‘First. That on October 23, in the city of New York, your relator was arrested by divers persons claiming to be acting by authority of the government of the United States, and was by said persons conveyed to the United States immigration station at Ellis island, in the harbor of New York, and is now there imprisoned by the commissioner of immigration of the port of New York.

Common-sense folk psychology and mainstream philosophy of action agree about decisions: these are under an agent's direct control, and are thus intentional actions for which agents can be held responsible. I begin this paper by presenting a problem for this view. In short, since the content of the motivational attitudes that drive deliberation and decision remains open-ended until the moment of decision, it is unclear how agents can be thought to exercise control over what they decide at the moment of (...) deciding. I note that this problem might motivate a non-actional view of deciding?a view that decisions are not actions, but are instead passive events of intention acquisition. For without an understanding of how an agent might exercise control over what is decided at the moment of deciding, we lack a good reason for maintaining commitment to an actional view of deciding. However, I then offer the required account of how agents exercise control over decisions at the moment of deciding. Crucial to this account is an understanding of the relation of practical deliberation to deciding, an understanding of skilled deliberative activity, and the role of attention in the mental action of deciding. (shrink)

We study concepts of decidability for subsets of Euclidean spaces ℝk within the framework of approximate computability . A new notion of approximate decidability is proposed and discussed in some detail. It is an effective variant of F. Hausdorff's concept of resolvable sets, and it modifies and generalizes notions of recursivity known from computable analysis, formerly used for open or closed sets only, to more general types of sets. Approximate decidability of sets can equivalently be expressed by (...) computability of the characteristic functions by means of appropriately working oracle Turing machines. The notion fulfills some natural requirements and is hereditary under canonical embeddings of sets into spaces of higher dimensions. However, it is not closed under binary union or intersection of sets. We also show how the framework of resolvability and approximate decidability can be applied to investigate concepts of reducibility for subsets of Euclidean spaces. (shrink)

A bounded monotone sequence of reals without a limit is called a Specker sequence. In Russian constructive analysis, Church's Thesis permits the existence of a Specker sequence. In intuitionistic mathematics, Brouwer's Continuity Principle implies it is false that every bounded monotone sequence of real numbers has a limit. We claim that the existence of Specker sequences crucially depends on the properties of intuitionistic decidable sets. We propose a schema about intuitionistic decidability that asserts “there exists an intuitionistic enumerable set (...) that is not intuitionistic decidable” and show that the existence of a Specker sequence is equivalent to ED. We show that ED is consistent with some certain well known axioms of intuitionistic analysis as Weak Continuity Principle, bar induction, and Kripke Schema. Thus, the assumption of the existence of a Specker sequence is conceivable in intuitionistic analysis. We will also introduce the notion of double Specker sequence and study the existence of them. (shrink)

This paper presents proofs of completeness and decidability of a non-temporal fragment of an Xstit logic. This shows a distinction between the non-temporal fragments of Xstit logic and regular stit logic since the latter is undecidable. The proof of decidability is via the finite model property. The finite model property is shown to hold by constructing a filtration. However, the set that is used to filter the models isn’t simply closed under subformulas, it has more complex closure conditions. (...) The filtration set is akin to the Fischer–Ladner closure of a set used to show completeness and decidability of propositional dynamic logic. (shrink)

The signature of the formal language of mereology contains only one binary predicate P which stands for the relation “being a part of”. Traditionally, P must be a partial ordering, that is, ${\forall{x}Pxx, \forall{x}\forall{y}((Pxy\land Pyx)\to x=y)}$ and ${\forall{x}\forall{y}\forall{z}((Pxy\land Pyz)\to Pxz))}$ are three basic mereological axioms. The best-known mereological theory is “general extensional mereology”, which is axiomatized by the three basic axioms plus the following axiom and axiom schema: (Strong Supplementation) ${\forall{x}\forall{y}(\neg Pyx\to \exists z(Pzy\land \neg Ozx))}$ , where Oxy means ${\exists (...) z(Pzx\land Pzy)}$ , and (Fusion) ${\exists x\alpha \to \exists z\forall y(Oyz\leftrightarrow \exists x(\alpha \land Oyx))}$ , for any formula α where z and y do not occur free. In this paper, I will show that general extensional mereology is decidable, and will also point out that the decidability of the first-order approximation of the theory of complete Boolean algebras can be shown in the same way. (shrink)

The species of indeterminist tense logic called Peircean by A. N. Prior is proved to be recursively decidable.

We give a decision procedure for the ∀∃-theory of the weak truth-table (wtt) degrees of the recursively enumerable sets. The key to this decision procedure is a characterization of the finite lattices which can be embedded into the r.e. wtt-degrees by a map which preserves the least and greatest elements: a finite lattice has such an embedding if and only if it is distributive and the ideal generated by its cappable elements and the filter generated by its cuppable elements are (...) disjoint. We formulate general criteria that allow one to conclude that a distributive upper semi-lattice has a decidable two-quantifier theory. These criteria are applied not only to the weak truth-table degrees of the recursively enumerable sets but also to various substructures of the polynomial many-one (pm) degrees of the recursive sets. These applications to the pm degrees require no new complexity-theoretic results. The fact that the pm-degrees of the recursive sets have a decidable two-quantifier theory answers a question raised by Shore and Slaman in [21]. (shrink)

For $\Bbb {F}$ the field of real or complex numbers, let $CG(\Bbb {F})$ be the continuous geometry constructed by von Neumann as a limit of finite dimensional projective geometries over $\Bbb {F}$ . Our purpose here is to show the equational theory of $CG(\Bbb {F})$ is decidable.

Let HF be the collection of the hereditarily finite well-founded sets and let the primitive language of set theory be the first-order language which contains binary symbols for equality and membership only. As announced in a previous paper by the authors, "Truth in V for ∃*∀∀-sentences is decidable," truth in HF for ∃*∀∀-sentences of the primitive language is decidable. The paper provides the proof of that claim.

Mereological theories are theories based on a binary predicate ‘being a part of’. It is believed that such a predicate must at least define a partial ordering. A mereological theory can be obtained by adding on top of the basic axioms of partial orderings some of the other axioms posited based on pertinent philosophical insights. Though mereological theories have aroused quite a few philosophers’ interest recently, not much has been said about their meta-logical properties. In this paper, I will look (...) into whether those theories are decidable or not. Besides, since theories of Boolean algebras are in some sense upper bounds of mereological theories which can be found in the literature, I shall also make some observations about the possibility of getting mereological theories beyond Boolean algebras. (shrink)

Qualitative Reasoning (QR) is an area of research within Artificial Intelligence that automates reasoning and problem solving about the physical world. QR research aims to deal with representation and reasoning about continuous aspects of entities without the kind of precise quantitative information needed by conventional numerical analysis techniques. Order-of-magnitude Reasoning (OMR) is an approach in QR concerned with the analysis of physical systems in terms of relative magnitudes. In this paper we consider the logic OMR_N for order-of-magnitude reasoning with the (...) bidirectional negligibility relation. It is a multi-modal logic given by a Hilbert-style axiomatization that reflects properties and interactions of two basic accessibility relations (strict linear order and bidirectional negligibility). Although the logic was studied in many papers, nothing was known about its decidability. In the paper we prove decidability of OMR N by showing that the logic has the strong finite model property. (shrink)

This paper presents two general results of decidability concerning logics based on an indeterministic metric tense logic, which can be applied to, among others, logics combining knowledge, time and agency. We provide a general Kripke semantics based on a variation of the notion of synchronized Ockhamist frames. Our proof of the decidability is by way of the finite frame property, applying subframe transformations and a variant of the filtration technique.

As this passage from a recent book on the psychology of decision-making indicates, deciding seems to be part of our daily lives. But what is it to decide to do something? It may be true, as some philosophers have claimed, that to decide to A is to perform a mental action of a certain kind – specifically, an action of forming an intention to A. (Henceforth, the verb ‘form’ in this context is to be understood as an action verb.) Even (...) if this is so, we are faced with pressing questions. Do we form all of our intentions? If not, how does forming an intention differ from other ways of acquiring one? Do we ever, in fact, form intentions, or do we rather merely acquire them in something like the way we acquire beliefs or desires? These are some of the questions that will occupy me here. My aim is to clarify the nature of deciding to act and to make a case for the occurrence of genuine acts of intention formation. (shrink)

We provide a simple translation of the satisfiability problem for regular grammar logics with converse into GF2, which is the intersection of the guarded fragment and the 2-variable fragment of first-order logic. The translation is theoretically interesting because it translates modal logics with certain frame conditions into first-order logic, without explicitly expressing the frame conditions. It is practically relevant because it makes it possible to use a decision procedure for the guarded fragment in order to decide regular grammar logics with (...) converse. The class of regular grammar logics includes numerous logics from various application domains. A consequence of the translation is that the general satisfiability problem for every regular grammar logics with converse is in EXPTIME. This extends a previous result of the first author for grammar logics without converse. Other logics that can be translated into GF2 include nominal tense logics and intuitionistic logic. In our view, the results in this paper show that the natural first-order fragment corresponding to regular grammar logics is simply GF2 without extra machinery such as fixed-point operators. (shrink)

The purpose of this paper is to prove the decidability ofstit theory (a logic of seeing to it that) with a single agent andRefref Equivalence. This result is obtained through an axiomatization of the theory and a proof that it has thefinite model property. A notion ofcompanions to stit formulas is introduced and extensively used in the proof.

We study the decidability problem for metric and layered temporal logics. The logics we consider are suitable to model time granularity in various contexts, and they allow one to build granular temporal models by referring to the "natural scale" in any component of the model and by properly constraining the interactions between differently-grained components. A monadic second-order language combining operators such as temporal contextualization and projection, together with the usual displacement operator of metric temporal logics, is considered, and the (...) theory of finitely-layered metric temporal structures is shown to be decidable. (shrink)

To decide to A, as I conceive of it, is to perform a momentary mental action of forming an intention to A. I argue that ordinary instances of practical deciding, so conceived, falsify the following...

Presented here is a new result concerning the computational power of so-called SADn computers, a class of Turing-machine-based computers that can perform some non-Turing computable feats by utilising the geometry of a particular kind of general relativistic spacetime. It is shown that SADn can decide n-quantifier arithmetic but not (n+1)-quantifier arithmetic, a result that reveals how neatly the SADn family maps into the Kleene arithmetical hierarchy. Introduction Axiomatising computers The power of SAD computers Remarks regarding the concept of computability.

We provide algebraic conditions ensuring the decidability of the theory of modules over effectively given Prüfer (in particular Bézout) domains whose localizations at maximal ideals have dense value groups. For Bézout domains, these conditions are also necessary.

We show that various fragments of the intuitionistic/constructive theory of the reals are decidable.

Since Plato’s massive critique of the Sophists rhetoric’s ill repute runs through the history of western philosophy denunciating methods of rhetoric as in large part dishonest persuasion strategies which are at most marginally interested in dealing with truths. This judgement falls way too short insofar as it distorts the historically grown stock labeled “rhetoric” not only in the Aristotelian work. With reference to Olaf Müller’s philosophical book addressing the “controversy” between Goethe and Newton about the nature of light, I will (...) study the different rhetorical models and methods used by Newton and Goethe and also Müller himself. It becomes apparent that even works attempting to decide who was right in the long run do far more than trying to evoke true “representations” of facts or truths in the reader. The specific use of language patterns provides deeper insight into an author’s mindset towards the subject area discussed in his work and generally speaking the investigation of the rhetoric of science and philosophy leads to a better understanding of different epistemic cultures both in philosophy and science. (shrink)

In this paper, we introduce a new fragment of the first-order temporal language, called the monodic fragment, in which all formulas beginning with a temporal operator have at most one free variable. We show that the satisfiability problem for monodic formulas in various linear time structures can be reduced to the satisfiability problem for a certain fragment of classical first-order logic. This reduction is then used to single out a number of decidable fragments of first-order temporal logics and of two-sorted (...) first-order logics in which one sort is intended for temporal reasoning. Besides standard first-order time structures, we consider also those that have only finite first-order domains, and extend the results mentioned above to temporal logics of finite domains. We prove decidability in three different ways: using decidability of monadic second-order logic over the intended flows of time, by an explicit analysis of structures with natural numbers time, and by a composition method that builds a model from pieces in finitely many steps. (shrink)

We consider the decision problem for cases of first-order temporal logic with function symbols and without equality. The monadic monodic fragment with flexible functions can be decided with EXPSPACE-complete complexity. A single rigid function is sufficient to make the logic not recursively enumerable. However, the monadic monodic fragment with rigid functions, where no two distinct terms have variables bound by the same quantifier, is decidable and EXPSPACE-complete.

We consider McCarthy's notions of predicate circumscription and formula circumscription. We show that the decision problems “does θ have a countably infinite minimal model” and “does φ hold in every countably infinite minimal model of θ” are complete Σ 1 2 and complete π 1 2 over the integers, for both forms of circumscription. The set of structures definable as first order definable subsets of countably infinite minimal models is the set of structures which are Δ 1 2 over the (...) integers, for both forms of circumscription. Thus, restricted to countably infinite structures, predicate and formula circumscription define the same sets and have equally difficult decision problems. With general formula circumscription we can define several infinite cardinals, so the decidability problems are dependent upon the axioms of set theory. (shrink)

Can we decide to trust? Sometimes, yes. And when we do, we need not believe that our trust will be vindicated. This paper is motivated by the need to incorporate these facts into an account of trust. Trust involves reliance; and in addition it requires the taking of a reactive attitude to that reliance. I explain how the states involved here differ from belief. And I explore the limits of our ability to trust. I then turn to the idea of (...) trusting what others say. I suggest that we sometimes decide to trust people to be sincere and knowledgeable; and that having taken this attitude towards them, we come to believe what they say. I spell out some consequences that this has for an account of testimony, and for van Fraassen's decision theoretic principle of Reflection. (shrink)

This book is the most comprehensive treatment available of one of the most urgent - and yet in some respects most neglected - problems in bioethics: decision-making for incompetents. Part I develops a general theory for making treatment and care decisions for patients who are not competent to decide for themselves. It provides an in-depth analysis of competence, articulates and defends a coherent set of principles to specify suitable surrogate decisionmakers and to guide their choices, examines the value of advance (...) directives, and investigates the role that considerations of cost ought to play in decisions concerning incompetents. Part II applies this theoretical framework to the distinctive problems of three important classes of individuals, many of whom are incompetent: minors, the elderly and psychiatric patients. The authors' approach combines a probing analysis of fundamental issues in ethical theory with a sensitive awareness of the concrete realities of health care institutions and the highly personal and individual character of difficult practical problems. Its broad scope will appeal to health professionals, moral philosophers and lawyers alike. (shrink)

In this paper we show that the usual intuitionistic characterization of the decidability of the propositional function B prop [x : A], i. e. to require that the predicate ∨ ¬ B) is provable, is equivalent, when working within the framework of Martin-Löf's Intuitionistic Type Theory, to require that there exists a decision function ψ: A → Boole such that = Booletrue) ↔ B). Since we will also show that the proposition x = Booletrue [x: Boole] is decidable, we (...) can alternatively say that the main result of this paper is a proof that the decidability of the predicate Bprop [x : A] can be effectively reduced by a function ψ A → Boole to the decidability of the predicate ψ = Booletrue [x : A]. All the proofs are carried out within the Intuitionistic Type Theory and hence the decision function ψ, together with a proof of its correctness, is effectively constructed as a function of the proof of ∨ ¬ B). (shrink)

Justification logics are modal logics that include justifications for the agent's knowledge. So far, there are no decidability results available for justification logics with negative introspection. In this paper, we develop a novel model construction for such logics and show that justification logics with negative introspection are decidable for finite constant specifications.

In this paper we shall introduce a simple temporal logic suitable for reasoning about the temporal aspects of parallel universes, parallel processes, distributed systems, or multiple agents. We will use a variant of the mosaic method to prove decidability of this logic. We also show that the logic does not have the finite model property. This shows that the mosaic method is sometimes a stronger way of establishing decidability.

We prove that if an n-element algebra generates the variety \ which is actively structurally complete, then the cardinality of the carrier of each subdirectly irreducible algebra in \ is at most \\cdot n^{2\cdot n}}\). As a consequence, with the use of known results, we show that there exist algorithms deciding whether a given finite algebra \ generates the structurally complete variety \\) in the cases when \\) is congruence modular or \\) is congruence meet-semidistributive or \ is a semigroup.

We provide a simple translation of the satisfiability problem for regular grammar logics with converse into GF2, which is the intersection of the guarded fragment and the 2-variable fragment of first-order logic. The translation is theoretically interesting because it translates modal logics with certain frame conditions into first-order logic, without explicitly expressing the frame conditions. It is practically relevant because it makes it possible to use a decision procedure for the guarded fragment in order to decide regular grammar logics with (...) converse. The class of regular grammar logics includes numerous logics from various application domains. A consequence of the translation is that the general satisfiability problem for every regular grammar logics with converse is in EXPTIME. This extends a previous result of the first author for grammar logics without converse. Other logics that can be translated into GF2 include nominal tense logics and intuitionistic logic. In our view, the results in this paper show that the natural first-order fragment corresponding to regular grammar logics is simply GF2 without extra machinery such as fixed-point operators. (shrink)

During the first wave of the COVID-19 pandemic in Italy, practitioners had to make tragic decisions regarding the allocation of scarce resources in the ICU. The Italian debate has paid a lot of attention to identifying the specific regulatory criteria for the allocation of resources in the ICU; in this paper, however, we argue that deciding such criteria is not enough for the implementation of fair and transparent allocative decisions. In this respect, we discuss three ethical issues: (a) in the (...) Italian context, the treating physician, rather than a separate committee, was generally the one responsible for the allocation decision; (b) although many allocative guidelines have supported moral equivalence between withholding and withdrawing treatments, some health professionals have continued to consider it a morally problematic aspect; and (c) the health workers who have had to make the aforementioned decisions or even only worked in ICU during the pandemic often experienced moral distress. We conclude by arguing that, even if these problems are not directly related to the above-mentioned issues of distributive justice, they can nevertheless directly affect the quality and ethics of the implementation of allocative criteria, regardless of those chosen. (shrink)

In this paper, John Tillson defends an approach to deciding the aims and content of public schooling from the critique of Public Reason Liberalism. The approach that he defends is an unrestricted pairing of the Epistemic Criterion and of the Momentousness Criterion. On the Epistemic Criterion, public schooling should align students' credence with credibility. On the Momentousness Criterion, public schooling ought to include content that it is costly for children to lack the correct view about, where they are otherwise unlikely (...) to have it. Public Reason Liberals seek to restrict both the Epistemic and Momentousness Criteria to within a range that is acceptable to politically reasonable citizens. In response, Tillson argues, first, that the considerations that encourage Public Reason Liberalism instead motivate unrestricted versions of the Epistemic and Momentousness Criteria; and, second, that Public Reason Liberalism faces a dilemma, that it either entails absurd consequences or must undermine itself in addressing these. (shrink)

The paper describes a decidable class of verification problems expressed in first order timed logic. To specify programs we useState Machines. It is known that Abstract State Machines and first order timed logic are two very powerful formalisms apt to represent verification problems for timed distributed systems. However, the general verification problem represented in this way is undecidable. Prior, some decidable classes of verification problems were described in semantical properties that are in their turn undecidable. The decidable class of the (...) present paper is described in syntactical terms. Though it admits no functions, only predicates, it is of practical interest and we give an example illustrating possible applications. (shrink)

Mele's chapter addresses two primary aims. The first is to develop an experimentally useful conception of conscious deciding. The second is to challenge a certain source of skepticism about free will: the belief that conscious decisions and intentions are never involved in producing corresponding overt actions. The challenge Mele develops has a positive dimension that accords with the aims of this volume: It sheds light on a way in which some conscious decisions and intentions do seem to be efficacious.

Considering humbler creatures than ourselves provides a better chance of uncovering what really matters for perceptual consciousness. This chapter discusses a series of broad classes of behaving systems, starting with ‘pure reflex systems’ and ending with ‘deciders’: systems with the ‘basic package’, capable of choosing between alternative courses of action, if only in a rudimentary sense. The idea of the basic package provides a framework for thinking about behaving systems, which advances the overall project of understanding perceptual consciousness. This framework (...) helps to bridge the gap between descriptions in purely physical or biomechanical terms, and descriptions in psychological terms. (shrink)

We determine precisely those locally finite varieties of unary algebras of finite type which, when augmented by a ternary discriminator, generate a variety with a decidable theory.

In this paper, the positive fragment of the logic math formula of contraction-less relevant implication is extended with the addition of a paraconsistent negation connective similar to the strong negation connective in Nelson's paraconsistent four-valued logic math formula. This extended relevant logic is called math formula, and it has the property of constructible falsity which is known to be a characteristic property of math formula. A Gentzen-type sequent calculus math formula for math formula is introduced, and the cut-elimination and (...) class='Hi'>decidability theorems for math formula are proved. Two extended Routley-Meyer semantics are introduced for math formula, and the completeness theorems with respect to these semantics are proved. (shrink)

An AE-sentence is a sentence in prenex normal form with all universal quantifiers preceding all existential quantifiers, and the AE-theory of a structure is the set of all AE-sentences true in the structure. We show that the AE-theory of \, \cap, \cup, 0, 1)\) is decidable by giving a procedure which, for any AE-sentence in the language, determines the truth or falsity of the sentence in our structure.

This paper is primarily concerned with ℵ 0 -categoricity of theories of partially ordered sets. It contains some general conjectures, a collection of known results and some new theorems on ℵ 0 -categoricity. Among the latter are the following. Corollary 3.3. For every countable ℵ 0 -categorical U there is a linear order of A such that $(\mathfrak{U}, is ℵ 0 -categorical. Corollary 6.7. Every ℵ 0 -categorical theory of a partially ordered set of finite width has a decidable theory. (...) Theorem 7.7. Every ℵ 0 -categorical theory of reticles has a decidable theory. There is a section dealing just with decidability of partially ordered sets, the main result of this section being. Theorem 8.2. If $(P, is a finite partially ordered set and K P is the class of partially ordered sets which do not embed $(P, , then Th(K P ) is decidable iff K P contains only reticles. (shrink)

If X is set and L a lattice, then an L-subset or fuzzy subset of X is any map from X to L, [11]. In this paper we extend some notions of recursivity theory to fuzzy set theory, in particular we define and examine the concept of almost decidability for L-subsets. Moreover, we examine the relationship between imprecision and decidability. Namely, we prove that there exist infinitely indeterminate L-subsets with no more precise decidable versions and classical subsets whose (...) unique shaded decidable versions are the L-subsets almost-everywhere indeterminate. (shrink)

Krynicki, M. and M. Mostowski, Decidability problems in languages with Henkin quantifiers, Annals of Pure and Applied Logic 58 149–172.We consider the language L with all Henkin quantifiers Hn defined as follows: Hnx1…xny1…yn φ iff f1…fnx1. ..xn φ, ...,fn). We show that the theory of equality in L is undecidable. The proof of this result goes by interpretation of the word problem for semigroups.Henkin quantifiers are strictly related to the function quantifiers Fn defined as follows: Fnx1…xny1…yn φ iff fx1…xn (...) φ,...,f). In contrast with the first result we show that the theory of equality with all quantifiers Fn is decidable.We also consider decidability problems for other theories in languages L and L. (shrink)

Fluted logic is the restriction of pure predicate logic to formulas in which variables play no essential role. Although fluted logic is significantly weaker than pure predicate logic, it is of interest because it seems closely to parallel natural logic, the logic that is conducted in natural language. It has been known since 1969 that if conjunction in fluted formulas is restricted to subformulas of equal arity, satisfiability is decidable. However, the decidability of sublogics lying between this restricted (homogeneous) (...) fluted logic and full predicate logic remained unknown. In 1994 it was shown that the satisfiability of fluted formulas without restriction is decidable, thus reducing the unknown region significantly. This paper further reduces the unknown region. It shows that fluted logic with the logical identity is decidable. Since the reflection functor can be defined in fluted logic with identity, it follows that fluted logic with the reflection functor also lies within the region of decidability. Relevance to natural logic is increased since the identity permits definition of singular predicates, which can represent anaphoric pronouns. (shrink)

In classical propositional logic, a theory T is prime iff it is complete. In Łukasiewicz infinite-valued logic the two notions split, completeness being stronger than primeness. Using toric desingularization algorithms and the fine structure of prime ideal spaces of free ℓ -groups, in this paper we shall characterize prime theories in infinite-valued logic. We will show that recursively enumerable prime theories over a finite number of variables are decidable, and we will exhibit an example of an undecidable r.e. prime theory (...) over countably many variables. (shrink)

Rybakov proved that the admissible rules of $\mathsf {IPC}$ are decidable. We give a proof of the same theorem, using the same core idea, but couched in the many notions that have been developed in the mean time. In particular, we illustrate how the argument can be interpreted as using refinements of the notions of exactness and extendibility.

Rybakov (1984a) proved that the admissible rules of IPC are decidable. We give a proof of the same theorem, using the same core idea, but couched in the many notions that have been developed in the mean time. In particular, we illustrate how the argument can be interpreted as using refinements of the notions of exactness and extendibility.

From 1929 through 1944, Wittgenstein endeavors to clarify mathematical meaningfulness by showing how (algorithmically decidable) mathematical propositions, which lack contingent "sense," have mathematical sense in contrast to all infinitistic "mathematical" expressions. In the middle period (1929-34), Wittgenstein adopts strong formalism and argues that mathematical calculi are formal inventions in which meaningfulness and "truth" are entirely intrasystemic and epistemological affairs. In his later period (1937-44), Wittgenstein resolves the conflict between his intermediate strong formalism and his criticism of set theory by requiring (...) that a mathematical calculus (vs. a "sign-game") must have an extrasystemic, real world application, thereby returning to the weak formalism of the Tractatus. (shrink)