Within legal contexts, a claim to knowledge requires that the evidence appealed to fulfills conditions established by legal norms that belong to the so-called Evidence Law. In addition, the epistemic justification presupposed by this kind of knowledge is temporally affected by the changing character of the evidence and the judicial discretion usually exercised in the applications of the Evidence Law. As such, these two scenarios involve three modalities: temporal, epistemic, and deontic. The present paper philosophically discusses the links between these modal dimensions in the above legal context. Based on the discussion, a formal tense-epistemic-deontic logic is developed. That is, the paper characterizes a formal language formally representing the three dimensions through specific logical operators, constructs a formal semantics for the language, and specifies a formal logical system. Finally, the soundness and completeness of the system with respect to the semantics is proved.

Let T be a countable complete first-order theory with a definable, infinite, discrete linear order. We prove that T has continuum-many countable models. The proof is purely first order, but it raises the question of Borel completeness of T.

In the vertex pursuit game of cops and robbers on finite graphs, the cop has a winning strategy if and only if the graph admits a dominating order. Such graphs are called constructible in the graph theory literature. This equivalence breaks down for infinite graphs, and variants of the game have been proposed to reestablish partial connections between constructibility and being cop-win. We answer an open question of Lehner about one of these variants by giving examples of weak cop-win graphs which are not constructible. We show that the index set of computable constructible graphs is ${\Pi }_1^1$ hard and the index set of computable constructible locally finite graphs is ${\Pi }_4^0$ hard. Finally, we give an example of a computable constructible graph for which every dominating order computes $0^{″}$.

Consider those many-valued logic models in which the truth values are a lattice that supplies interpretations for the logical connectives of conjunction and disjunction, and which has a De Morgan involution supplying an interpretation for negation. Assume that the set of designated truth values is a prime filter in the lattice. Each of these structures determines a simple many-valued logic. We show that there is a single Smullyan-style signed tableau system appropriate for all of the logics these structures determine. Differences between the logics are confined entirely to tableau branch closure rules. Completeness, soundness, and interpolation can be proved in a uniform way for all cases. Since branch closure rules have a limited number of variations, in fact all the semantic structures determine just four different logics, all well-known ones. Asymmetric logics such as strict/tolerant, $\mathsf{ST}$, also share all the same tableau rules, but differ in what constitutes an initial tableau. It is also possible to capture the notion of antivalidity using the same set of tableau rules. Thus a simple set of tableau rules serves as a unifying and classifying device for a natural and simple family of many-valued logics.

In set theory without the axiom of choice ($\mathsf{AC}$), we study the relative strength of the principle “No decreasing sequence of cardinals,” that is, “There is no function f on ω such that $|f(n+1)|<|f(n)|$ for all $n\in \mathit{\omega }$” ($\mathsf{NDS}$) with regard to its position in the hierarchy of weak choice principles. We establish the following results: (1) The Boolean prime ideal theorem plus countable choice does not imply $\mathsf{NDS}$ in $\mathsf{ZF}$; (2) “Every non-well-orderable set has a well-orderable partition into denumerable sets” (${\mathsf{D}}_{{\mathrm{\aleph }}_0}$) plus the axiom of choice for well-ordered families of nonempty finite sets does not imply $\mathsf{NDS}$ ∨ “The axiom of choice for countable families of nonempty countable sets” in $\mathsf{ZFA}$; and (3) The axiom of choice for well-ordered families of nonempty sets, each of cardinality not greater than or equal to $2^{{\mathrm{\aleph }}_0}$, does not imply $\mathsf{NDS}$ ∨ “countable union theorem” in $\mathsf{ZFA}$. The above results answer corresponding questions left open in works by Howard and Rubin (1998) and Howard and Tachtsis (2016). We also show the following: (4) “Every non-well-orderable set has a well-orderable partition into infinite well-orderable sets” ($D_w$) is strictly weaker than ${\mathsf{D}}_{{\mathrm{\aleph }}_0}$ in $\mathsf{ZFA}$. This resolves an open problem from Keremedis and Tachtsis (2003).

Let E be an analytic equivalence relation on a Polish space. We introduce a framework for studying the possible “reasonable” complete classifications and the complexity of possible classifying invariants for E, such that: (1) the standard results and intuitions regarding classifications by countable structures are preserved in this framework; (2) this framework respects Borel reducibility; and (3) this framework allows for a precise study of the possible invariants of certain equivalence relations which are not classifiable by countable structures, such as $E_1$. In this framework we show that $E_1$ can be classified, with classifying invariants which are κ-sequences of $E_0$-classes where $\mathrm{\kappa }=\mathfrak{b}$, and it cannot be classified in such a manner if $\mathrm{\kappa }<\mathbf{add}(\mathcal{B})$. These results depend on analyzing the following submodel of a Cohen-real extension. Let $⟨c_n:n<\mathit{\omega }⟩$ be a generic sequence of Cohen reals, and define the tail intersection model

$$M=\bigcap _{n<\mathit{\omega }}V[⟨c_m:m\ge n⟩].$$

An analysis of reals in M will provide lower bounds for the possible invariants for $E_1$. We also extend a characterization of turbulence, due to Larson and Zapletal, in terms of intersection models.

The use of the primitive notion of mereological fusion (also known as composition and sum) has been considered by various philosophers and logicians, including Aristotle, G. Leibniz, S. Leśniewski, K. Fine, J. Ketland, T. Schindler, and S. Kleishmid. The problem of finding an axiomatization of Classical Mereology with primitive fusion, instead of the primitive notion of being a part, is quite old and was formally considered by C. Lejewski. Lejewski somehow axiomatized classical mereology using primitive fusion (1962, and also later in 1984). However, his mereology is formulated in a very rich nonclassical language of Leśniewski’s ontology with the essential use of quantification over function symbols, including function symbols “part of” and “sum of.” Lejewski’s approach is not expressible in modern mereologies, and we do not use it in any way. We prove that classical mereology is axiomatizable using primitive mereological fusion within the framework of two-sorted logic. The theory presented here is the first contemporary axiomatization of classical mereology with primitive mereological fusion.