This paper is concerned with the interaction between formal semantics and the foundations of mathematics. We introduce a formal theory of truth, TLR, which extends the classical first order theory of pure combinators with a primitive truth predicate and a family of truth approximations, indexed by a directed partial ordering. TLR naturally works as a theory of partial classifications, in which type-free comprehension coexists with functional abstraction. TLR provides an inner model for a well known subsystem $\mbox{ATR}_0$ of second order arithmetic; indeed, TLR is proof-theoretically equivalent to Predicative Analysis.

We consider the modal logic of non-contingency in a general setting, without making special assumptions about the accessibility relation. The basic logic in this setting is axiomatized, and some of its extensions are discussed, with special attention to the expressive weakness of the language whose sole modal primitive is non-contingency (or equivalently, contingency), by comparison with the usual language based on necessity (or equivalently, possibility).

Simple finite axiomatizations are given for versions of the modal logics K and K4 with non-contingency (or contingency) as the sole modal primitive. This answers two questions of I. L. Humberstone.

An approach to the treatment of inference in the presence of uncertain truth values is described, based on representing uncertainties by sets of ordinary (certain) truth values. Both the algebraic and the logical aspects are studied for a variety of lattices used as truth value spaces in the domain of many-valued logic.

We give a characterization for those stable theories whose $\omega_{1}$-saturated models have a "Shelah-style" structure theorem. We use this characterization to prove that if a theory is countable, stable, and 1-based without dop or didip, then its $\omega_{1}$-saturated models have a structure theorem. Prior to us, this is proved in a paper of Hart, Pillay, and Starchenko (in which they also count the number of models, which we do not do here). Some other remarks are also included.

Ontologically minimal truth law semantics are provided for various branches of formal logic (classical propositional logic, S5 modal propositional logic, intuitionistic propositional logic, classical elementary predicate logic, free logic, and elementary arithmetic). For all of them logical validity/truth is defined in an ontologically minimal way, that is, not via truth value assignments or interpretations. Semantical soundness and completeness are proved (in an ontologically minimal way) for a calculus of classical elementary predicate logic.

A language in which we can express arithmetic and which contains its own satisfaction predicate (in the style of Kripke's theory of truth) can be formulated using just two nonlogical primitives: $'$ (the successor function) and Sat (a satisfaction predicate).

We establish the decidability, with respect to open formulas in the first order language with equality =, the membership relation $\in$, the constant $\emptyset$ for the empty set, and a binary operation w which, applied to any two sets x and y, yields the results of adding y as an element to x, of the theory NW having the obvious axioms for $\emptyset$ and w. Furthermore we establish the completeness with respect to purely universal sentences of the theory $\mbox{NW}+\mbox{E}+\mbox{R}$, obtained from NW by adding the Extensionality Axiom E and the Regularity Axiom R, and of the theory $\mbox{NW}+\mbox{AFA}'$ obtained by adding to NW (a slight variant of) the Antifoundation Axiom AFA.

We consider a deterministic rewrite system for combinatory logic over combinators $S,K,I,B,C,S',B'$, and $C'$. Terms will be represented by graphs so that reduction of a duplicator will cause the duplicated expression to be "shared" rather than copied. To each normalizing term we assign a weighting which is the number of reduction steps necessary to reduce the expression to normal form. A lambda-expression may be represented by several distinct expressions in combinatory logic, and two combinatory logic expressions are considered equivalent if they represent the same lambda-expression (up to $\beta $-$\eta $-equivalence). The problem of minimizing the number of reduction steps over equivalent combinator expressions (i.e., the problem of finding the "fastest running" combinator representation for a specific lambda-expression) is proved to be NP-complete by reduction from the "Hitting Set" problem.