The aim of this article is to discuss pure variable inclusion logics, that is, logical systems where valid entailments require that the propositional variables occurring in the conclusion are included among those appearing in the premises, or vice versa. We study the subsystems of Classical Logic satisfying these requirements and assess the extent to which it is possible to characterise them by means of a single logical matrix. In addition, we semantically describe both of these companions to Classical (...) class='Hi'>Logic in terms of appropriate matrix bundles and as semilattice-based logics, showing that the notion of consequence in these logics can be interpreted in terms of truth (or non-falsity) and meaningfulness (or meaninglessness) preservation. Finally, we use Płonka sums of matrices to investigate the pure variable inclusion companions of an arbitrary finitary logic. -/- . (shrink)

In this article, we explore the semantic characterization of the (right) pure refined variable inclusion companion of all logics, which is a further refinement of the nowadays well-studied pure right variable inclusion logics. In particular, we will focus on giving a characterization of these fragments via a single logical matrix, when possible, and via a class of finite matrices, otherwise. In order to achieve this, we will rely on extending the semantics of the logics whose companions we will be discussing (...) with infectious values in direct and in more subtle ways. This further establishes the connection between infectious logics and variable inclusion logics. (shrink)

Inclusion logic is a variant of dependence logic that was shown to have the same expressive power as positive greatest fixed‐point logic. Inclusion logic is not axiomatisable in full, but its first order consequences can be axiomatized. In this paper, we provide such an explicit partial axiomatization by introducing a system of natural deduction for inclusion logic that is sound and complete for first order consequences in inclusion logic.

The paper aims at studying, in full generality, logics defined by imposing a variable inclusion condition on a given logic \. We prove that the description of the algebraic counterpart of the left variable inclusion companion of a given logic \ is related to the construction of Płonka sums of the matrix models of \. This observation allows to obtain a Hilbert-style axiomatization of the logics of left variable inclusion, to describe the structure of their reduced models, and (...) to locate them in the Leibniz hierarchy. (shrink)

On the one hand, the absence of contraction is a safeguard against the logical (property theoretic) paradoxes; but on the other hand, it also disables inductive and recursive definitions, in its most basic form the definition of the series of natural numbers, for instance. The reason for this is simply that the effectiveness of a recursion clause depends on its being available after application, something that is usually assured by contraction. This paper presents a way of overcoming this problem within (...) the framework of a logic based on inclusion and unrestricted abstraction, without any form of extensionality. (shrink)

We investigate several consequences of inclusion relations between quantified provability logics. Moreover, we give a necessary and sufficient condition for the inclusion relation between quantified provability logics with respect to \ arithmetical interpretations.

In this paper, first, we determine the number of sublogics of variable inclusion of an arbitrary finitary logic ⊢ with a composition term. Then, we investigate their position into the lattice of co...

Some authors have studied in an ad hoc fashion the inclusive logics, that is the logics which admit or include objects or sets without element. These logics have been recently brought into the limelight because of the use of arbitrary topoi for interpreting languages. (In topoi there are usually many objects without element.)The aim of the paper is to present, for some inclusive logics, an axiomatization as natural and as simple as possible. Because of the intended applications to (...) category theory, the logics studied are many-sorted and intuitionistic. (shrink)

In this essay, I argue for a view that inclusive positivists share with Ronald Dworkin. According to the Moral Incorporation Thesis (MIT), it is logically possible for a legal system to incorporate moral criteria of legality (or “grounds of law,” as Dworkin puts it). Up to this point, the debate has taken the shape of attacks on the coherence of MIT with the defender of MIT merely attempting to refute the attacking argument. I give a positive argument for MIT. (...) I begin with an explanation of the logic of establishing possibility claims, such as MIT. At the outset, it is worth noting that the logic of establishing possibility claims is very different from the logic of establishing contingent descriptive claims or necessary claims. For this reason, some explication of the relevant features of the semantics of modal logic will be necessary here. Once the structural framework is adequately developed, the argument for MIT will be grounded on the strength of a thought experiment of a surprisingly simple kind. Indeed, the argument is inspired by a Razian argument for the possibility of a legal system without coercive enforcement machinery; on his view, a society of angels could still have a system of law without any coercive machinery. My argument will possess two theoretically important qualities that are also possessed by Raz’s powerfully simple, but ultimately unsuccessful, argument. (shrink)

We show how Richard Jeffrey’s The Logic of Decision provides the proper formalism for calculating expected fitness for correlated encounters in general. As an illustration, some puzzles about kin selection are resolved.

We show how Richard Jeffrey's The Logic of Decision provides the proper formalism for calculating expected fitness for correlated encounters in general. As an illustration, some puzzles about kin selection are resolved.

ABSTRACTProbability judgements entail a conjunction fallacy if a conjunction is estimated to be more probable than one of its conjuncts. In the context of predication of alternative logical hypothesis, Bayesian logic provides a formalisation of pattern probabilities that renders a class of pattern-based CFs rational. BL predicts a complete system of other logical inclusion fallacies. A first test of this prediction is investigated here, using transparent tasks with clear set inclusions, varying in observed frequencies only. Experiment 1 uses data (...) where BL makes dominant predictions; Experiment 2's predictions were less clear, and we additionally investigated judgements about the second-most probable hypotheses. The results corroborated a pattern-probability account and cannot be easily explained by other theories of CFs. IFs were not limited to conjunctions, but rather occurred systematically for several logical connectives. Thus, pattern-based prob... (shrink)

Inclusivity is widely considered a requirement of defensible environmental risk consultations and is often either mandated or recommended to help ensure attention to stakeholders’ diverse views. Experience suggests the opposite: the emphasis on an inclusive consultation process often makes it impossible for decision makers to listen carefully to stakeholders and for citizens’ views to influence the design and choice of proposed actions. This paper briefly reviews the promise of environmental risk consultations before outlining several of the more serious problems (...) associated with an emphasis on inclusivity: long lists of undifferentiated concerns, facts tainted by stakeholders’ perspectives and worldviews, little access to clarifying dialogue or tests of expertise, few opportunities to scrutinize knowledge quality, avoidance of controversial issues, and an overwhelming abundance of information. As a result, the promotion of inclusivity often serves as a convenient excuse for decision makers to silence citizens by substituting quantity for quality, breadth for depth, and an adversarial approach for dialogue and informed understanding. (shrink)

In this paper we study automated reasoning in the modal logic CPDLreg which is a combination of CPDL (Propositional Dynamic Logic with Converse) and REGc (Regular Grammar Logic with Converse). The logic CPDL is widely used in many areas, including program verification, theory of action and change, and knowledge representation. On the other hand, the logic REGc is applicable in reasoning about epistemic states and ontologies (via Description Logics). The modal logic CPDLreg can serve (...) as a technical foundation for reasoning about agency. Even very rich multi-agent logics can be embedded into CPDLreg via a suitable translation. Therefore, CPDLreg can serve as a test bed to implement and possibly verify new ideas without providing specific automated reasoning techniques for the logic in question. This process can to a large extent be automated. The idea of embedding various logics into CPDLreg is illustrated on a rather advanced logic TEAMLOGK designed to specify teamwork in multi-agent systems. Apart from defining informational and motivational attitudes of groups of agents, TEAMLOGK allows for grading agents' beliefs, goals and intentions. The current paper is a companion to our paper (Dunin-Kęplicz et al., 2010a). The main contribution are proofs of soundness and completeness of the tableau calculus for CPDLreg provided in (Dunin-Kęplicz et al., 2010a). (shrink)

Logic is one of the most male-dominated areas within the already hugely male-dominated subject of philosophy. Popular hypotheses for this disparity include a preponderance of confident, mathematically-minded male students in the classroom, the historical association between logic and maleness, and the lack of female role-models for students, though to date none of these have been empirically tested. In this paper I discuss the effects of various attempts to address these potential causes whilst teaching second-year formal and philosophical (...) class='Hi'>logic courses at different universities in the UK. I found the most noticeable positive effect came from assigning a good proportion of reading by female authors presenting an original point of view. I go on to suggest some implementations for incorporating more texts by female authors into our arsenal of tools for teaching logic. (shrink)

We propose a nonmonotonic Description Logic of typicality able to account for the phenomenon of combining prototypical concepts, an open problem in the fields of AI and cognitive modelling. Our logic extends the logic of typicality ALC + TR, based on the notion of rational closure, by inclusions p :: T(C) v D (“we have probability p that typical Cs are Ds”), coming from the distributed semantics of probabilistic Description Logics. Additionally, it embeds a set of cognitive (...) heuristics for concept combination. We show that the complexity of reasoning in our logic is EXPTIME-complete as in ALC. (shrink)

This study is concerned with the acquisition of the disjunction operator, or, in English. Two mutually inconsistent claims have been made about the acquisition of disjunction. One claim is that the acquisition of the adult truth conditions for logical connectives, including disjunction, is a late and not fully universal, achievement. With particular reference to disjunction, the findings from several studies are interpreted as showing that only the truth conditions associated with exclusive-or are available to young children (e.g., Beilin and Lust (...) 1975; Braine and Rumain 1981, 1983; Paris 1973). There is a related claim, that even when children respond as if they have access to a broader range of truth conditions, namely those associated with inclusive-or, children's adult-like responses are the result of a failure to distinguish or from and (Paris 1973). (shrink)

A universally free logic is a system of quantification theory, with or without identity, whose theses remain logically true if the domain of quantification is empty and some of the singular terms present in the language do not denote existing objects. In the West, logics satisfying and ones satisfying were developed starting in the 1950s. But Stanisław Jaśkowski preceded all this work by some twenty years: his paper “On the Rules of Supposition in Formal Logic” of 1934 can (...) be regarded as containing, at least implicitly, the first universally free logic. The system proposed there is an inclusive logic as it is, and a straightforward extension of it is also free. (shrink)

We present a formal system for reasoning about inclusion and exclusion in natural language, following work by MacCartney and Manning. In particular, we show that an extension of the Monotonicity Calculus, augmented by six new type markings, is sufficient to derive novel inferences beyond monotonicity reasoning, and moreover gives rise to an interesting logic of its own. We prove soundness of the resulting calculus and discuss further logical and linguistic issues, including a new connection to the classes of weak, (...) strong, and superstrong negative polarity items. (shrink)

A nonnegative interger is called a number, a collection of numbers a set and a collection of sets a class. We write ε for the set of all numbers, o for the empty set, N(α) for the cardinality of $\alpha, \subset$ for inclusion and $\subset_+$ for proper inclusion. Let α, β 1 ,...,β k be subsets of some set ρ. Then α' stands for ρ-α and β 1 ⋯ β k for β 1 ∩ ⋯ ∩ β k . For (...) subsets α 1 ,..., α r of ρ we write: $\alpha_\sigma - \{x \in v \ (\nabla i) \lbrack i \in \sigma \Rightarrow x \in \alpha_i\rbrack\} \text{for} \sigma \subset (1, \ldots, r),\\ s_i = \sum \{N(\alpha_\sigma) \mid \sigma \subset (1,\ldots, r) \& N(\sigma) = i\}, \text{for} 0 \leqq i \leqq r$ . Note that α 0 = v, hence s 0 = N(v). If the set v is finite, the classical inclusion-exclusion principle (abbreviated IEP) states $(a) N(\alpha_1 \cup \cdots \cup \alpha_r) = \sum^r_{t=1} (-1)^{t-1} s_t = \sum_{o \subset_+\sigma \subset (1,\ldots,r)}$ (b) N(α' 1 ⋯ α' r ) = ∑ r t=0 (-1) t s t = ∑ (-1) N(σ) N (α σ ). In this paper we generalize (a) and (b) to the case where α 1 ,⋯, α r are subsets of some countable but isolated set v. Then the role of the cardinality N(α) of the set α is played by the RET (recursive equivalence type) Req α of α. These generalization of (a) and (b) are proved in § 3. Since they involve recursive distinctness, this notion is discussed in § 2. In § 4l we consider a natural extension of "the sum of the elements of a finite set σ" to the case where σ is countable. § 5 deals with valuations, i.e., certain mappings μ from classes of isolated sets into the collection λ of all isols which permit us to further generalize IEP by substituting μ(α) for $\operatorname{Req} \alpha$. (shrink)

This study is concerned with the acquisition of the disjunction operator, or, in English. Two mutually inconsistent claims have been made about the acquisition of disjunction. One claim is that the acquisition of the adult truth conditions for logical connectives, including disjunction, is a late and not fully universal, achievement. With particular reference to disjunction, the findings from several studies are interpreted as showing that only the truth conditions associated with exclusive-or are available to young children (e.g., Beilin and Lust (...) 1975; Braine and Rumain 1981, 1983; Paris 1973). There is a related claim, that even when children respond as if they have access to a broader range of truth conditions, namely those associated with inclusive-or, children's adult-like responses are the result of a failure to distinguish or from and (Paris 1973). A different conclusion was reached in a recent study by Chierchia, Crain, Guasti and Thornton (1998), who conducted a series of experiments which revealed children's adult-like knowledge of logical connectives, including disjunction. To reconcile the findings from the Chierchia et al. study and those of previous research, Chierchia et al. (1998) draw upon a distinction between semantic and pragmatic knowledge. They argue that the acquisition of logical connectives is simply a process of mapping the appropriate truth conditions (associated with logical connectives in classical logic) onto the particular words of the language to which the child is exposed. In understanding sentences, however, children are also influenced by the pragmatic norms they follow. In many.. (shrink)

How does causation in the physical world relate to implication in logic? This article presents implication as fundamentally a relation of inclusion between propositions. Given this, it is argued that an event cannot “causally imply” another, also given the laws of nature. Then, by applying the notion of inclusion to physical objects, a relation “within the possibilities of” is developed, which generates a partial order on sets of entities and is independent of time. Based on this, it is shown (...) that changes of physical objects in time (at any rate, a great many of them) imply, and thus counterfactually depend on, what we call “causes” – an asymmetric dependence which is robust despite the perspectival nature of the concept of “cause”. (shrink)

Description logic knowledge bases can be used to represent knowledge about a particular domain in a formal and unambiguous manner. Their practical relevance has been shown in many research areas, especially in biology and the Semantic Web. However, the tasks of constructing knowledge bases itself, often performed by human experts, is difficult, time-consuming and expensive. In particular the synthesis of terminological knowledge is a challenge that every expert has to face. Because human experts cannot be omitted completely from the (...) construction of knowledge bases, it would therefore be desirable to at least get some support from machines during this process. To this end, we shall investigate in this work an approach which shall allow us to extract terminological knowledge in the form of general concept inclusions from factual data, where the data is given in the form of vertex- and edge-labelled graphs. Because such graphs appear naturally within the scope of the Semantic Web in the form of sets of Resou... (shrink)

Does the recent success of Podemos and Syriza herald a new era of inclusive, egalitarian left populism? Because leaders of both parties are former students of Ernesto Laclau and cite his account of populism as guiding their political practice, this essay considers whether his theory supports hope for a new kind of populism. For Laclau, the essence of populism is an “empty signifier” that provides a means by which anyone can identify with the people as a whole. However, the (...) concept of the empty signifier is not as neutral as he assumes. As I show by analyzing the role of race in his theory, some subjects are constituted in a way that prevents their unmediated identification with the people. Consequently, Laclau’s view should be read as symptomatic of the problems with populist logic if its adherents are to avoid reproducing its exclusions and practice a more inclusive politics. (shrink)

In this paper I defend three points: (1) God loves and desires the salvation of every human person, (2) saving grace is available outside of the Christian church to those who do not hear the gospel but pursue moral goodness and (3) most, if not all, human persons will be saved. I argue that soteriological restrictivism is logically incoherent since its two ideas—every person is loved by God and only those who hear and believe the Christian gospel can be saved—cannot (...) both be true. Throughout I integrate insights from Karl Rahner’snotion of “anonymous Christianity.”. (shrink)

Gaining information can be modelled as a narrowing of epistemic space . Intuitively, becoming informed that such-and-such is the case rules out certain scenarios or would-be possibilities. Chalmers’s account of epistemic space treats it as a space of a priori possibility and so has trouble in dealing with the information which we intuitively feel can be gained from logical inference. I propose a more inclusive notion of epistemic space, based on Priest’s notion of open worlds yet which contains only (...) those epistemic scenarios which are not obviously impossible. Whether something is obvious is not always a determinate matter and so the resulting picture is of an epistemic space with fuzzy boundaries. (shrink)

This much-needed book provides a thorough account of temporal logic, one of the most important areas of logic in computer science today. The book begins with a solid introduction to semantical and axiomatic approaches to temporal logic. It goes on to cover predicate temporal logic, meta-languages, general theories of axiomatization, many dimensional systems, propositional quantifiers, expressive power, Henkin dimension, temporalization of other logics, and decidability results. With its inclusion of cutting-edge results and unifying methodologies, this book (...) is an indispensable reference for both the pure logician and the theoretical computer scientist. (shrink)

Ample sociological evidence demonstrates that binary gender ideologies are an intractable part of formal organizations and that transgender issues tend to be marginalized by a wide range of social institutions. Yet, in the last 15 years, more than 200 colleges and universities have attempted to ameliorate such realities by adopting gender-inclusive facilities in which students of any gender can share residential and restroom spaces. What cultural logics motivate these transformations? How can their emergence be reconciled with the difficulty of (...) altering the gender order? Using an original sample of 2,036 campus newspaper articles, I find that support for inclusive facilities frames such spaces as a resource through which an institution can claim improved standing in the field of higher education. This process of engendering reputation allows traditional gender separation in residential arrangements to be overcome, but it also situates institutional responsiveness to transgender issues as a means of enhancing a college or university’s public prestige. This, in turn, produces novel status systems in the field of higher education—albeit ones that perpetuate familiar forms of institutional and cultural exclusion. (shrink)

RESUMEN: Durante mucho tiempo las investigaciones sociológicas se centraron en el término exclusión. Existe, sin embargo, un abuso del término designando como tales, situaciones que en realidad responden a la vulnerabilidad creada por la degradación de las relaciones de trabajo, por la precarización o la marginación. Éstas son propiamente situaciones bajo amenaza de exclusión pero no son exclusión propiamente dicha, pueden desembocar en ella pero dependen de otra lógica. La lógica de la exclusión procede por discriminaciones oficiales, la marginación se (...) produce por procesos de desestabilización. En este trabajo se realizará un análisis comparado entre dos perspectivas tan disímiles como las subyacentes en las políticas de inclusión aplicadas en Argentina y Leonardo Polo pues de la confrontación fructífera emergerán las condiciones de posibilidad para una auténtica inclusión social en las organizaciones. SUMMARY: From longtime ago, sociological researches focused on the Word exclusion abusing on it. The fact is that many situations considered as exclusion really are the effect of vulnerability due to degrade of job´s relationships, or due to marginal and precarious situations. Those are situations under danger of exclusion but they are not properly exclusion, those could to end in exclusion but those depend on a different logic which runs by official discriminations. Marginality is consequence of process of unbalances. This article intend to analyzes two different perspectives, that from political plans from Argentinian government and that from Leonardo Polo because it is possible to confront in a positive way and then, many possibilities for authentic social inclusion in organizations will emerge. (shrink)

The Knowability Paradox is a logical argument that, starting from the plainly innocent assumption that every true proposition is knowable, reaches the strong conclusion that every true proposition is known; i.e. if there are unknown truths, there are unknowable truths. The paradox has been considered a problem for every theory assuming the Knowability Principle, according to which all truths are knowable and, in particular, for semantic anti-realist theories. A well known criticism to the Knowability Paradox is the so called restriction (...) strategy. It bounds the scope of the universal quantification in (KP) to a set of formulas whose logical form avoids the paradoxical conclusion. Specifically, Tennant suggests to restrict the quantifier in (KP) to propositions whose knowledge is provably inconsistent. He calls them Anti-Cartesian propositions and distinguished them in three kinds. In this paper we will not be concerned with the soundness of the restriction proposal and the criticisms it has received. Rather, we are interested in analyzing the proposed distinction. We argue that Tennant’s distinction is problematic because it is not completely clear, it is not grounded on an adequate logical analysis, and it is incomplete. We suggest an alternative distinction, and we give some reasons for accepting it: it results logically grounded and more complete than Tennant’s one, inclusive of it and independent from non-epistemic notions. (shrink)

Ever since [4], systems of spheres have been considered to give an intuitive and elegant way to give a semantics for logics of theory- or belief- change. Several authors [5, 11] have considered giving up the rather strong assumption that systems of spheres be linearly ordered by inclusion. These more general structures are called hypertheories after [8]. It is shown that none of the proposed logics induced by these weaker structures are compact and thus cannot be given a strongly complete (...) axiomatization in a finitary logic. Complete infinitary axiomatizations are given for several intuitive logics based on hypertheories that are not linearly ordered by inclusion. (shrink)

Despite increasing evidence that the traditional lecture is inefficient for student learning, such methods remain the central paradigm for teaching logic. In this paper, I identify the deficits of the lecture model and outline the many benefits of flipping the logic classroom—namely that students can absorb information at their own pace, freeing classroom time for active learn- ing activities, and allowing the students to come prepared to actively engage in deeper levels of learning. I provide advice for curricular (...) change from the traditional model, and guidance for flipped classroom implementation. I also offer suggestions for the best use of newly available class time, and advice for keeping students accountable for learning the information prior to class. Last, I consider common challenges with the flipped classroom model. I acknowledge possible obstacles to flipping the undergraduate logic course and address these challenges with potential solutions. (shrink)

Axel Honneth reconstructs Hegel’s social and political philosophy on the basis of the concept of recognition. For Honneth, recognition is a constitutive relation between individuals that is in principle symmetrical. By conceiving recognition through symmetry, Honneth effectively bans the inclusion of power within recognitive relation. He thus regards the relations of power as cases of non-recognition or misrecognition. In this paper, I develop an alternative theory of the constitutive relation between individuals for Hegel, one that is based on the asymmetrical (...) relation of power. To this aim, I focus on the chapter of “determinations of reflection” in the Science of Logic. Through a close analysis of Hegel’s Logic, I argue that the most fundamental form of relation between individuals is the relation of opposition, that individuals are solely constituted in and through the relation of opposition, and that the relation of opposition is essentially asymmetrical. Together, these claims establish... (shrink)

By formalizing recent syntactic theories for natural languages Stabler shows how their complexity can be handled without guesswork or oversimplification. By formalizing recent syntactic theories for natural languages in the tradition of Chomsky's Barriers, Stabler shows how their complexity can be handled without guesswork or oversimplification. He introduces logical representations of these theories together with special deductive techniques for exploring their consequences that will provide linguists with a valuable tool for deriving and testing theoretical predictions and for experimenting with alternative (...) formulations of grammatical principles. Stabler's novel approach allows results to be deduced with straightforward calculations and provides a systematic framework for tackling the problem of how speakers can infer the properties of an utterance from principles of the grammar. The special treatment of equality, induction principles, and inclusion of a general method for collecting structures from proofs means that sophisticated linguistic arguments can be carried out in detail, giving a rich perspective to issues in linguistic theory and parsing. (shrink)

Assertion plays a crucial dual role in Frege's conception of logic, a formal and a transcendental one. A recurrent complaint is that Frege's inclusion of the judgement-stroke in the Begriffsschrift is either in tension with his anti-psychologism or wholly superfluous. Assertion, the objection goes, is at best of merely psychological significance. In this paper, I defend Frege against the objection by giving reasons for recognising the central logical significance of assertion in both its formal and its transcendental role.

In lieu of an abstract, here is a brief excerpt of the content:Beauty, Transcendence, and the Inclusive Hierarchy of Creation1Thomas Joseph White, O.P.Interpreters of Thomas Aquinas have long argued about whether he holds that beauty is a “transcendental,” a feature of reality coextensive with all that exists, like unity, goodness, and truthfulness.2 In the first part of this article, I will argue that Aquinas can [End Page 1215] be read to affirm in an implicit way that beauty is a (...) transcendental. In the second part, I will consider what it might mean from a Thomistic point of view to speak of a transcendent divine beauty, given Aquinas’s metaphysical commitments, particularly with respect to his doctrine of divine simplicity. In the final part, I will treat the question of how the beauty of the creation both manifests and conceals divine beauty.Beauty as a Transcendental Feature of RealityAquinas does not list beauty as a transcendental term in texts on transcendental notions. Perhaps, then, one should simply exclude it from a responsible account of his teaching on this subject. However, at least two well-known texts should give us reason to pause before reaching such a conclusion. One is found in his Commentary on Dionysius’ Divine Names, chapter 4, lectio 5. The other is in his discussion of the beauty of the eternal Son of God in a discussion of the Holy Trinity in the Summa theologiae I, q. 39, a. 8.In the first of these texts, Aquinas is commenting on Dionysius. The extended text is analytically dense. Aquinas is discussing ways in which one might say that God is beautiful and in what ways one might not say so. I will return to his topic below. Here, however, it is pertinent to consider Aquinas’s discussion of the presence of beauty in all that exists. He makes six main points.3 First, all beauty comes [End Page 1216] from God insofar as God is the cause of all that exists. Second, he gives a first definition of beauty: beauty can be defined ontologically as the splendor (claritas) that results from form; everything has a formal determination of some kind insofar as it has existence (esse); therefore, insofar as anything exists (and has some formal ontological content) it has some degree of beauty. Third, the splendor of the form in created things is a participation in the divine splendor from which it originates. The divine nature is the transcendent exemplar of beauty in diverse finite created realities. Fourth, then (and perhaps most importantly), “ex divina pulchritudine esse omnium derivatur”: literally, the existence of everything originates from divine beauty. Fifth, a second definition of beauty is considered: beauty can be defined ontologically as a property of being that emerges from proportion or harmony (consonantia). For example, authentic relationships of personal friendship imply spiritual harmony or concord and are beautiful and noble in this respect. Sixth, then, the concord or beautiful harmonies we find in the created order are expressive of the wisdom of God, who is the author of creation.Evidently, if the existence of everything derives from divine beauty, and if everything that has existence is in some way beautiful by virtue of its intrinsic form, then it would seem to follow logically that beauty, for Aquinas, is a characteristic of being that is coextensive with all that exists. We see a similar idea expressed in the aforementioned passage of ST. Here, however, Aquinas gives a more synthetic definition of beauty in things that combines both the definitions found in our previous discussion, claritas and proportio, but it also adds a third, integritas: ontological integrity or wholeness.Species or beauty has a likeness to the property of the Son. For beauty includes three conditions, “integrity” or “perfection,” [End Page 1217] since those things which are impaired are by the very fact ugly; due “proportion” or “harmony”; and lastly, “brightness” or “clarity,” whence things are called beautiful which have a bright color.4The implication of this point of view is readily apparent. God is essentially beautiful, and God has created all that exists in light of the eternal Word and Wisdom of God, who is the Son. Consequently, all that exists and that... (shrink)

We present a plural logic that is as expressively strong as it can be without sacrificing axiomatisability, axiomatise it, and use it to chart the expressive limits set by axiomatisability. To the standard apparatus of quantification using singular variables our object-language adds plural variables, a predicate expressing inclusion (is/are/is one of/are among), and a plural definite description operator. Axiomatisability demands that plural variables only occur free, but they have a surprisingly important role. Plural description is not eliminable in favour (...) of quantification; on the contrary, quantification is definable in terms of it. Predicates and functors (function signs) can take plural as well as singular terms as arguments, and both many-valued and single-valued functions are expressible. The system accommodates collective as well as distributive predicates, and the condition for a predicate to be distributive is definable within it; similarly for functors. An essential part of the project is to demonstrate the soundness and completeness of the calculus with respect to a semantics that does without set-theoretic domains and in which the use of settheoretic extensions of predicates and functors is replaced by the sui generis relations and functions for which the extensions were at best artificial surrogates. Our metalanguage is designed to solve the difficulties involved in talking plurally about individuals and about the semantic values of plural items. (shrink)