A Henkin-Style Completeness Proof for the Modal Logic S5

The propositional modal logic is obtained by adding the necessity operator □ to the propositional logic. Each formula in the propositional logic is equivalent to a formula in the disjunctive normal form. In order to obtain the correspondence between the propositional modal logic and the propositional logic, we add the axiom \(\Box\varphi \leftrightarrow\Diamond\varphi \) to K and get a new system K + . Each formula in such a logic is equivalent to a formula in the disjunctive normal form, where □ k (k ≥ 0) only occurs before an atomic formula p, and \(\lnot\) only occurs before a pseudo-atomic formula of form □ k p. Maximally consistent sets of K + have a property holding in the propositional logic: a set of pseudo-atom-complete formulas uniquely determines a maximally consistent set. When a pseudo-atomic formula □ k p i (k,i ≥ 0) is corresponding to a propositional variable q ki , each formula in K + then can be corresponding to a formula in the propositional logic P + . We can also get the correspondence of models between K + and P + . Then we get correspondences of theorems and valid formulas between them. So, the soundness theorem and the completeness theorem of K + follow directly from those of P + .

Hilbert aimed to justify acceptance and use of arithmetic,analysis, and set theory. A natural way of approaching this task is to look for indubitable truths to serve as axioms for a comprehensive theory, along with inference rules that are indubitably truth-preserving. This may have been Frege’s approach, but following discovery of the contradiction, Russell concluded that some element of doubt about the truth of the axioms is unavoidable. The best one could hope for, on Russell’s view, was that the axioms were inductively well supported by their known consequences. Accordingly, Russell distinguished between epistemological foundations and deductive foundations. In making this move, Russell had performed a half-revolution. The epistemological basis was now regarded as comprising the simplest and most immediately evident truths of mathematics, together with quasi-empirical induction. The goal, however, was still to establish the truth of the axioms, beyond reasonable doubt if not beyond all doubt. For this goal the permitted means were inadequate. Russell never found a sufficiently powerful set of axioms that he felt met the required standard. Had he abandoned the goal of a comprehensive theory and decided to treat the subjects of real analysis and transfinite sets separately, he would have fared no better: recall the troubles over Dedekind’s Continuity Axiom, the Axiom of Infinity, and the Axiom of Choice.

In this paper we give an abstract representation of propositional Horn logic (finite or infinitary) in terms of set-valued functions F : (sic) (PR) -> (sic) (PR); here PR is the collection of atomic propositions. We investigate the properties of set-valued functions and establish that they satisfy the axioms of weak lazy Kleene algebra. These axioms and other properties are used to prove uniform left interpolation for propositional Horn logic. Then we introduce thin set-valued functions, which enable us to prove polynomial interpolation for propositional Horn logic. For the infinite case, the Axiom of Choice is used.

In this paper, we investigate what kind of modal logics can be constructed into the Gentzen-type formulation which enjoys the Cut elimination theorem. We treat normal, regular and monotonic modal logics. A normal modal logic is the modal logic with the Rule of Necessitation and the axiom K:□(A⊃B)⊃(□A⊃□B), a regular modal logic with the rule of inference RR :A,Г→B□A,□Г→□B and a monotonic modal logic with the rule of inference RM :A→B□A→□B. We provide Gentzen-type formulation and prove the cut elimination theorem for the following modal logics: RT4 (a regular modal logic with the axioms T:□A⊃A and 4:□A⊃□□A), MT4 (a monotonic modal logic with the axioms T and 4), MNT4 (a monotonic modal logic with the Rule of Necessitation and the axioms T and 4), RT (a regular modal logic with the axiom T), MT (a monotonic modal logic with the axiom T), MNT (a monotonic modal logic with the Rule of Necessitation and the axiom T), KL (a normal modal logic with the axiom L:□A⊃(A∨□B)) and RTA (a regular modal logic with axioms T and A:□A&B⊃□B).

: Modern modal logic begins with the work of C.I. Lewis early on in the present century. We can think of Lewis thinking to himself as follows: Well, I can't analyze the notions of metaphysical or logical possibility and necessity; but I can sure formulate alternative axiomatizations of such notions. I can then compare and contrast such axiomatic systems and see what I learn. Thus were born the Lewis Systems, S1-S5, axiomatizing increasingly strong conceptions of necessity. Another 40 or so years went by before the purely axiomatic approach was properly systematized and rendered fit for human consumption. In current lore, a certain axiomatic system, K, is central. The standard presentation of K consists of infinitely many axioms plus one axiom scheme and two rules of inference... It may be, then, that to take epistemic/doxastic logics seriously, one must both be working from within that conceptualization of cognitive states according to which they are either essentially or importantly language involving and, further, conceive of the language(s) in question on the model of standard formal languages, as consisting, that is, of eternal sentences only. This could be taken as an argument to the effect that the proper home of epistemic/doxastic logic is theoretical computer science -- precisely the locus of its greatest current vitality.

Formal verification involves the use of logical and computational methods to establish claims that are expressed in precise mathematical terms. These can include ordinary mathematical theorems, as well as claims that pieces of hardware or software, network protocols, and mechanical and hybrid systems meet their specifications. In practice, there is not a sharp distinction between verifying a piece of mathematics and verifying the correctness of a system: formal verification requires describing hardware and software systems in mathematical terms, at which point establishing claims as to their correctness becomes a form of theorem proving. Conversely, the proof of a mathematical theorem may require a lengthy computation, in which case verifying the truth of the theorem requires verifying that the computation does what it is supposed to do. The gold standard for supporting a mathematical claim is to provide a proof, and twentieth-century developments in logic show most if not all conventional proof methods can be reduced to a small set of axioms and rules in any of a number of foundational systems. With this reduction, there are two ways that a computer can help establish a claim: it can help find a proof in the first place, and it can help verify that a purported proof is correct. Automated theorem proving focuses on the “finding” aspect. Resolution theorem provers, tableau theorem provers, fast satisfiability solvers, and so on provide means of establishing the validity of formulas in propositional and first-order logic. Other systems provide search procedures and decision procedures for specific languages and domains, such as linear or nonlinear expressions over the integers or the real numbers. Architectures like SMT (“satisfiability modulo theories”) combine domain-general search methods with domain-specific procedures. Computer algebra systems and specialized mathematical software packages provide means of carrying out mathematical computations, establishing mathematical bounds, or finding mathematical objects. A calculation can be viewed as a proof as well, and these systems, too, help establish mathematical claims. Automated reasoning systems strive for power and efficiency, often at the expense of guaranteed soundness. Such systems can have bugs, and it can be difficult to ensure that the results they deliver are correct. In contrast, interactive theorem proving focuses on the “verification” aspect of theorem proving, requiring that every claim is supported by a proof in a suitable axiomatic foundation. This sets a very high standard: every rule of inference and every step of a calculation has to be justified by appealing to prior definitions and theorems, all the way down to basic axioms and rules. In fact, most such systems provide fully elaborated “proof objects” that can be communicated to other systems and checked independently. Constructing such proofs typically requires much more input and interaction from users, but it allows us to obtain deeper and more complex proofs. The Lean Theorem Prover aims to bridge the gap between interactive and automated theorem proving, by situating automated tools and methods in a framework that supports user interaction and the construction of fully specified axiomatic proofs. The goal is to support both mathematical reasoning and reasoning about complex systems, and to verify claims in both domains.

Some Remarks on Discussive Logic

We investigate logical consequence in temporal logics in terms of logical consecutions, i.e., inference rules. First, we discuss the question: what does it mean for a logical consecution to be ‘correct’ in a propositional logic. We consider both valid and admissible consecutions in linear temporal logics and discuss the distinction between these two notions. The linear temporal logicLDTL, consisting of all formulas valid in the frame 〈L≤, ≥〉 of all integer numbers, is the prime object of our investigation. We describe consecutions admissible inLDTLin a semantic way—via consecutions valid in special temporal Kripke/Hintikka models. Then we state that any temporal inference rule has a reduced normal form which is given in terms of uniform formulas of temporal degree 1. Using these facts and enhanced semantic techniques we construct an algorithm, which recognizes consecutions admissible inLDTL. Also, we note that using the same technique it follows that the linear temporal logicL(N) of all natural numbers is also decidable w.r.t. inference rules. So, we prove that both logicsLDTLandL(N) are decidable w.r.t. admissible consecutions. In particular, as a consequence, they both are decidable (known fact), and the given deciding algorithms are explicit.

In classical quantificational logic, every individual constant is assigned a value from the domain of discourse, thus ensuring that every instance of \(\exists x(x=\alpha )\) is valid and so a theorem of a complete logic. Standard tense and modal logics validate a rule of necessitation, according to which, crudely, every theorem is always and necessarily true. Combining these two generates the result that everything always and necessarily exists. In a number of works from the late 1950s through to his death in 1969, Prior worked to develop tense and modal logics that avoided these results. Prior’s key idea was to reject the rule of necessitation. In this essay, I present and criticize Prior’s account of the contingency of existence and offer a more satisfying account that shares many of the virtues of Prior’s account.

A complete proof system for propositional projection temporal logic

We present the propositional logic LEC for the two epistemic modalities of current and stable knowledge used by an agent who system-atically enriches his language. A change in the linguistic resources of an agent as a result of certain cognitive processes is something that commonly happens. Our system is based on the logic LC intended to formalize the idea that the occurrence of changes induces the passage of time. Here, the primitive operator C read as: it changes that, defines the temporal succession of states of the world. The notion of current knowledge concerns variable components of the world and it may change over time. We represent it by the primitive operator k read as: the agent currently knows that, and assume that it has S5 properties. The second type of knowledge, symbolized by the primitive operator K read as: the agent stably knows that, relates to constant components of the world and it does not change. As a result of the axiomatic entanglement of C, K and k we show that stable knowledge satisfies axioms of S4.3. K and k modalities are not mutually definable, stable knowledge implies the current one and if the latter never changes, then it comes to be stable. The combination of K and k with the idea of an expanding language allows questioning of the so-called perfect recall principle. It cannot be maintained for both types of knowledge just because of changes in the vocabulary of the agent and possibly the growing spectrum of possible states of the world. We interpret LEC in the semantics of histories of epistemic changes and show that it is complete. Finally, we compare our logic with selected epistemic logics based on the concept of linear discrete time.

Focused proofs are sequent calculus proofs that group inference rules into alternating positive and negative phases. These phases can then be used to define macro-level inference rules Gentzen’s original and tiny introduction and structural rules. We show here that the inference rules of labeled proof systems for modal logics can similarly be described as pairs of such phases within the LKF focused proof system for first-order classical logic. We consider the system $${ G3K} $$ of Negri for the modal logic $$K$$ and define a translation from labeled modal formulas into first-order polarized formulas and show a strict correspondence between derivations in the two systems, i.e., each rule application in $${ G3K} $$ corresponds to a bipole—a pair of a positive and a negative phases—in $${ LKF} $$ . Since geometric axioms (when properly polarized) induce bipoles, this strong correspondence holds for all modal logics whose Kripke frames are characterized by geometric properties. We extend these results to present a focused labeled proof system for this same class of modal logics and show its soundness and completeness. The resulting proof system allows one to define a rich set of normal forms of modal logic proofs.

We consider the relationship between the algorithmic properties of the validity problem for a modal or superintuitionistic propositional logic and the size of the smallest Kripke countermodels for non-theorems of the logic. We establish the existence, for every degree of unsolvability, of a propositional logic whose validity problem belongs to the degree and whose every non-theorem is refuted on a Kripke frame that validates the logic and has the size linear in the length of the non-theorem. Such logics are obtained among the normal extensions of the propositional modal logics $\textbf {KTB}$, $\textbf {GL}$ and $\textbf {Grz}$ as well as in the lattice of superintuitionistic propositional logics. This shows that the computational complexity of a modal or superintuitionistic propositional logic is, in general, not related to the size of the countermodels for its non-theorems.

As a dominant method in evidential reasoning, Bayesian network has been proved powerful in discrete fields. Although Bayesian network performs reliable in continuous variables and interval estimations, it relies on discretizing continuous variables or building an approximate model to conduct, which causes information loss and accuracy reduction. In order to bridge this gap, this paper introduces two inference rules combined with four inference rules proposed by other scholars. Then we propose a concept of uncertain inference network that consists of six basic structures matching inference rules to represent relationships and logic connection among the evidence. Evidence is represented by uncertain sets that can apply to continuous variables using membership functions to represent vague concepts. Furthermore, a numeric experiment for a forensic investigation of fire incidents is given to compare the results of uncertain inference network and Bayesian network. We found three merits in the case study. First, an uncertain inference network has simpler data access for each node because Bayesian network depends on conditional probability tables while uncertain inference network only relies on membership function. Second, an uncertain inference network has a more wide application because it can perform continuous variables with certain mathematical formulas without discretizing or approximating. Third, an uncertain inference network has a more accurate result because Bayesian network gives a point estimation with a 0–1 value while uncertain inference network conducts an interval estimation with a range value.

JUST AS music history is a species of history, so no doubt is music theory a species of theory. Hence music theory should have some structure in common with other theories. The question is, then, what is this structure? And of what species of theory is music theory? To specify a theory in logically acceptable terms is usually no easy matter. Every theory involves an underlying logic, and many alternative logics clamor here for attention. Even after underlying logic has been agreed upon to be of such and such a kind, we then must worry about (i) special domain of objects allowed as values for variables and (ii) basic classes (or domains or sorts) into which these objects may be grouped as well as basic relations whereby they may be related. Item (i) has to do with choice of an ontology for theory and (ii) with choice of suitable non-logical predicate or relational constants as primitive or undefined. Only after (i) and (ii) have been well settled is it of any interest to start search for axioms. In process of getting a theory, gaining of axioms comes at very end.1 Let us begin then at beginning with (i) and (ii), ontology and primitive predicates. This much might be called the proto-theory of music. We should first note that proto-theory need not involve complexities exhibited by many scientific or philosophical theories. For example, music theory need not, it seems, involve modal logic, or threevalued logic. In modal logic, it will be recalled, notions of necessity and possibility play fundamental roles, and in three-valued logic a value intermediary between truth-values truth and falsehood is admitted. It is very doubtful that such notions need play any role in music theory. Nor for that matter need music theory assume such complicated mathematical axioms as axiom of choice, or, say, continuum hypothesis. In fact it is not clear that music theory need even