George Boolos

Self-Reference and Modal Logic

0. Introduction.- 1. The Incompleteness Theorems.- 2. Self-Reference.- 3. Things to Come.- 4. The Theory PRA.- 5. Encoding Syntax in PRA.- 6. Additional Arithmetic Prerequisites.- I. The Logic of Provability.- 1. Provability as Modality.- 1. A System of Basic Modal Logic.- 2. Provability Logic(s).- 3. Self-Reference in PRL.- 4. Avoiding R2.- 2. Modal Model Theory.- 1. Model Theory for BML.- 2. Model Theory for PRL.- 3. Models and Self-Reference.- 4. Another Provability Logic.- 3. Arithmetic Interpretations of PRL.- 1. Solovays First Completeness Theorem.- 2. Solovays Second Completeness Theorem.- 3. Generalisations, Refinements, and Analogues.- II. Multi-Modal Logic and Self-Reference.- 4. Bi-Modal Logics and Their Arithmetic Interpretations.- 1. Bi-Modal Self-Reference.- 2. Kripke Models.- 3. Carlson Models.- 4. Carlsons Arithmetic Completeness Theorem.- 5. Fixed Point Algebras.- 1. Boolean and Diagonalisable Algebras.- 2. Fixed Point Algebras.- 3. Discussion.- III. Non-Extensional Self-Reference.- 6. Rosser Sentences.- 1. Modal Systems for Rosser Sentences.- 2. Arithmetic Interpretations.- 3. Inequivalent Rosser Sentences.- 7. An Ubiquitous Fixed Point Calculation.- 1. An Ubiquitous Fixed Point Calculation.- 2. Applications.- 3. Relativisation to a Partial Truth Definition.- 4. Svejdars Self-Referential Formulae.

Degrees of Unsolvability of Constructible Sets of Integers

Why the Post-Kleene arithmetical hierarchy of degrees of (recursive) unsolvability was extended into the transfinite is not clear. Perhaps it was thought that if a hierarchy of sufficiently fine structure could be described that would include all sets of integers, some light might be thrown on the Continuum Hypothesis, and its truth or falsity possibly even ascertained. There is also some evidence in the 1955 papers of Kleene (cf. Kleene [2], [3], [4]) that it was once hoped that a theorem for the analytical hierarchy analogous to the result of Post and Kleene

A curious inference

Z is an inference in the first-order predicate calculus with identity and function signs. (s is a l-place, f a 2-place function sign.) Z is small: it contains 60 symbols or so, fairly evenly distributed among its five premisses and conclusion. And Z is logically valid; the Frege Russell definition of natural number enables us to see that there is a derivation of (6) from (l)-(5) in any standard axiomatic formulation of second-order logic, e.g. the one given in Chapter 5 of Church’s Introduction to Mathematical Logic.’ A sketch of a second-order derivation of (6) from (l)-(5) is given in the appendix, and it should be evident from the sketch that there is a derivation of (6) from (l)-(5) in any standard axiomatic system of second-order logic whose every symbol can easily be written down. But it is well beyond the bounds of physical possibility that any actual or conceivable creature or device should ever write down all the symbols of a complete derivation in a standard system ofjirstorder logic of (6) from (1) (5): there are far too many symbols in any such derivation for this to be possible. Of course in every standard

An incomplete system of modal logic

Over the last decade a number of logicians have devoted a considerable amount of attention to a system of propositional modal logic known variously as GL, G, L, and provability logic [1]. In this paper our attention will be devoted to a system of modal logic that is a proper subsystem of the system GL. We shall call this system GH. The axioms of GH are the tautologies, the distribution axioms, and the sentences

Extremely Undecidable Sentences

Let ‘ ϕ ’, ‘χ’, and ‘ψ’ be variables ranging over functions from the sentence letters P 0 , P 1 , … P n , … of (propositional) modal logic to sentences of P(eano) Arithmetic), and for each sentence A of modal logic, inductively define A ϕ by [and similarly for other nonmodal propositional connectives]; and where Bew( x ) is the standard provability predicate for PA and ⌈ F ⌉ is the PA numeral for the Godel number of the formula F of PA. Then for any ϕ , (−□⊥) ϕ = −Bew(⌈⊥⌉), which is the consistency assertion for PA; a sentence S is undecidable in PA iff both and , where ϕ ( p 0 ) = S . If ψ ( p 0 ) is the undecidable sentence constructed by Godel, then ⊬ PA (−□⊥→ −□ p 0 & − □ − p 0 ) ψ and ⊢ PA ( P 0 ↔ −□⊥) ψ . However, if ψ ( p 0 ) is the undecidable sentence constructed by Rosser, then the situation is the other way around: ⊬ PA ( P 0 ↔ −□⊥) ψ and ⊢ PA (−□⊥→ −□− p 0 & −□− p 0 ) ψ . We call a sentence S of PA extremely undecidable if for all modal sentences A containing no sentence letter other than p 0 , if for some ψ , ⊬ PA A ψ , then ⊬ PA A ϕ , where ϕ ( p 0 ) = S . (So, roughly speaking, a sentence is extremely undecidable if it can be proved to have only those modal-logically characterizable properties that every sentence can be proved to have.) Thus extremely undecidable sentences are undecidable, but neither the Godel nor the Rosser sentence is extremely undecidable. It will follow at once from the main theorem of this paper that there are infinitely many inequivalent extremely undecidable sentences.

The Degree of the Set of Sentences of Predicate Provability Logic that are True Under Every Interpretation

The formalism of P(redicate) P(rovability) L(ogic) is the result of adjoining the unary operator □ to first-order logic without identity, constants, or function symbols. The term “provability” indicates that □ is to be “read” as “it is provable in P(eano) A(rithmetic) that…” and that the formulae of predicate provability logic are to be interpreted via formulae of PA as follows. Pr( x ), alias Bew( x ), is the standard provability predicate of PA. For any formula F of PA, Pr[ F ] is the formula of PA that expresses the PA-provability of F “of” the values of the variables free in F , i.e., it is the formula of PA with the same free variables as F that expresses the PA-provability of the result of substituting for each variable free in F the numeral for the value of that variable. For the details of the construction of Pr[ F ], the reader may consult [B2, p. 42]. If F is a sentence of PA, then Pr[ F ] = Pr(‘ F ’), the sentence that expresses the PA-provability of F . Let υ 1 , υ 2 ,… be an enumeration of the variables of PA. An interpretation * of a formula ϕ of PPL is a function which assigns to each predicate symbol P of ϕ a formula P * of the language of arithmetic whose free variables are the first n variables of PA, where n is the degree of P .

On «syllogistic inference»

What is one to make of a paper on syllogistic inference whose authors suppose that “All A are B; no B are C; therefore, some C are not A” is a valid form of inference (cf. Johnson-Laird and Bara, 1984, p. 12)? Let US assure ourselves that, because there might be no Cs at all, the inference is ~tot valid: suppose that ‘A’ abbreviates ‘dogs located on earth’; ‘B,’ ‘animals located on earth’; and ‘C,’ ‘unicorns located on earth.’ Then the premisses assert that all dogs located on earth are animals located on earth and that no animals located on earth are unicorns located on earth: these statements are true. Now if it is true that some unicorns located on earth are not dogs located on earth, then it is true that there are some unicorns located on earth (indeed. some that are not dogs located on earth). This is false. Thus the conclusion, that some unicorns located on earth are not dogs located on earth, is not true, and therefore does not follow from the premisses, which are true. It is hard to see how Johnson-Laird and Bara can believe that the premisses “All A are B” and “No B are C” “yield validly” the conclusion “Some C are not A,” unless they suppose that “No B are C” implies the non-emptiness of C, i.e., that there are some things to which C applies. This is not a view one should take seriously: ‘No women are witches’ states something true. Nineteenth century logicians used to fret about an issue called the question of ‘existential import’: whether or not the syllogistic form ‘All A are B’ implies the non-emptiness of A, i.e. the existence of at least one thing to which A applies. Lewis C roll, for one, thought it does. Of course, ‘Some A are B’ and ‘Some A are not B’ both imply the non-emptiness of A. But with the possible exception of Aristotle, whlo, according to one distinguished commentator, J.L. Ackrill, may have intended that all temz in a syllogism were to be assumed non-empty, no one else I knGw of thought or thinks that ‘No A are B’ implies the non-emptiness of either A or B.

Constructing cantorian counterexamples

Cantor’s diagonal argument provides an indirect proof that there is no one-one function from the power set of a set A into A. This paper provides a somewhat more constructive proof of Cantor’s theorem, showing how, given a function f from the power set of A into A, one can explicitly define a counterexample to the thesis that f is one-one.

The analytical completeness of Dzhaparidze's polymodal logics

Abstract The bimodal provability logics of analysis (second-order arithmetic) for ordinary provability and provability by (unrestricted application of) the ω-rule are shown to be fragments of certain ‘polymodal’ logics introduced by G.K. Dzhaparidze. In addition to modal axiom schemes expressing Lobs theorem for the two kinds of provability, the logics treated here contain a scheme expressing that if a statement is consistent, then the statement that it is consistent is provable by the ω-rule.

Omega-consistency and the diamond

G is the result of adjoining the schema □ (□qA→A)→□qA to K; the axioms of G* are the theorems of G and the instances of the schema □qA→A and the sole rule of G* is modus ponens. A sentence is ω-provable if it is provable in P(eano) A(rithmetic) by one application of the ω-rule; equivalently, if its negation is ω-inconsistent in PA. Let ω-Bew(x) be the natural formalization of the notion of ω-provability. For any modal sentence A and function ϕ mapping sentence letters to sentences of PA, inductively define Aωϕ by: pωϕ = ϕ(p) (p a sentence letter); ⊥ωϕ= ⊥; (A→B)suωϕ}= (Aωϕ→Bωϕ); and (□qA)ωϕ= ω-Bew(⌜Aωϕ⌝)(⌜S⌝) is the numeral for the Gödel number of the sentence S). Then, applying techniques of Solovay (Israel Journal of Mathematics 25, pp. 287–304), we prove that for every modal sentence A,⊢GA iff for all ϕ, ⊢PAAωϕ; and for every modal sentence A, ⊢G*A iff for all ϕ, Aωϕ is true.