Provable Formulas Research Articles

Time-based security explained: Provable security models and formulas for the practitioner and vendor

Time-based security explained: Provable security models and formulas for the practitioner and vendor

Fuzzy Logic and Arithmetical Hierarchy III

Fuzzy logic is understood as a logic with a comparative and truth-functional notion of truth. Arithmetical complexity of sets of tautologies (identically true sentences) and satisfiable sentences (sentences true in at least one interpretation) as well of sets of provable formulas of the most important systems of fuzzy predicate logic is determined or at least estimated.

The global properties of valid formulas in modal logic K

Global property is the necessary condition which must be satisfied by the provable formulas. It can help to find out some unprovable formula that does not satisfy some global property before proving it using formal automated reasoning systems, thus the efficiency of the whole system is improved. This paper presents some global properties of valid formulas in modal logic K. Such properties are structure characters of formulas, so they are simple and easy to check. At the same time, some global properties of K unsatisfiable formula set are also given.

Intersection Types as Logical Formulae

The aim of this paper is to investigate, in the Curry-Howard isomorphism approach, a logical characterization for the intersection-type discipline First a novel formulation of the intersection type inference for combinatory logic is presented, such that it is equivalent to the original version of the system, while the intersection operator is no longer dealt with as a proof-theoretical connective Then a Hilbert-style axiomatization is defined and proved to totally parallel lntersecOon-denvability, in such a way that inhabited intersection-types are all and only the provable formulae in the logic system

Uniqueness of normal proofs of minimal formulas

AbstractA minimal formula is a formula which is minimal in provable formulas with respect to the substitution relation. This paper shows the following: (1) A β-normal proof of a minimal formula of depth 2 is unique in NJ. (2) There exists a minimal formula of depth 3 whose βη-normal proof is not unique in NJ. (3) There exists a minimal formula of depth 3 whose βη-normal proof is not unique in NK.

The number of proofs for a BCK-formula

In this note, we give a necessary and sufficient condition for a BCK-formula to have the unique normal form proof.We call implicational propositional formulas formulas for short. BCK-formulas are the formulas which are derivable from axioms B = (a → b) → (c → a) → c → b, C = (a→b→c)→b→a→c, and K = a→b→a by substitution and modus ponens. It is known that the property of being a BCK-formula is decidable (Jaskowski [11, Theorem 6.5], Ben-Yelles [3, Chapter 3, Theorem 3.22], Komori [12, Corollary 6]). The set of BCK-formulas is identical to the set of provable formulas in the natural deduction system with the following two inference rules.Here γ occurs at most once in (→I). By the formulae-as-types correspondence [10], this set is identical to the set of type-schemes of closed BCK-λ-terms. (See [5].) A BCK-λ-term is a λ-term in which no variable occurs twice. Basic notion concerning the type assignment system can be found [4]. Uniqueness of normal form proofs has been known for balanced formulas. (See [2,14].) It is related to the coherence theorem in cartesian closed categories. A formula is balanced when no variable occurs more than twice in it. It was shown in [8] that the proofs of balanced formulas are BCK-proofs. Relevantly balanced formulas were defined in [9], and it was proved that such formulas have unique normal form proofs. Balanced formulas are included in the set of relevantly balanced formulas.

Two recursion theoretic characterizations of proof speed-ups

Smullyan in [Smu61] identified the recursion theoretic essence of incompleteness results such as Gödel's first incompleteness theorem and Rosser's theorem. Smullyan (improving upon [Kle50] and [Kle52]) showed that, for sufficiently complex theories, the collection of provable formulae and the collection of refutable formulae are effectively inseparable—where formulae and their Gödel numbers are identified. This paper gives a similar treatment for proof speed-up. We say that a formal system S1is speedable over another system S0on a set of formulaeAiff, for each recursive functionh, there is a formulaαinAsuch that the length of the shortest proof ofαin S0is larger thanhof the shortest proof ofαin S1. (Here we equate the length of a proof with something like the number of characters making it up,notits number of lines.) We characterize speedability in terms of the inseparability by r.e. sets of the collection of formulae which are provable in S1but unprovable in S0from the collectionA–where again formulae and their Gödel numbers are identified. We provide precise definitions of proof length, speedability and r.e. inseparability below.We follow the terminology and notation of [Rog87] with borrowings from [Soa87]. Below,ϕis an acceptable numbering of the partial recursive functions [Rog87] andΦa (Blum) complexity measure associated withϕ[Blu67], [DW83].

Super-Łukasiewicz implicational logics

In the traditional study of Łukasiewicz propositional logic, the finite-valued or infinite-valued linearly ordered model exists at the start, and then the axiomatization of the set of all formulas valid in its model are studied. On the other hand, we are in a point of view such that the set of provable formulas is important and models are no more than means to characterize the set.

On simple, weak and strong models of propositional calculi

In this paper we will be concerned primarily with weak, strong and simple models of a propositional calculus, simple models being structures of a certain type in which all provable formulae of the calculus are valid. It is shown that the finite model property defined in terms of simple models holds for all calculi. This leads to a new proof of the fact that there is no general effective method for testing, given a finite structure and a calculus, whether or not the structure is a simple model of the calculus.

Note on a problem of Porte

Porte (1), p. 117, conjectures that the positive implicational propositional calculus has no finite characteristic matrix. The proof of this conjecture is a straightforward modification of Gödel's proof (2) that the intuitionistic propositional calculus has no finite characteristic matrix (see e.g. Church(3), ex. 26.12). Writing (A ∨ B) for ((A ⊃ B) ⊃ B) and Xij for (pj ⊃ pi) (i, j = 1,2,…), we define, for n > l, the formulawhere the terms associate to the left. Since provable formulae take the value n for all systems of values of the variables in the matrix {1,…,n} where x ⊃ y is n when x ≤ y and y otherwise, whereas Gn takes the value n − 1 for the system of values pi = i (i = 1,…,n), it follows that Gn is not provable. On the other hand, since A ⊢ A ∨ B and B ⊢ A ∨ B, it is easily seen that is provable whenever r ≠ s (r, s = 1,…,n). The result follows from these two remarks.

Concerning formulas of the types A→B ν C,A →(Ex)B(x) in intuitionistic formal systems

In a previous paper [1] it was proved, among other results, that a closed disjunction of intuitionistic elementary number theory N can be proved if and only if at least one of its disjunctands is provable and that a closed formula of the type (Ex)B(x) is provable in N if and only if B(n) is provable for some numeral n. The method of proof was to show that, as far as closed formulas are concerned, N is equivalent to a calculus N1 for which the result is immediate. The main step in the proof consisted in showing that the set of provable formulas of N1 is closed under modus ponens. This was done by obtaining a subset of the set which is closed under modus ponens and contains all members of the original set, with which it is therefore identical.

Extensions of the Lewis system S5

Dugundji has proved that none of the Lewis systems of modal logic, S1 through S5, has a finite characteristic matrix. The question arises whether there exist proper extensions of S5 which have no finite characteristic matrix. By an extension of a sentential calculus S, we usually refer to any system S′ such that every formula provable in S is provable in S′. An extension S′ of S is called proper if it is not identical with S. The answer to the question is trivially affirmative in case we make no additional restrictions on the class of extensions. Thus the extension of S5 obtained by adding to the provable formulas the additional formula p has no finite characteristic matrix (indeed, it has no characteristic matrix at all), but this extension is not closed under substitution—the formula q is not provable in it. McKinsey and Tarski have defined normal extensions of S4* by imposing three conditions. Normal extensions must be closed under substitution, must preserve the rule of detachment under material implication, and must also preserve the rule that if α is provable then ~◊~α is provable. McKinsey and Tarski also gave an example of an extension of S4 which satisfies the first two of these conditions but not the third. One of the results of this paper is that every extension of S5 which satisfies the first two of these conditions also satisfies the third, and hence the above definition of normal extension is redundant for S5. We shall therefore limit the extensions discussed in this paper to those which are closed under substitution and which preserve the rule of detachment under material implication. These extensions we shall call quasi-normal. The class of quasi-normal extensions of S5 is a very broad class and actually includes all extensions which are likely to prove interesting. It is easily shown that quasi-normal extensions of S5 preserve the rules of replacement, adjunction, and detachment under strict implication. It is the purpose of this paper to prove that every proper quasi-normal extension of S5 has a finite characteristic matrix and that every quasi-normal extension of S5 is a normal extension of S5 and to describe a simple class of characteristic matrices for S5.

Axiom schemes for m-valued functional calculi of first order. Part I. Definition of axiom schemes and proof of plausibility

A calculus, or its formalization, is m-valued when its truth-functions are allowed to take truth-values ranging from 1 through m. It is customary to divide the m truth-values that are possible into those that are “designated” and those that are “undesignated.” Furthermore, it is usually desired of a formalization that provable formulas and only provable formulas take designated truth-values exclusively. For our present purposes, we shall suppose that the truth-values 1, …, 8 are designated and the truth-values 8+1, …, m are undesignated. When calculi differ in respect to the number of their designated truth-values, we shall consider them different calculi, even if they are otherwise similar.

Axiom schemes for m-valued propositions calculi

In an m-valued propositional calculus, or a formalization of such a calculus, truth-value functions are allowed to take any truth-value t where 1 ≦ t ≦ m and m ≧ 2. In working with such calculi, or formalizations thereof, it has been decided to distinguish those truth-values which it is desirable for provable formulas to have from those which it is not desirable for provable formulas to have. The first class of truth-values is called designated and the second undesignated. This specification of certain of the m truth-values as designated and the remainder as undesignated is one of the distinguishing characteristics of m-valued propositional calculi, and it should be observed at the outset that two m-valued propositional calculi will be considered to differ even if they differ only in respect to the number of truth-values which are taken as designated.

Note on a property of matrices for Lewis and Langford's calculi of propositions

The object of this note is to show that there is no finite characteristic matrix for any one of Lewis and Langford's systems.Theorem I. There is no finite characteristic matrix for any one of the systems S1–S5.Proof. Let M be a matrix with less than n elements for which all the provable formulas in S1, or S2, or S3, or S4, or S5, are satisfied. Let Fn represent the formulawhere ∑ stands for a ∨-chain, and the pi, are variables in any one of the calculi. Using M, there is always at least one summand in Fn where pi and pk have the same value. Therefore, Fn can always be written in the form (a = a)∨ B, and thus will give, for any B, a “designated” value, since the formula (p = p) ∨ q is provable in any one of the systems S1–S5.Give to any one of the systems S1–S5, the following matrix, due to Henle:1. Elements: all possible classes formed from the integers 1, 2, 3, …, n.2. “Designated” element: the class {1, 2, 3, …,n}.3. Boole-Schröder algebra on the elements.4. ◊N = N (N the null class)◊A = {1, 2, …, n} (A any non-null class).5

Extensions of some theorems of Gödel and Church

We shall say that a logic is “simply consistent” if there is no formula A such that both A and ∼ A are provable. “ω-consistent” will be used in the sense of Gödel. “General recursive” and “primitive recursive” will be used in the sense of Kleene, so that what Gödel calls “rekursiv” will be called “primitive recursive.” By an “Entscheidungsverfahren” will be meant a general recursive function ϕ(n) such that, if n is the Gödel number of a provable formula, ϕ(n) = 0 and, if n is not the Gödel number of a provable formula, ϕ(n) = 1. In specifying that ϕ must be general recursive we are following Church in identifying “general recursiveness” and “effective calculability.”First, a modification is made in Gödel's proofs of his theorems, Satz VI (Gödel, p. 187—this is the theorem which states that ω-consistency implies the existence of undecidable propositions) and Satz XI (Gödel, p. 196—this is the theorem which states that simple consistency implies that the formula which states simple consistency is not provable). The modifications of the proofs make these theorems hold for a much more general class of logics. Then, by sacrificing some generality, it is proved that simple consistency implies the existence of undecidable propositions (a strengthening of Gödel's Satz VI and Kleene's Theorem XIII) and that simple consistency implies the non-existence of an Entscheidungsverfahren (a strengthening of the result in the last paragraph of Church).