### Unary interpretability logic

Maarten de Rijke
1992 Notre Dame Journal of Formal Logic
Let T be an arithmetical theory. We introduce a unary modal operator T to be interpreted arithmetically as the unary interpretability predicate over T. We present complete axiomatizations of the (unary) interpretability principles underlying two important classes of theories. We also prove some basic modal results about these new axiomatizations. (P) A>B-+Π(A>B) (M) A> B-+ (AΛΠC) > (BΛΠC). MAARTEN de RIJKE We use ILX to denote the system IL + X, where X is the name of some axiom schema. ILMP
more » ... iom schema. ILMP denotes the system IL + M+ P plus the additional axiom A > B -» A A (C > D) > B A (C > D). Let ILS be one of the systems introduced above; the system ILS ω has as axioms all theorems of ILS plus all instances of the schema of reflection: ΠA -*A. Its sole rule of inference is Modus Ponens. Recall that an L-frame is a pair (W,R) with R Q W 2 transitive and conversely well-founded, and that an L-model is given by an L-frame T together with a forcing relation Ih that satisfies the usual clauses for -• and Λ, while u Ih ΠA iff Vv (uRv => v tA). A (Veltman-) frame for IL is a triple , where < W, R) is an Z-frame, and S = {S w : w G W\ is a collection of binary relations on W satisfying 1. S w is a relation on wR ( = {v: wRv}) 2. S w is reflexive and transitive 3. if w\ w" G wR, and w'Rw" then w'S w w". An IL-model is given by a Veltman-frame T for IL together with a forcing relation Ih that satisfies the above clauses for ->, Λ, and D, where u\VA > B&W(uRv and vtA => 3w(iλS w wand wlh£)). An ILP-model is an /L-model that satisfies the extra condition: if wRw'RuS w v then uS w υ. An /LM-model is an /L-model satisfying the extra condition: if uS w vRz then uRz. A model is an /LMP-model if it is both an ILM-and an /LPmodel, and it also satisfies the condition: if xRyS x zRuS y υ then uS z v. In the sequel, T denotes a theory which has a reasonable notion of natural numbers and finite sequences. The theories we consider are either Σ?-sound essentially reflexive theories (like PA), or Σ?-sound finitely axiomatized sequential theories (like GB). An arithmetical interpretation ( ) Γ of <£(D,>) in the language of T is a map which assigns to every proposition letter/? a sentencep τ in the language of T, and which is defined on other modal formulas as follows: 1. U) Γ is'0=l'; 2. ( ) r commutes with -ι and Λ; (ΏA) T is a formalization of 'T h (A) Ti ; 4. (A > B) τ is a formalization of Ύ + (A) τ interprets T + (B) τ \