§1. Introduction. In 1964 Shepherdson [6] introduced a weak system of arithmetic, Open Induction (OI), in which the Tennenbaum phenomenon does not hold. More precisely, if we restrict induction just to open formulas (with parameters), then we have a recursive nonstandard model. Since then several authors have studied Open Induction and its related fragments of arithmetic. For instance, since Open Induction is too weak to prove many true statements of number theory (It cannot even prove the irrationality of √2), a number of algebraic first order properties have been suggested to be added to OI in order to obtain closer systems to number theory. These properties include: Normality [9] (abbreviated by N), having the GCD property [8], being a Bezout domain [3, 8] (abbreviated by Bez), and so on. We mention that GCD is stronger than N, Bez is stronger than GCD and Bez is weaker than IE1 (IE1 is the fragment of arithmetic based on the induction scheme for bounded existential formulas and by a result of Wilmers [11], does not have a recursive nonstandard model). Boughattas in [1, 2] studied the non-finite axiomatizability problem and established several new results, including: (1) OI is not finitely axiomatizable, (2) OI + N is not finitely axiomatizable.

Substitution for predicate letters is a syntactically intricate inference procedure that is derivable in standard systems of non-modal quantificational logic. It allows a new theorem to be deduced from a given theorem ϕ by replacing an atomic formula Pτ1 ··· τn within ϕ by another formula ψ, of arbitrary complexity, involving the terms τ1, …, τn. The intricacy comes in stating the restrictions that must be placed on free variables of ϕ and ψ for the substitution to be allowed.

Church [1956, p. 289] gives some historical notes on this rule, pointing out that it was inadequately stated in early works of Hilbert and Ackerman, Carnap, and Quine; and first correctly stated, but not in full generality, in Hilbert and Bernays's Grundlagen der Mathematik in 1934. Church himself calls the predicate letter P a functional variable, viewing it as a variable whose values are propositional functions of individuals. In non-modal logic, this means that P is interpreted as an n-ary function Un → {truth, falsehood} from individuals to truth-values.

Intuitively, a logic that is closed under substitution for predicate letters is one whose theorems represent “universal laws”, expressing properties that hold of all predicates, i.e. hold no matter what interpretation is given to the predicate letters, hence hold no matter what formulas are (correctly) substituted for them.

This book is about the possible-worlds semantic analysis of systems of logic that have quantifiers binding individual variables. Our approach is based on a notion of “admissible” model that places a restriction on which sets of worlds can serve as propositions. We show that admissible models provide semantic characterisations of a wide range of logical systems, including many for which the well-known model theory of Kripke [1963b] is incomplete. The key to this is an interpretation of quantification that takes into account the admissibility of propositions.

This is a subject that bristles with choices and challenges. Should terms be treated as rigid designators, or should their denotations vary from world to world? Should individual constants and variables be treated the same in this respect, or differently? Should each world have its own domain of existing individuals over which the quantifiable variables range, or should there be just a single domain of individuals? If there are varying domains, how should they relate to each other? Can any function from worlds to individuals be regarded as the “meaning” of some individual concept? Should an arbitrary mapping from individuals to propositions be admissible as a propositional function? Can we deductively axiomatise the class of valid formulas determined by each answer to these questions?

We now extend our language for quantified modal logic by adding a predicate symbol ≈ for an identity relation, allowing us to express assertions about the identity of individuals. A two-sorted language is developed, with one sort of term standing for individual concepts, represented as partial functions from worlds to individuals, and the other sort specialising to concepts that are rigid, i.e. have the same value in accessible worlds. The identity predicate allows the existence predicate ε to be defined, taking ετ to be the self-identity formula τ ≈ τ, whose corresponding proposition/truth set is the domain of the partial function interpreting τ. The use of admissibility is extended from propositions to individuals by requiring each model to have a designated set of admissible individual concepts, within which there is a designated set of admissible rigid ones.

We axiomatise the set of formulas that are valid in these models, using a new inference rule that allows deduction of assertions of non-existence (¬ετ). The logic characterised by Kripkean models is then treated separately. The final section of the chapter gives a semantic analysis of Russelian definite description terms ιx.ϕ in this context, and shows how to construct canonical models and axiomatisations for logics in languages that have these description terms as well as the identity predicate.

The main aim of this chapter is to set out a new kind of admissible model theory for the propositional relevant logic R and its quantified extension RQ. First we review the relational semantics for R of Routley and Meyer [1973], and its adaptation by Mares and Goldblatt [2006] to an admissible semantics for RQ. Then we introduce an alternative kind of structure, called a cover system, motivated by topological ideas about “local truth” from the Kripke-Joyal semantics for intuitionistic logic in topos theory. These are combined with a modelling of negation by a binary world-relation of orthogonality, or incompatibility, as in [Goldblatt 1974], and an operation of combination, or “fusion”, of worlds to interpret relevant implication. Characteristic model systems for R have an algebra Prop of admissible propositions, while those for RQ have a set PropFun of admissible propositional functions as well.

We then show that by conservatively adding an intuitionistic implication connective to R it is possible to characterise that logic by models in which all possible propositions are admissible. The prospects for a similar analysis of RQ are considered at the end.

Routley-Meyer Models for R

The subject of relevant logic (also known as relevance logic) is based on the view that an implication A → B can only be true if the meaning of A is relevant to the meaning of B.

These lecture notes originated in a seminar on Model Theory that I gave in the academic years 2005–06 and 2006–07 at the Department of Logic, History and Philosophy of Science of the University of Barcelona. I had presented some previous work on the basic notions of simple theories in July 2002 in the Simpleton Workshop held at the Centre International de Rencontres Mathématiques, Luminy (Marseille), which was subsequently published as [8]. A more extended version, including the exposition of stable theories, was the topic of a tutorial entitled Advanced Stability Theory that I taught in the Modnet Summer School that took place at the University of Freiburg in April 2006. And in preparing the material, I also drew on some courses on these topics given at the Universidad de los Andes, Bogotá, in August 2000 and in August 2004.

The notes are based on the work of many model theorists. The names of John T. Baldwin, Ehud Hrushovski, Byunghan Kim, Daniel Lascar, Ludomir Newelski, Anand Pillay, Bruno Poizat, Saharon Shelah, Frank O. Wagner, and Martin Ziegler deserve special mention. I learned stability theory from Martin Ziegler and I have made as much use as I could of his short and elegant proofs, presented in his courses and in his unpublished lecture notes.

This book is not as ambitious as Frank O. Wagner's book on simple theories [41], but its pace might be more comfortable for the beginner.

Definition 7.1. Let M ⊆ A and p(x) ∈ S(A). We say that p is an heir of p ↾ M or that p inherits from M if for every φ(x, y) ∈ L(M) if φ(x, a) ∈ p for some tuple a ∈ A, then φ(x, m) ∈ p for some tuple m ∈ M. We say that p is a coheir of p ↾ M or that p coinherits from M if p is finitely satisfiable in M. The same definitions apply to global types, i.e., to the case A = ℭ. These definitions also make sense for types in infinitely many variables.

Remark 7.2. tp(a/Mb) inherits from M if and only if tp(b/Ma) coinherits from M.

Proof. It is just a matter of writing down the definitions.

Lemma 7.3.

1. 1. If p(x) ∈ S(M), then p inherits and coinherits from M.

2. 2. If M ⊆ A and p(x) ∈ S(A) coinherits from M, then for every B ⊇ A there is some q(x) ∈ S(B) such that p ⊆ q and q coinherits from M.

3. 3. If M ⊆ A and p(x) ∈ S(A) inherits from M, then for every B ⊇ A there is some q(x) ∈ S(B) such that p ⊆ q and q inherits from M.