Online research in philosophy

We investigate several approaches to resolution based automated theoremproving in classical higher-order logic (based on Church's simply typed-calculus) and discuss their requirements with respect to Henkincompleteness and full extensionality. In particular we focus on Andrews'higher-order resolution (Andrews 1971), Huet's constrained resolution (Huet1972), higher-order E-resolution, and extensional higher-order resolution(Benzmüller and Kohlhase 1997). With the help of examples we illustratethe parallels and differences of the extensionality treatment of these approachesand demonstrate that extensional higher-order resolution is the sole approach thatcan completely avoid additional (...) extensionality axioms. (shrink)

The history of building automated theorem provers for higher-order logic is almost as old as the field of deduction systems itself. The first successful attempts to mechanize and implement higher-order logic were those of Huet [13] and Jensen and Pietrzykowski [17]. They combine the resolution principle for higher-order logic (first studied in [1]) with higher-order unification. The unification problem in typed λ-calculi is much more complex than that for first-order terms, since it has to take the theory of αβη-equality into (...) account. As a consequence, the higher-order unification problem is undecidable and sets of solutions need not even always have most general elements that represent them. Thus the mentioned calculi for higher-order logic have take special measures to circumvent the problems posed by the theoretical complexity of higher-order unification. In this paper, we will exemplify the methods and proof- and model-theoretic tools needed for extending first-order automated theorem proving to higherorder logic. For the sake of simplicity take the tableau method as a basis (for a general introduction to first-order tableaux see part I.1) and discuss the higherorder tableau calculi HT and HTE first presented in [19]. The methods in this paper also apply to higher-order resolution calculi [1, 13, 6] or the higher-order matings method of Peter [3], which extend their first-order counterparts in much the same way. Since higher-order calculi cannot be complete for the standard semantics by Gödel’s incompleteness theorem [11], only the weaker notion of Henkin models [12] leads to a meaningful notion of completeness in higher-order logic. It turns out that the calculi in [1, 13, 3, 19] are not Henkin-complete, since they fail to capture the extensionality principles of higher-order logic. We will characterize the deductive power of our calculus HT (which is roughly equivalent to these calculi) by the semantics of functional Σ-models. To arrive at a calculus that is complete with respect to Henkin models, we build on ideas from [6] and augment HT with tableau construction rules that use the extensionality principles in a goal-oriented way.. (shrink)

Thus, despite the di culty of higher-order automated theorem proving, which has to deal with problems like the undecidability of higher-order uni - cation (HOU) and the need for primitive substitution, there are proof problems which lie beyond the capabilities of rst-order theorem provers, but instead can be solved easily by an higher-order theorem prover (HOATP) like Leo. This is due to the expressiveness of higher-order Logic and, in the special case of Leo, due to an appropriate handling of the (...) extensionality principles (functional extensionality and extensionality on truth values). (shrink)

Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using three-valued logic decades ago, but there has not been a satisfactory mechanization. Recent years have seen a thorough investigation of the framework of many-valued truth-functional logics. However, strong Kleene logic, where quantification is restricted and therefore not truthfunctional, does not fit the framework directly. We solve this problem (...) by applying recent methods from sorted logics. This paper presents a resolution calculus that combines the proper treatment of partial functions with the efficiency of sorted calculi. (shrink)

This paper introduces a multi-valued variant of higher-order resolution and proves it correct and complete with respect to a variant of Henkin’s general model semantics. This resolution method is parametric in the number of truth values as well as in the particular choice of the set of connectives (given by arbitrary truth tables) and even substitutional quantifiers. In the course of the completeness proof we establish a model existence theorem for this logical system. The work reported in this paper provides (...) a basis for developing higherorder mechanizations for many non-classical logics. (shrink)

Symbolic logic is a marvelous thing. It allows for an explicit expression of existence, viz. by means of the existential quantifier, and by it only. This is the true gist in Quine’s slogan “to be is to be a value of a bound variable.” Accordingly, one can also formulate explicitly the thesis of nominalism in terms of such logic. What this thesis says is that all the values of existential quantifiers we need in our language are particular objects, not higher-order (...) objects such as properties, relations, functions and sets. This requirement is satisfied by the first-order languages using the received first-order logic. The commonly used basic logic is therefore nominalistic. But this result does not tell anything, for the received first-order logic is far too weak to capture all we need in mathematics or science. According to conventional wisdom, we need for this purpose either higher-order logic or set theory. Now both of them deal with higher-order entities and hence violate the canons of nominalism. This does not refute nominalism, however. For I will show that both set theory and higher-order logic can be made dispensable by developing a more powerful first-order logic that can do the same job as they do. Moreover, there are very serious problems connected with both of them. This constitutes an additional reason for dispensing with them in the foundations of mathematics. I will show how we can do just that. But we obviously need a better first-order logic for the purpose. Hence my first task is to develop one. But is this a viable construal of the problem of nominalism? The very distinction between particular and higher-order entities might perhaps seem to be hard to capture in logical terms — harder than has been indicated so far. Logicians like Jouko Väänänen (2001) have emphasized the complexities involved in trying to distinguish first-order logic from higher-order logic.. (shrink)

Classical logic has proved inadequate in various areas of computer science, artificial intelligence, mathematics, philosopy and linguistics. This is an introduction to extensions of first-order logic, based on the principle that many-sorted logic (MSL) provides a unifying framework in which to place, for example, second-order logic, type theory, modal and dynamic logics and MSL itself. The aim is two fold: only one theorem-prover is needed; proofs of the metaproperties of the different existing calculi can be avoided by borrowing them from (...) MSL. To make the book accessible to readers from different disciplines, whilst maintaining precision, the author has supplied detailed step-by-step proofs, avoiding difficult arguments, and continually motivating the material with examples. Consequently this can be used as a reference, for self-teaching or for first-year graduate courses. (shrink)

Higher order unification is a way of combining information (or equivalently, solving equations) expressed as terms of a typed higher order logic. A suitably restricted form of the notion has been used as a simple and perspicuous basis for the resolution of the meaning of elliptical expressions and for the interpretation of some non-compositional types of comparative construction also involving ellipsis. This paper explores another area of application for this concept in the interpretation of sentences containing intonationally marked focus, or (...) various semantic constructs which are sensitive to focus.Similarities and differences between this approach, and theories using alternative semantics, structured meanings, or flexible categorial grammars, are described. The paper argues that the higher order unification approach offers descriptive advantages over these alternatives, as well as the practical advantage of being capable of fairly direct computational implementation. (shrink)