This paper presents Multi-Entity Bayesian Networks (MEBN), a first-order language for specifying probabilistic knowledge bases as parameterized fragments of Bayesian networks.
As the foregoing example shows, in a second-order language for arithmetic, we can say that the natural numbers are well ordered.
In fact, no condition can be stated in a first-order language for a predicate to express identity, rather than mere indiscernibility by the resources of the language.
The aim of this paper is to introduce a higher-order language for the study of co-authorship data and illustrate the type of novel information it provides.
A first-order language for describing structures of this sort has infinitely many suites of variables, one suite for each type, and a membership symbol ('∈' as usual) which can be placed only between variables of consecutive types: 'xn ∈ yn+1' is wellformed but 'xn ∈ yn' is not.
In fact, worries of this kind have prompted the proposal of a different kind of notions of validity (for Fregean languages), in which set-theoretic structures are replaced with suitable values of higher-order variables in a higher-order language for set theory, e.g. with "plural interpretations" (see Boolos 1985, Rayo and Uzquiano 1999, Williamson 2003; see also the entry on plural quantification).
In familiar first-order languages, for instance, it is pointless to define, say, a one-place predicate \(G\) by a homogeneous definition.
Another, equivalent, coding structure for the first-order language of arithmetic is the structure[5] ⟨H, ∈ ⨡ H⟩ of hereditarily finite sets, where a set x is hereditarily finite if x, its members, its members of members, etc., are all finite.
At around the same time as he was writing the 1933 paper, Tarski gave a truth definition by quantifier elimination for the first-order language of the field of real numbers.
Another notable cut from the original executive order — language stipulating preferential treatment for religious minorities fleeing persecution (i.e., Christian refugees in Muslim-majority countries).
Above we spoke of a first-order language appropriate for a structure \(\langle\bD, \bR_1, \ldots, \bR_n\rangle\), enhanced with clauses, but these clauses were only shown via examples.
