This paper introduces two languages and associated logics designed to afford perspicuous representations of a range of natural language arguments involving adverbs and the like: first-order logic with basic adverbs (FOL-BA) and first-order logic with scoped adverbs (FOL-SA). The guiding logical idea is that an adverb can come between a term and the rest of the statement it is a part of, resulting in a logically stronger statement. I explain various interesting challenges that arise (...) in the attempt to implement the guiding idea, and provide solutions for some but not all of them. I conclude by outlining some directions for further research. (shrink)

The volume includes the proceedings from the conference FOL75 -- 75 Years of First-Order Logic held at Humboldt University, Berlin, September 18 - 21, 2003 on the occasion of the anniversary of the publication of Hilbert's and Ackermann's Grundzuge der theoretischen Logik. The papers provide analyses of the historical conditions of the shaping of FOL, discuss several modern rivals to it, and show the importance of FOL for interdisciplinary research. While there is no doubt that the celebrated (...) book marks a most important step in the development of logic, the volume in hand proves the actuality of the question "Which logic is the right logic." The volume contains articles by: H. Andreka, J. X. Madarasz, I. Nemeti, A. Avron, K. Brunnler, A. Guglielmi, G. Englebretsen, W. Ewald, P. Hajek, J. Hintikka, W. Hodges, M. Kracht, R. Lanzet, H. Ben-Yami, C. Toke, S. P. Odintsov, H. Wansing, J. A. Robinson, M. Rossberg, M. Thielscher, D. E. Willard, andJ. Wole 'nski. (shrink)

Starting from certain metalogical results, I argue that first-order logical truths of classical logic are a priori and necessary. Afterwards, I formulate two arguments for the idea that first-order logical truths are also analytic, namely, I first argue that there is a conceptual connection between aprioricity, necessity, and analyticity, such that aprioricity together with necessity entails analyticity; then, I argue that the structure of natural deduction systems for FOL displays the analyticity of its truths. (...) Consequently, each philosophical approach to these truths should account for this evidence, i.e., that first-order logical truths are a priori, necessary, and analytic, and it is my contention that the semantic account is a better candidate. (shrink)

To explore the extent of embeddability of Leibnizian infinitesimal calculus in first-order logic (FOL) and modern frameworks, we propose to set aside ontological issues and focus on pro- cedural questions. This would enable an account of Leibnizian procedures in a framework limited to FOL with a small number of additional ingredients such as the relation of infinite proximity. If, as we argue here, first order logic is indeed suitable for developing modern proxies for the (...) inferential moves found in Leibnizian infinitesimal calculus, then modern infinitesimal frameworks are more appropriate to interpreting Leibnizian infinitesimal calculus than modern Weierstrassian ones. (shrink)

We de ne an executable process interpretation for dynamic rst order logic and show that it is a faithful approximation of a dynamic interpre tation procedure for rst order formulas familiar from natural language semantics extended with constructs for bounded choice and bounded it eration This new interpretation of extended dynamic FOL is inspired by an executable interpretation for standard FOL proposed by Apt and Bezem The relation to the Apt Bezem style execution process and the advantages (...) of taking dynamic FOL rather than standard FOL as one s point of reference are discussed at some length Our results relate computational interpretation of FOL to a research tra dition from natural language semantics We discuss some example pro grams in Dynamo a simple language for dynamic logic programming based on the executable process interpretation for dynamic FOL.. (shrink)

The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we offer in the first part of the paper a series of new completeness results for salient classical systems of first order modal logic. Among other results we show that it is possible to prove strong completeness results for (...) normal systems without the Barcan Formula (like FOL + K)in terms of neighborhood frames with constant domains. The first order models we present permit the study of many epistemic modalities recently proposed in computer science as well as the development of adequate models for monadic operators of high probability. Models of this type are either difficult of impossible to build in terms of relational Kripkean semantics [40].We conclude by introducing general first order neighborhood frames with constant domains and we offer a general completeness result for the entire family of classical first order modal systems in terms of them, circumventing some well-known problems of propositional and first order neighborhood semantics (mainly the fact that many classical modal logics are incomplete with respect to an unmodified version of either neighborhood or relational frames). We argue that the semantical program that thus arises offers the first complete semantic unification of the family of classical first order modal logics. (shrink)

The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we offer in the first part of the paper a series of new completeness results for salient classical systems of first order modal logic. Among other results we show that it is possible to prove strong completeness results for (...) normal systems without the Barcan Formula (like FOL + K)in terms of neighborhood frames with constant domains. The first order models we present permit the study of many epistemic modalities recently proposed in computer science as well as the development of adequate models for monadic operators of high probability. Models of this type are either difficult of impossible to build in terms of relational Kripkean semantics [40].We conclude by introducing general first order neighborhood frames with constant domains and we offer a general completeness result for the entire family of classical first order modal systems in terms of them, circumventing some well-known problems of propositional and first order neighborhood semantics (mainly the fact that many classical modal logics are incomplete with respect to an unmodified version of either neighborhood or relational frames). We argue that the semantical program that thus arises offers the first complete semantic unification of the family of classical first order modal logics. (shrink)

The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we offer in the first part of the paper a series of new completeness results for salient classical systems of first order modal logic. Among other results we show that it is possible to prove strong completeness results for (...) normal systems without the Barcan Formula (like FOL + K)in terms of neighborhood frames with constant domains. The first order models we present permit the study of many epistemic modalities recently proposed in computer science as well as the development of adequate models for monadic operators of high probability. Models of this type are either difficult of impossible to build in terms of relational Kripkean semantics [40].We conclude by introducing general first order neighborhood frames with constant domains and we offer a general completeness result for the entire family of classical first order modal systems in terms of them, circumventing some well-known problems of propositional and first order neighborhood semantics (mainly the fact that many classical modal logics are incomplete with respect to an unmodified version of either neighborhood or relational frames). We argue that the semantical program that thus arises offers the first complete semantic unification of the family of classical first order modal logics. (shrink)

We investigate how the sentence choice semantics for propositional superposition logic developed in Tzouvaras could be extended so as to successfully apply to first-order superposition logic. There are two options for such an extension. The apparently more natural one is the formula choice semantics based on choice functions for pairs of arbitrary formulas of the basis language. It is proved however that the universal instantiation scheme of first-order logic, $\varphi \rightarrow \varphi $, is (...) false, as a scheme of tautologies, with respect to FCS. This causes the total failure of FCS as a candidate semantics. Then we turn to the other option, which is a variant of SCS, since it uses again choice functions for pairs of sentences only. This semantics however presupposes that the applicability of the connective | is restricted to quantifier-free sentences, and thus the class of well-formed formulas and sentences of the language is restricted too. Granted these syntactic restrictions, the usual axiomatizations of FOLS turn out to be sound and conditionally complete with respect to this second semantics, just like the corresponding systems of PLS. (shrink)

We investigate the expressive power of second-order logic over finite structures, when two limitations are imposed. Let SAA ) be the set of second-order formulas such that the arity of the relation variables is bounded by k and the number of alternations of second-order quantification is bounded by n . We show that this imposes a proper hierarchy on second-order logic, i.e. for every k , n there are problems not definable in AA but (...) definable in AA for some c 1 , d 1 . The method to show this is to introduce the set AUTOSAT of formulas in F which satisfy themselves. We study the complexity of this set for various fragments of second-order loeic. For first-order logic FOL with unbounded alternation of quantifiers AUTOSAT is PSpace complete. For first-order logic FOL n with alternation of quantifiers bounded by n , AUTOSAT is definable in AA . AUTOSAT) is definable in AA for some c 1 , d 1. (shrink)

It is almost universally acknowledged that first-order logic (FOL), with its clean, well-understood syntax and semantics, allows for the clear expression of philosophical arguments and ideas. Indeed, an argument or philosophical theory rendered in FOL is perhaps the cleanest example there is of “representing philosophy”. A number of prominent syntactic and semantic properties of FOL reflect metaphysical presuppositions that stem from its Fregean origins, particularly the idea of an inviolable divide between concept and object. These presuppositions, taken (...) at face value, reflect a significant metaphysical viewpoint, one that can in fact hinder or prejudice the representation of philosophical ideas and arguments. Philosophers have of course noticed this and have, accordingly, sought to alter or extend traditional FOL in novel ways to reflect a more flexible and egalitarian metaphysical standpoint. The purpose of this paper, however, is to document and discuss how similar “adaptations” to FOL—culminating in a standardized framework known as Common Logic —have evolved out of the more practical and applied encounter of FOL with the problem of representing, sharing, and reasoning upon information on the World Wide Web. (shrink)

We study the foundation of space-time theory in the framework of first-order logic (FOL). Since the foundation of mathematics has been successfully carried through (via set theory) in FOL, it is not entirely impossible to do the same for space-time theory (or relativity). First we recall a simple and streamlined FOL-axiomatization Specrel of special relativity from the literature. Specrel is complete with respect to questions about inertial motion. Then we ask ourselves whether we can prove the (...) usual relativistic properties of accelerated motion (e.g., clocks in acceleration) in Specrel. As it turns out, this is practically equivalent to asking whether Specrel is strong enough to “handle” (or treat) accelerated observers. We show that there is a mathematical principle called induction (IND) coming from real analysis which needs to be added to Specrel in order to handle situations involving relativistic acceleration. We present an extended version AccRel of Specrel which is strong enough to handle accelerated motion, in particular, accelerated observers. Among others, we show that~the Twin Paradox becomes provable in AccRel, but it is not provable without IND. (shrink)

In this paper I try to answer four basic questions: (1) How the concept of God is to be represented? (2) Are there any logical principles governing it? (3) If so, what kind of logic lies behind them? (4) Can there be a logic of the concept of God? I address them by presenting a formal-logical account to the concept of God. I take it as a methodological desideratum that this should be done within the simplest existing logical (...) formalism. I start with first-order logic (FOL) with identity, and then show that its simplest modal extension (SQML, or the simplest quantified modal logic) is enough for us to formalize a minimally satisfactory theory of the concept of God. I focus exclusively on the monotheistic concept of God. (shrink)

forall x: Calgary is a full-featured textbook on formal logic. It covers key notions of logic such as consequence and validity of arguments, the syntax of truth-functional propositional logic TFL and truth-table semantics, the syntax of first-order (predicate) logic FOL with identity (first-order interpretations), symbolizing English in TFL and FOL, and Fitch-style natural deduction proof systems for both TFL and FOL. It also deals with some advanced topics such as modal logic, (...) soundness, and functional completeness. Exercises with solutions are available. It is provided in PDF (for screen reading, printing, and a special version for dyslexics), HTML, and in LaTeX source code. (shrink)

We extend natural deduction for first-order logic (FOL) by introducing diagrams as components of formal proofs. From the viewpoint of FOL, we regard a diagram as a deductively closed conjunction of certain FOL formulas. On the basis of this observation, we first investigate basic heterogeneous logic (HL) wherein heterogeneous inference rules are defined in the styles of conjunction introduction and elimination rules of FOL. By examining what is a detour in our heterogeneous proofs, we discuss (...) that an elimination-introduction pair of rules constitutes a redex in our HL, which is opposite the usual redex in FOL. In terms of the notion of a redex, we prove the normalization theorem for HL, and we give a characterization of the structure of heterogeneous proofs. Every normal proof in our HL consists of applications of introduction rules followed by applications of elimination rules, which is also opposite the usual form of normal proofs in FOL. Thereafter, we extend the basic HL by extending the heterogeneous rule in the style of general elimination rules to include a wider range of heterogeneous systems. (shrink)

One of the fundamental properties inclassical equational reasoning isLeibniz's principle of substitution. Unfortunately, this propertydoes not hold instandard epistemic logic. Furthermore,Herbrand's lifting theorem which isessential to thecompleteness ofresolution andParamodulation in theclassical first order logic (FOL), turns out to be invalid in standard epistemic logic. In particular, unlike classical logic, there is no skolemization normal form for standard epistemic logic. To solve these problems, we introduce anintensional epistemic logic, based on avariation of Kripke's (...) possible-worlds semantics that need not have a constant domain. We show how a weaker notion of substitution through indexed terms can retain the Herbrand theorem. We prove how the logic can yield a satisfibility preserving skolemization form. In particular, we present an intensional principle for unifing indexed terms. Finally, we describe asound andcomplete inference system for a Horn subset of the logic withequality, based onepistemic SLD-resolution. (shrink)

Semantic networks were developed in cognitive science and artificial intelligence studies as graphical knowledge representation and inference tools emulating human thought processes. Formal analysis of the representation and inference capabilities of the networks modeled them as subsets of standard first-order logic (FOL), restricted in the operations allowed in order to ensure the tractability that seemed to characterize human reasoning capabilities. The graphical network representations were modeled as providing a visual language for the logic. Sub-sets of (...) FOL targeted on knowledge representation came to be called description logics, and research on these logics has focused on issues of tractability of subsets with differing representation capabilities, and on the implementation of practical inference systems achieving the best possible performance. Semantic network research has kept pace with these developments, providing visual languages for knowledge entry, editing, and presenting the results of inference, that translate unambiguously to the underlying description logics. This paper discusses the design issues for such semantic network formalisms, and illustrates them through detailed examples of significant generic knowledge structures analyzed in the literature, including determinables, contrast sets, genus/differentiae, taxonomies, faceted taxonomies, cluster concepts, family resemblances, graded concepts, frames, definitions, rules, rules with exceptions, essence and state assertions, opposites and contraries, relevance, and so on. Such examples provide important test material for any visual language formalism for logic. (shrink)

We briefly overview some of the historical landmarks on the path leading to the reduction of the number of logical connectives in classical logic. Relying on the duality inherent in Boolean algebras, we introduce a new operator ( Nallor ) that is the dual of Schönfinkel’s operator. We outline the proof that this operator by itself is sufficient to define all the connectives and operators of classical first-order logic ( Fol ). Having scrutinized the proof, we (...) pinpoint the theorems of Fol that are needed in the proof. Using the insights gained from the proof, we show that there are four binary operators that each can serve as the only undefined logical constant for Fol . Finally, we show that from every n -ary connective that yields a functionally complete singleton set of connectives two Schönfinkel-type operators are definable, and all the latter are so definable. (shrink)

A history of logic -- Patterns of reasoning -- A language and its meaning -- A symbolic language -- 1850-1950 mathematical logic -- Modern symbolic logic -- Elements of set theory -- Sets, functions, relations -- Induction -- Turning machines -- Computability and decidability -- Propositional logic -- Syntax and proof systems -- Semantics of PL -- Soundness and completeness -- First order logic -- Syntax and proof systems of FOL -- Semantics of (...) FOL -- More semantics -- Soundness and completeness -- Why is first order logic "First Order"? (shrink)

Many efforts have been made in recent years to construct formal systems for mechanizing general mathematical reasoning. Most of these systems are based on logics which are stronger than first-order logic. However, there are good reasons to avoid using full second-order logic for this task. In this work we investigate a logic which is intermediate between FOL and SOL, and seems to be a particularly attractive alternative to both: ancestral logic. This is the (...) logic which is obtained from FOL by augmenting it with the transitive closure operator. While the study of this logic has so far been mostly model-theoretical, this work is devoted to its proof theory. Two natural Gentzen-style proof systems for ancestral logic are presented: one for the reflexive transitive closure, and one for the non-reflexive one. We show that these systems are sound for ancestral logic and provide evidence that they indeed encompass all forms of reasoning for this logic that are used in practice. The two systems are shown to be equivalent by providing translation algorithms between them. We end with an investigation of two main proof-theoretical properties: cut elimination and constructive consistency proof. (shrink)

Inspired by Hintikka’s ideas on constructivism, we are going to ‘effectivize’ the game-theoretic semantics for independence-friendly first-order logic, but in a somewhat different way than he did in the monograph ‘The Principles of Mathematics Revisited’. First we show that Nelson’s realizability interpretation—which extends the famous Kleene’s realizability interpretation by adding ‘strong negation’—restricted to the implication-free first-order formulas can be viewed as an effective version of GTS for FOL. Then we propose a realizability interpretation for (...) IF-FOL, inspired by the so-called ‘trump semantics’ which was discovered by Hodges, and show that this trump realizability interpretation can be viewed as an effective version of GTS for IF-FOL. Finally we prove that the trump realizability interpretation for IF-FOL appropriately generalises Nelson’s restricted realizability interpretation for the implication-free first-order formulas. (shrink)

This paper presents an extended version of the Quantified Argument Calculus (Quarc). Quarc is a logic comparable to the first-order predicate calculus. It employs several nonstandard syntactic and semantic devices, which bring it closer to natural language in several respects. Most notably, quantifiers in this logic are attached to one-place predicates; the resulting quantified constructions are then allowed to occupy the argument places of predicates. The version presented here is capable of straightforwardly translating natural-language sentences involving (...) defining clauses. A three-valued, model-theoretic semantics for Quarc is presented. Interpretations in this semantics are not equipped with domains of quantification: they are just interpretation functions. This reflects the analysis of natural-language quantification on which Quarc is based. A proof system is presented, and a completeness result is obtained. The logic presented here is capable of straightforward translation of the classical first-order predicate calculus, the translation preserving truth values as well as entailment. The first-order predicate calculus and its devices of quantification can be seen as resulting from Quarc on certain semantic and syntactic restrictions, akin to simplifying assumptions. An analogous, straightforward translation of Quarc into the first-order predicate calculus is impossible. (shrink)

We re-examine the problem of existential import by using classical predicate logic. Our problem is: How to distribute the existential import among the quantified propositions in order for all the relations of the logical square to be valid? After defining existential import and scrutinizing the available solutions, we distinguish between three possible cases: explicit import, implicit non-import, explicit negative import and formalize the propositions accordingly. Then, we examine the 16 combinations between the 8 propositions having the first (...) two kinds of import, the third one being trivial and rule out the squares where at least one relation does not hold. This leads to the following results: (1) three squares are valid when the domain is non-empty; (2) one of them is valid even in the empty domain: the square can thus be saved in arbitrary domains and (3) the aforementioned eight propositions give rise to a cube, which contains two more (non-classical) valid squares and several hexagons. A classical solution to the problem of existential import is thus possible, without resorting to deviant systems and merely relying upon the symbolism of First-order Logic (FOL). Aristotle’s system appears then as a fragment of a broader system which can be developed by using FOL. (shrink)

forall x: Dortmund is an adaptation and German translation of forall x: Calgary. As such, it is a full-featured textbook on formal logic. It covers key notions of logic such as consequence and validity, the syntax of truth-functional (propositional) logic and truth-table semantics, the syntax of first-order (predicate) logic with identity and first-order interpretations, formalizing German in TFL and FOL, and Fitch-style natural deduction proof systems for both TFL and FOL. It also (...) deals with some advanced topics such as the syntax and (Kripke-)semantics of modal logic. The book is provided in PDF and in LaTeX source code. A booklet with solutions for the exercises in the book is available. (shrink)

We aim to compile some means for a rational reconstruction of a named part of the start-over of Baruch (Benedictus) de Spinoza's metaphysics in 'de deo' (which is 'pars prima' of the 'ethica, ordine geometrico demonstrata' ) in terms of 1st order model theory. In so far, as our approach will be judged successful, it may, besides providing some help in understanding Spinoza, also contribute to the discussion of some or other philosophical evergreen, e.g. 'ontological commitment'. For this text (...) we assume the reader familiar with 'de deo' as well as with some basic concepts and results of 1st order model theory. Before we start reconstruction, we will first revisit shortly the concept of 'attributum' (definitio IV) in it's setting in 'de deo' , next scan for formalizable aspects of 'in suo genere finita' ('de deo', definitio II), subsequently list the model theoretic constructs we will make use of. Then we begin reconstruction by stating "coordinative definitions" for the notions of 'attribute (of a substance)', 'modus (as conceived via an attribute)' and 'substance (as conceived via an attribute)', reasoning shortly for each of them. The "coordinative definitions", we will arrive at, must not be understood as literal translations of Spinoza's concepts - of course, there can't be such a thing as a literal translation - they are meant as formal analoga of these concepts, mapping some logical structure. But even with this caveat they may seem strange to the reader at this stage of discussion. Additional justification for them then should be found in our endeavour, to map some argumentation of Spinoza's proofs of some of his propositions from this starting point. -/- [ Version 1 of this paper originated from 2020-03-14. This is version 2 from 2021-09-12 with some readability amendments. ]. (shrink)

This paper describes a decision procedure for disjunctions of conjunctions of anti-prenex normal forms of pure first-order logic (FOLDNFs) that do not contain V within the scope of quantifiers. The disjuncts of these FOLDNFs are equivalent to prenex normal forms whose quantifier-free parts are conjunctions of atomic and negated atomic formulae (= Herbrand formulae). In contrast to the usual algorithms for Herbrand formulae, neither skolemization nor unification algorithms with function symbols are applied. Instead, a procedure is described (...) that rests on nothing but equivalence transformations within pure first-order logic (FOL). This procedure involves the application of a calculus for negative normal forms (the NNF-calculus) with A -||- A & A (= &I) as the sole rule that increases the complexity of given FOLDNFs. The described algorithm illustrates how, in the case of Herbrand formulae, decision problems can be solved through a systematic search for proofs that reduce the number of applications of the rule &I to a minimum in the NNF-calculus. In the case of Herbrand formulae, it is even possible to entirely abstain from applying &I. Finally, it is shown how the described procedure can be used within an optimized general search for proofs of contradiction and what kind of questions arise for a &I-minimal proof strategy in the case of a general search for proofs of contradiction. (shrink)

The interpretability of AI is just as important as its performance. In the LegalAI field, there have been efforts to enhance the interpretability of models, but a trade-off between interpretability and prediction accuracy remains inevitable. In this paper, we introduce a novel framework called LK-IB for compulsory measure prediction (CMP), one of the critical tasks in LegalAI. LK-IB leverages Legal Knowledge and combines an Interpretable model and a Black-box model to balance interpretability and prediction performance. Specifically, LK-IB involves three steps: (...) (1) inputting cases into the first module, where first-order logic (FOL) rules are used to make predictions and output them directly if possible; (2) sending cases to the second module if FOL rules are not applicable, where a case distributor categorizes them as either “simple” or “complex“; and (3) sending simple cases to an interpretable model with strong interpretability and complex cases to a black-box model with outstanding performance. Experimental results demonstrate that the LK-IB framework provides more interpretable and accurate predictions than other state-of-the-art models. Given that the majority of cases in LegalAI are simple, the idea of model combination has significant potential for practical applications. (shrink)

There exist valuable methods for theorem proving in non classical logics based on translation from these logics into first-order classical logic (abbreviated henceforth FOL). The key notion in these approaches istranslation from aSource Logic (henceforth abbreviated SL) to aTarget Logic (henceforth abbreviated TL). These methods are concerned with the problem offinding a proof in TL by translating a formula in SL, but they do not address the very important problem ofpresenting proofs in SL via a (...) backward translation. We propose a framework for presenting proofs in SL based on a partial backward translation of proofs obtained in a familiar TL: Order-Sorted Predicate Logic. The proposed backward translation transfers some formulasF TL belonging to the proof in TL into formulasF SL , such that the formulasF SL either (a) belong to a corresponding deduction in SL (in the best case) or, (b) are semantically related in some precise way, to formulas in the corresponding deduction in SL (in the worst case). The formulasF TL andF SL can obviously be considered aslemmas of their respective proofs. Therefore the transfer of lemmas of TL gives at least a skeleton of the corresponding proof in SL. Since the formulas of a proof keep trace of the strategy used to obtain the proof, clearly the framework can also help in solving another fundamental and difficult problem:the transfer of strategies from classical to non classical logics. We show how to apply the proposed framework, at least to S5, S4(p), K, T, K4. Two conjectures are stated and we propose sufficient (and in general satisfactory) conditions in order to obtain formulas in the proof in SL. Two particular cases of the conjectures are proved to be theorems. Three examples are treated in full detail. The main lines of future research are given. (shrink)

In this paper we analyse the conditions for attributing to AI autonomous systems the ontological status of “artificial moral agents”, in the context of the “distributed responsibility” between humans and machines in Machine Ethics (ME). In order to address the fundamental issue in ME of the unavoidable “opacity” of their decisions with ethical/legal relevance, we start from the neuroethical evidence in cognitive science. In humans, the “transparency” and then the “ethical accountability” of their actions as responsible moral agents is (...) not in contradiction with the unavoidable “opacity” (unawareness) of the brain process by which they perform their moral judgements on the right action to execute. In fact, the moral accountability of our actions depends on what is immediately before and after our “moral judgements” on the right action to execute (formally, deontic first order logic (FOL) decisions). I.e., our moral accountability depends on the “ethical constraints” we imposed to our judgement before performing it in an opaque way. Anyway, our moral accountability depends overall on the “ethical assessment” or explicit “moral reasoning” after and over the moral judgement before executing our actions (deontic higher order logic (HOL) assessment). In this way, in the light of the AI “imitation game”, the consistent attribution of the status of ethically accountable artificial moral agents to autonomous AI systems depends on two similar conditions. Firstly, it depends on the presence of “ethical constraints” to be satisfied in their Machine Learning (ML) supervised optimization algorithm during its training phase, to give the system ethical skills (“competences”) in its decisions. Secondly – and definitely—, it depends on the presence in an AI autonomous system of a deontic HOL “ethical reasoner” to perform an automatic, and fully transparent assessment (metalogical deontic valuation) about the decisions taken by the ethically skilled ML algorithm about the right action to execute, before executing it. Finally, we show that the proper deontic FOL and HOL for this class of artificial moral agents is Kripke’s modal relational logic, in its algebraic topological formalization. This is naturally implemented in the dissipative QFT unsupervised deep learning of our brains, based on the “doubling of the degrees of freedom” (DDF), and then in the so-called “deep-belief” artificial neural networks for the statistical data pre-processing. This unsupervised learning procedure is also compliant with the usage of the “maximin fairness principle”, used as a balancing aggregation principle of the statistical variables in Sen’s formal theory of fairness. (shrink)

A process-oriented model of belief is presented which permits the representation of nested propositional attitudes within first-order logic. The model (NIM, for nested intensional model) is axiomatized, sense-based (via intensions), and sanctions inferences involving nested epistemic attitudes, with different agents and different times. Because NIM is grounded upon senses, it provides a framework in which agents may reason about the beliefs of another agent while remaining neutral with respect to the syntactic forms used to express the latter (...) agent's beliefs. Moreover, NIM provides agents with a conceptual map, interrelating the concepts of knowledge, belief, truth, and a number of cognate concepts, such as infers, retracts, and questions. The broad scope of NIM arises in part from the fact that its axioms are represented in a novel extension of first-order logic, -FOL (presented herein). -FOL simultaneously permits the representation of truth ascriptions, implicit self-reference, and arbitrarily embedded sentences within a first-order setting. Through the combined use of principles derived from Frege, Montague, and Kripke, together with context-sensitive semantic conventions, -FOL captures the logic of truth inferences, while avoiding the inconsistencies exhibited by Tarski. Applications of -FOL and NIM to interagent reasoning are described and the soundness and completeness of -FOL are established herein. (shrink)

In this paper we make some observations about Natural Deduction derivations [Prawitz, 1965, van Dalen, 1986, Bell and Machover, 1977]. We assume the reader is familiar with it and with proof-theory in general. Our development will be simple, even simple-minded, and concrete. However, it will also be evident that general ideas motivate our examples, and we think both our specific examples and the ideas behind them are interesting and may be useful to some readers. In a sentence, the bare technical (...) content of this paper is: Extending natural deduction with global well-formedness conditions can neatly and cheaply capture classical and intermediate logics. The interest here is in the ‘neatly’ and ‘cheaply’. By ‘neatly’ we mean ‘preserving proof-normalisation’,1 and ‘maintaining the subformula property’, and by ‘cheaply’ we mean ‘preserving the formal structure of deductions’ (so that a deduction in the original system is still, formally, a deduction in the extended system, and in particular it requires no extra effort to write just because it is in the extended system). To illustrate what we have in mind consider intuitionistic first-order logic (FOL) [van Dalen, 1986] as a paradigmatic example of a formal notion of deduction. A natural deduction derivation (or deduction) is an inductively defined tree structure where each node contains an instance of a formula. A deduction is valid when each successive node follows from its predecessors in accordance with some predetermined inference rules. A particular attraction of Natural Deduction is its clean and economical presentation. Here for example are deduction (fragments) proving A ∧ B from A and B, and ∀x. (P (x) ∧ Q(x)) from ∀x. P (x) and ∀x. Q(x): ∀x. P (x) (∀E). (shrink)

and Data The essence of scope in natural language semantics can be characterized as follows: an expression e1 takes scope over an expression e2 iff the interpretation of the former affects the interpretation of the latter. Consider, for example, the sentence in (1) below, which is typical of the cases discussed in this paper in that it involves an indefinite and a universal (or, more generally, a non-existential) quantifier. (1) Everyx student in my class read ay paper about scope. How (...) can we tell whether the indefinite in (1) is in the scope of the universal or not? We can answer this question in two ways. From a dependence-based perspective, Q y is in the scope of Qx if the values of the variable y (possibly) covary with the values of x. From an independence-based perspective, Q y is outside the scope of Qx if y’s value is fixed relative to the values of x. This brings us to the first of our two central questions: should the scopal properties of ordinary, ‘unmarked’ indefinites be characterized in terms of dependence or in terms of independence? The difference between these two conceptualizations is that a dependence-based approach establishes which quantifier(s) Q y is dependent on, while an independence-based approach establishes which quantifier(s) Q y is independent of. Logical semantics has taken both paths to the notion of scope: compare the standard, dependence-based semantics of first-order logic (FOL) – or the dependence-driven Skolemization procedure – with the independence-based semantics of Independence-Friendly Logic (IFL, Hintikka 1973, Sandu 1993, Hodges 1997, Väänänen 2007 among others). Natural language semantics has only taken the dependence-based path. (shrink)

This paper reveals two fallacies in Turing's undecidability proof of first-order logic (FOL), namely, (i) an 'extensional fallacy': from the fact that a sentence is an instance of a provable FOL formula, it is inferred that a meaningful sentence is proven, and (ii) a 'fallacy of substitution': from the fact that a sentence is an instance of a provable FOL formula, it is inferred that a true sentence is proven. The first fallacy erroneously suggests that Turing's (...) proof of the non-existence of a circle-free machine that decides whether an arbitrary machine is circular proves a significant proposition. The second fallacy suggests that FOL is undecidable. (shrink)

In the context of proliferation of many logical systems in the area of mathematical logic and computer science, we present a generalization of forcing in institution-independent model theory which is used to prove two abstract results: Downward Löwenheim-Skolem Theorem and Omitting Types Theorem . We instantiate these general results to many first-order logics, which are, roughly speaking, logics whose sentences can be constructed from atomic formulas by means of Boolean connectives and classical first-order quantifiers. These (...) include first-order logic , logic of order-sorted algebras , preorder algebras , as well as their infinitary variants \ , \ , \ . In addition to the first technique for proving OTT, we develop another one, in the spirit of institution-independent model theory, which consists of borrowing the Omitting Types Property from a simpler institution across an institution comorphism. As a result we export the OTP from FOL to partial algebras and higher-order logic with Henkin semantics , and from the institution of \ to \ and \ . The second technique successfully extends the domain of application of OTT to non conventional logical systems for which the standard methods may fail. (shrink)

Kit Fine has argued that the Tarski Semantics for the language of first order logic is inadequate. A semantic theory for FOL is inadequate if there are formulae of FOL whose meanings or satisfaction conditions it cannot compositionally account for. It is argued here that Fine’s case against Tarski rests on a mistake.

My goal in this paper is, to tentatively sketch and try defend some observations regarding the ontological dignity of object references, as they may be used from within in a formalized language. -/- Hence I try to explore, what properties objects are presupposed to have, in order to enter the universe of discourse of an interpreted formalized language. -/- First I review Frege′s analysis of the logical structure of truth value definite sentences of scientific colloquial language, to draw (...) suggestions from his saturated vs. unsaturated sentence components paradigm. -/- Next try investigate, in how far reference to non pure math objects might allow for a role as argument of a truth value function. Object kinds to be considered are: common sense objects, technical objects, humanities object kinds (social, psychical, ...), objects of art, ... , be they abstract, concrete, or in this respect mixed objects. -/- Then have a comment on the just referenced label abstract objects. -/- Next try to get an idea wrt the ontological significance of the fact, that pure math objects and functions in some important classical cases can be uniquely defined by means of categorical theories. Here, in the course of my argument, I have a little corollary wrt the standard model of first order Peano arithmetic, reducing the epistemological significance of the existence of the non standard models. -/- Next, wrt a special concept of a formalized empirical theory, I care for whether the impure math objects and mixed objects described here, and complying best with truth value function mapping, are also reliable candidates for the ontological commitment of such theories: and discuss an alternative, which reduces the ontological significance of the universe of discourse of the theories intended models. -/- In the end I sum up what is - from my point of view - the status quo, according to my respective findings. (shrink)

Many formalisms for reasoning about knowing commit an agent to be logically omniscient. Logical omniscience is an unrealistic principle for us to use to build a real-world agent, since it commits the agent to knowing infinitely many things. A number of formalizations of knowledge have been developed that do not ascribe logical omniscience to agents. With few exceptions, these approaches are modifications of the possible-worlds semantics. In this paper we use a combination of several general techniques for building non-omniscient reasoners. (...) First we provide for the explicit representation of notions such as problems, solutions, and problem solving activities, notions which are usually left implicit in the discussions of autonomous agents. A second technique is to take explicitly into account the notion of resource when we formalize reasoning principles. We use the notion of resource to describe interesting principles of reasoning that are used for ascribing knowledge to agents. For us, resources are abstract objects. We make extensive use of ordering and inaccessibility relations on resources, but we do not find it necessary to define a metric. Using principles about resources without using a metric is one of the strengths of our approach.We describe the architecture of a reasoner, built from a finite number of components, who solves a puzzle, involving reasoning about knowing, by explicitly using the notion of resource. Our approach allows the use of axioms about belief ordinarily used in problem solving – such as axiom K of modal logic – without being forced to attribute logical omniscience to any agent. In particular we address the issue of how we can use resource-unbounded (e.g., logically omniscient) reasoning to attribute knowledge to others without introducing contradictions. We do this by showing how omniscient reasoning can be introduced as a conservative extension over resource-bounded reasoning. (shrink)