µ+-chromatic graph X on µ+ has an edge colouring c of X into µ colours for which every vertex colouring g of X into at most µ many colours has a g-colour class on which c takes every value.
After a couple of weeks I eventually got around to reading the preprint and started wondering about recasting the argument in my preferred formalism. I arrogantly assumed that this would allow one to smooth out parts of the proof and simplify the details of the definition of the forcing conditions (at the cost of taking the framework set out in §§1,2 below as given). However when I tried to write things down I found myself, to my chagrin, more or less (...) cornered in making my definitions into most of the intricacies that Friedman had been. I suppose that this shows that the original proof is in some sense the natural one (or one of a family of ‘natural’ ones). Nevertheless I obstinately persisted and this note is the result. (shrink)
We prove consistent, assuming there is a supercompact cardinal, that there is a singular strong limit cardinal $\mu$ , of cofinality $\omega$ , such that every $\mu^{+}$ -chromatic graph X on $\mu^{+}$ has an edge colouring c of X into $\mu$ colours for which every vertex colouring g of X into at most $\mu$ many colours has a g-colour class on which c takes every value. The paper also contains some generalisations of the above statement in which $\mu^{+}$ is replaced (...) by other cardinals < $\mu$. (shrink)
Conclusions reached using common sense reasoning from a set of premises are often subsequently revised when additional premises are added. Because we do not always accept previous conclusions in light of subsequent information, common sense reasoning is said to be nonmonotonic. But in the standard formal systems usually studied by logicians, if a conclusion follows from a set of premises, that same conclusion still follows no matter how the premise set is augmented; that is, the consequence relations of standard logics (...) are monotonic. Much recent research in AI has been devoted to the attempt to develop nonmonotonic logics. After some motivational material, we give four formal proofs that there can be no nonmonotonic consequence relation that is characterized by universal constraints on rational belief structures. In other words, a nonmonotonic consequence relation that corresponds to universal principles of rational belief is impossible. We show that the nonmonotonicity of common sense reasoning is a function of the way we use logic, not a function of the logic we use. We give several examples of how nonmonotonic reasoning systems may be based on monotonic logics. (shrink)
In this paper we examine the thesis that the probability of the conditional is the conditional probability. Previous work by a number of authors has shown that in standard numerical probability theories, the addition of the thesis leads to triviality. We introduce very weak, comparative conditional probability structures and discuss some extremely simple constraints. We show that even in such a minimal context, if one adds the thesis that the probability of a conditional is the conditional probability, then one trivializes (...) the theory. Another way of stating the result is that the conditional of conditional probability cannot be represented in the object language on pain of trivializing the theory. (shrink)
This paper concerns the theory of morasses. In the early 1970s Jensen defined (κ,α)-morasses for uncountable regular cardinals κ and ordinals $\alpha . In the early 1980s Velleman defined (κ, 1)-simplified morasses for all regular cardinals κ. He showed that there is a (κ, 1)-simplified morass if and only if there is (κ, 1)-morass. More recently he defined (κ, 2)-simplified morasses and Jensen was able to show that if there is a (κ, 2)-morass then there is a (κ, 2)-simplified morass. (...) In this paper we prove the converse of Jensen's result, i.e., that if there is a (κ, 2)-simplified morass then there is a (κ, 2)-morass. (shrink)
We show that the implicational fragment of intuitionism is the weakest logic with a non-trivial probabilistic semantics which satisfies the thesis that the probabilities of conditionals are conditional probabilities. We also show that several logics between intuitionism and classical logic also admit non-trivial probability functions which satisfy that thesis. On the other hand, we also prove that very weak assumptions concerning negation added to the core probability conditions with the restriction that probabilities of conditionals are conditional probabilities are sufficient to (...) trivialize the semantics. (shrink)
Fuzzy logics are systems of logic with infinitely many truth values. Such logics have been claimed to have an extremely wide range of applications in linguistics, computer technology, psychology, etc. In this note, we canvass the known results concerning infinitely many valued logics; make some suggestions for alterations of the known systems in order to accommodate what modern devotees of fuzzy logic claim to desire; and we prove some theorems to the effect that there can be no fuzzy logic which (...) will do what its advocates want. Finally, we suggest ways to accommodate these desires in finitely many valued logics. (shrink)
Almost every formal model of explanation thus far proposed has been demonstrated to be faulty. In this paper, a new model, proposed by Raimo Tuomela, is also demonstrated to be faulty. In particular, one condition of the model is shown to be too restrictive, and another condition of the model is shown to be too permissive.
The usual semantics for the modal systems T, S4, and S5 assumes that the set of possible worlds contains at least one member. Recently versions of these modal systems have been developed in which this assumption is dropped. The systems discussed here are obtained by slightly weakening the liberated versions of T and S4. The semantics does not assume the existence of possible worlds, and the accessibility relation between worlds is only required to be quasi-reflexive instead of reflexive. Completeness and (...) independence results are established. (shrink)
The intuitive notion behind the usual semantics of most systems of modal logic is that of ?possible worlds?. Loosely speaking, an expression is necessary if and only if it holds in all possible worlds; it is possible if and only if it holds in some possible world. Of course, contradictory expressions turn out to hold in no possible worlds, and logically true expressions turn out to hold in every possible world. A method is presented for transforming standard modal systems into (...) systems of modal logic for impossible worlds. To each possible world there corresponds an impossible world such that an expression holds in the impossible world if and only if it does not hold in the possible world. One can then talk about such worlds quite consistently, and there seems to be no logical reason for excluding them from consideration. (shrink)
