Annals of Pure and Applied Logic最新文献

The article studies two forms of responsibility in the setting of strategic games with imperfect information. They are referred to as seeing-to-it responsibility and counterfactual responsibility. It shows that counterfactual responsibility is definable through seeing-to-it, but not the other way around. The article also proposes a sound and complete bimodal logical system that describes the interplay between the seeing-to-it modality and the individual ex ante knowledge modality.

The standard comparison lemma of inner model theory is deficient, in that it does not in general produce a comparison of all the relevant inputs. How two mice compare can depend upon which iteration strategies are used to compare them. We shall outline here a method for comparing iteration strategies that removes this defect.

We introduce a general framework of generalized tree forcings, GTF for short, that includes the classical tree forcings like Sacks, Silver, Laver or Miller forcing. Using this concept we study the cofinality of the ideal $I\left(Q\right)$ associated with a GTF Q. We show that if for two GTF's $Q_0$ and $Q_1$ the consistency of $\mathrm{add}\left(I\right(Q_0\left)\right)<\mathrm{add}\left(I\right(Q_1\left)\right)$ holds, then we can obtain the consistency of $\mathrm{cof}\left(I\right(Q_1\left)\right)<\mathrm{cof}\left(I\right(Q_0\left)\right)$. We also show that $\mathrm{cof}\left(I\right(Q\left)\right)$ can consistently be any cardinal of cofinality larger than the continuum.

“Eggleston's dichotomy” is a “one of a kind” unique observation which broadly tells us that the characterized subgroups of the circle group (characterized by a sequence of positive integers $\left(a_n\right)$) are either countable or of cardinality $c$ depending on the asymptotic behavior of the sequence of the ratios $\frac{a_n}{a_{n-1}}$. One should note that these subgroups are generated by using the notion of usual convergence which is nothing but a special case of the more general notion of ideal convergence for the ideal Fin. It has been recently established that “Eggleston's dichotomy” fails in the case of modified versions of characterized subgroups when the ideal Fin is replaced by the natural density ideal $I_d$, or more generally, by ideals which are now known as simple density and modular simple density ideals. As all the ideals mentioned above are analytic P-ideals, a natural question arises as to whether one can isolate some appropriate property of ideals which enforces the dichotomy or the failure of it. In this article we are able to isolate that particular feature of an ideal and come out with a new class of ideals which we call, “strongly non-translation invariant ideals” (in short snt-ideals). In particular, we are able to establish that for a sequence of positive integers $\left(a_n\right)$, be it arithmetic or arising from the continued fraction expansion of an irrational number:

• (i)

  For non-snt analytic P ideals, the size of the corresponding characterized subgroups is always $c$ even if the sequence $\left(a_n\right)$ is b-bounded (i.e. the sequence of the ratios $\frac{a_n}{a_{n-1}}$ is bounded) which implies the breaking down of “Eggleston's dichotomy”.

• (ii)

  For snt analytic P ideals, the corresponding characterized subgroups are always countable if the sequence $\left(a_n\right)$ is b-bounded which means “Eggleston's dichotomy” holds.

We give four different independence relations on any exponential field. Each is a canonical independence relation on a suitable Abstract Elementary Class of exponential fields, showing that two of these are NSOP₁-like and non-simple, a third is stable, and the fourth is the quasiminimal pregeometry of Zilber's exponential fields, previously known to be stable (and uncountably categorical). We also characterise the fourth independence relation in terms of the third, strong independence.

If $B$ is a relational structure, define $P\left(B\right)$ the partial order of all substructures of $B$ that are isomorphic to it. Improving a result of Kurilić and the second author, we prove that if $R$ is the random graph, then $P\left(R\right)$ is forcing equivalent to $S⁎\dot{R}$, where $S$ is Sacks forcing and $\dot{R}$ is an ω-distributive forcing that is not forcing equivalent to a σ-closed one. We also prove that ${P(H}_3)$ is forcing equivalent to a σ-closed forcing, where $H_3$ is the generic triangle-free graph.

The Gödel-McKinsey-Tarski embedding allows to view intuitionistic logic through the lenses of modal logic. In this work, an extension of the modal embedding to infinitary intuitionistic logic is introduced. First, a neighborhood semantics for a family of axiomatically presented infinitary modal logics is given and soundness and completeness are proved via the method of canonical models. The semantics is then exploited to obtain a labelled sequent calculus with good structural properties. Next, soundness and faithfulness of the embedding are established by transfinite induction on the height of derivations: the proof is obtained directly without resorting to non-constructive principles. Finally, the modal embedding is employed in order to relate classical, intuitionistic and modal derivability in infinitary logic extended with axioms.

We provide a general preservation theorem for preserving selective independent families along countable support iterations. The theorem gives a general framework for a number of results in the literature concerning models in which the independence number $i$ is strictly below $c$, including iterations of Sacks forcing, Miller partition forcing, h-perfect tree forcings, coding with perfect trees. Moreover, applying the theorem, we show that $i={\aleph }_1$ in the Miller Lite model. An important aspect of the preservation theorem is the notion of “Cohen preservation”, which we discuss in detail.

We generically construct a model in which the ${\Pi }_3^1$-reduction property is true and the ${\Pi }_3^1$-uniformization property is false, thus producing a model which separates these two principles for the first time.