To the axioms of Peano arithmetic formulated in a language with an additional unary predicate symbol T we add the rules of necessitation $\phi/T\,\overline{\phi }$ and conecessitation T $\,\overline{\phi }/\phi $ and axioms stating that T commutes with the logical connectives and quantifiers. By a result of McGee this theory is $\omega$-inconsistent, but it can be approximated by models obtained by a kind of rule-of-revision semantics. Furthermore we prove that FS is equivalent to a system already studied by Friedman and Sheard and give an analysis of its proof theory.

Assuming Martin's Axiom, we compute the value of the cofinality of the symmetric group on the natural numbers. We also show that Martin's Axiom does not decide the value of the covering number of a related Mycielski ideal.

Hájek and Montagna proved that the modal propositional logic ILM is the logic of $\Pi_1$-conservativity over sound theories containing I$\Sigma_1$ (PA with induction restricted to $\Sigma_1$ formulas). I give a simpler proof of the same fact.

The present work, which was inspired by Kripke and McCarthy, is about a non-classical predicate logic system containing a truth predicate symbol. In this system, each sentence A is referred to not by a Gödel number but by its quotation name 'A'.

In this paper two deductive systems (i.e., two consequence relations) associated with relevance logic are studied from an algebraic point of view. One is defined by the familiar, Hilbert-style, formalization of R; the other one is a weak version of it, called WR, which appears as the semantic entailment of the Meyer-Routley-Fine semantics, and which has already been suggested by Wójcicki for other reasons. This weaker consequence is first defined indirectly, using R, but we prove that the first one turns out to be an axiomatic extension of WR. Moreover we provide WR with a natural Gentzen calculus (of a classical kind). It is proved that both deductive systems have the same associated class of algebras but different classes of models on these algebras. The notion of model used here is an abstract logic, that is, a closure operator on an abstract algebra; the abstract logics obtained in the case of WR are also the models, in a natural sense, of the given Gentzen calculus.

Minimality ranks in the style of Deissler are one way of measuring the structural complexity of minimal extensions of first-order structures. In particular, positive Deissler rank measures the complexity of the injective envelope of a module as an extension of that module. In this paper we solve a problem of the second author by showing that certain injective envelopes have the maximum possible positive Deissler rank complexity. The proof shows that this complexity naturally reflects the internal structure of the injective extension in the form of the levels of the Matlis hierarchy.

Within a weak subsystem of second-order arithmetic $WKL_{0}$, that is $\Pi^0_2$-conservative over $PRA$, we reformulate Kreisel's proof of the Second Incompleteness Theorem and Boolos' proof of the First Incompleteness Theorem.

The deductive relationships between six statements are examined in set theory without the axiom of choice. Each of these statements follows from the axiom of choice and involves linear orderings in some way.

We consider the structure $IT_S$ of all labeled trees, called also infinite terms, in the first order language ${\cal L}$ with function symbols in a recursive signature $S$ of cardinality at least two and at least a symbol of arity two, with equality and a binary relation symbol $\sqsubseteq$ which is interpreted to be the subtree relation. The existential theory over ${\cal L}$ of this structure is decidable (see Tulipani [9]), but more complex fragments of the theory are undecidable. We prove that the $ \exists\Delta$ theory of the structure is in $\Sigma^1_1$, where $ \exists\Delta$ formulas are those in prenex form consisting of a string of unbounded existential quantifiers followed by a string of arbitrary quantifiers all bounded with respect to $\sqsubseteq$. Since the fragment of the theory was already known to be $\Sigma^1_1$-hard (see Marongiu and Tulipani [5]), it is now established to be $\Sigma^1_1$-complete.

While accepting that distribution is a coherent notion, I argue that it is nevertheless irrelevant to the working of the syllogism. Instead, I propose: (i) that a term's being distributed or undistributed in a proposition is its capacity to be replaced in a truth-preserving substitution with a narrower or a wider term; (ii) that which capacity the term has is determined by whether it occurs as the predicate of a negative or of an affirmative statement of the proposition; and (iii) that it is only the term's occurrence as the predicate of a negative or an affirmative statement–rather than its distribution value–that is relevant to syllogistic entailment.

This paper gives a new, purely semantic proof of the following theorem: if an intermediate propositional logic L has the disjunction property then a disjunction free formula is provable in L iff it is provable in intuitionistic logic. The main idea of the proof is to use the well-known semantic criterion of the disjunction property for "simulating" finite binary trees (which characterize the disjunction free fragment of intuitionistic logic) by general frames.

We show that the following conjecture about the universe V of all sets is wrong: for all set-theoretical (i.e., first order) schemata $\Sigma$ true in V there is a transitive set $\Sigma$ "reflecting" $\Sigma$ in such a way that the second order statement $\sigma$ corresponding to $\Sigma$ is true in $u$. More generally, we indicate the ontological commitments of any theory that exploits reflection principles in order to yield large cardinals. The disappointing conclusion will be that our only apparently good arguments for the existence of large cardinals have bad presuppositions.