It is well known that in prepositional logic not every propositional connective is definable in terms of equivalence nor indeed in terms of equivalence together with negation. It is the purpose of this note to show that in a certain sense this is, however, possible within predicate logic.

Let F1 be the class of predicate logical formulae containing only the logical connectives ¬ (not), ∨ (or), E (exists), and F2 the class of predicate logical formulae containing only the logical connectives ≡ (if and only if) and E.

Our object will be to find a method for reducing any formulae ϕ of F1 to a formula ψ of F2 such that ϕ is valid if and only if ψ is valid.

Note first that for every domain D containing at least two individuals we can for every n-place relation R defined over D find an n + 1-place relation S defined over D such that for all x1, …, xn ∈ D,

holds. For instance, let S(y, x1, …, xn) be the relation y = x1 & R(x1, …, xn).

Now let ϕ be an arbitrary formula of F1 and A1, …, A3 a complete list of predicate letters occurring in ϕ of degree g1, …, gi, respectively. Let ϕ′ be the formula obtained from ϕ by replacing every prime formula Ai(x1, …, xgi) by

Then it follows immediately from the previous observation that (a) ϕ is satisfiable in D if and only if ϕ′ is satisfiable in D.

Given a prime p, we exhibit a Horn sentence Hp which expresses “the cardinality is not p” and has size O(p5 log p).

We give upper and lower bounds for the consistency strength of the failure of a combinatorial principle introduced by Jensen. Square on singular cardinals.

Strengthening a theorem of Hjorth this paper gives a new characterization of which Polish groups admit compatible complete left invariant metrics. As a corollary it is proved that any Polish group without a complete left invariant metric has a continuous action on a Polish space whose associated orbit equivalence relation is not essentially countable.

We study a strengthening of the notion of a perfectly meager set. We say that a subset A of a perfect Polish space X is countably perfectly meager in X, if for every sequence of perfect subsets $\{P_n: n \in \mathbb N\}$ of X, there exists an $F_\sigma $ -set F in X such that $A \subseteq F$ and $F\cap P_n$ is meager in $P_n$ for each n. We give various characterizations and examples of countably perfectly meager sets. We prove that not every universally meager set is countably perfectly meager correcting an earlier result of Bartoszyński.

We introduce the notion of ${\cal C}$-system of filters, generalizing the standard definitions of both extenders and towers of normal ideals. This provides a framework to develop the theory of extenders and towers in a more general and concise way. In this framework we investigate the topic of definability of generic large cardinals properties.