The degree of the set of sentences of predicate provability logic that are true under every interpretation

1987 ◽  
Vol 52 (1) ◽  
pp. 165-171 ◽  
Author(s):  
George Boolos ◽  
Vann McGee
Keyword(s):  
Predicate Symbol ◽  
Order Logic ◽  
First Order Logic ◽  
Provability Logic ◽  
First Order ◽  
Free Variables ◽  
Unary Operator

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 υ1, υ2,… 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.

scholarly journals Boolean algebras with a unary operator

1986 ◽  
Vol 36 (2) ◽  
pp. 232-237
Author(s):  
Georges Hansoul
Keyword(s):  
Boolean Algebras ◽  
Unary Operator

Some calculi with strong negation primitive

1968 ◽  
Vol 33 (1) ◽  
pp. 97-100
Author(s):  
J. Jay Zeman
Keyword(s):  
Propositional Calculus ◽  
Strong Negation ◽  
Material Implication ◽  
Unary Operator ◽  
Modal Systems

We shall take “strong negation” to be a unary operator with some of the properties usually associated with negation, and such that the strong negation of a statement implies the “ordinary” negation ofthat statement, but not vice-versa. This paper will consider two varieties of strong negation: first, strong negation set in the context of the propositional calculus, specifically, the intuitionist PC, and secondly, strong negation in modal systems, that is, strong negation as impossibility. In the latter part of the paper, we shall present axiomatizations of several classical Lewis-modal systems having only material implication and impossibility primitive.

Undecidability in diagonalizable algebras

1997 ◽  
Vol 62 (1) ◽  
pp. 79-116 ◽  
Author(s):  
V. Yu. Shavrukov
Keyword(s):  
Wide Class ◽  
Formal Theory ◽  
First Order ◽  
Unary Operator ◽  
Diagonalizable Algebra

AbstractIf a formal theory T is able to reason about its own syntax, then the diagonalizable algebra of T is defined as its Lindenbaum sentence algebra endowed with a unary operator □ which sends a sentence φ to the sentence □φ asserting the provability of φ in T. We prove that the first order theories of diagonalizable algebras of a wide class of theories are undecidable and establish some related results.

Generalized Rough Logics with Rough Algebraic Semantics

2010 ◽  
Vol 4 (2) ◽  
pp. 35-49 ◽  
Author(s):  
Jianhua Dai
Keyword(s):  
Rough Set ◽  
Stone Algebra ◽  
Algebraic Semantics ◽  
Sequent Calculi ◽  
Approximation Space ◽  
Lower Approximation ◽  
Upper Approximation ◽  
Double Stone Algebra ◽  
Unary Operator ◽  
Soundness And Completeness

The collection of the rough set pairs <lower approximation, upper approximation> of an approximation (U, R) can be made into a Stone algebra by defining two binary operators and one unary operator on the pairs. By introducing a more unary operator, one can get a regular double Stone algebra to describe the rough set pairs of an approximation space. Sequent calculi corresponding to the rough algebras, including rough Stone algebras, Stone algebras, rough double Stone algebras, and regular double Stone algebras are proposed in this paper. The sequent calculi are called rough Stone logic (RSL), Stone logic (SL), rough double Stone logic (RDSL), and double Stone Logic (DSL). The languages, axioms and rules are presented. The soundness and completeness of the logics are proved.

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