The paper shows that in the Art of Thinking (The Port Royal Logic) Arnauld and Nicole introduce a new way to state the truth-conditions for categorical propositions. The definition uses two new ideas: the notion of distributive or, as they call it, universal term, which they abstract from distributive supposition in medieval logic, and their own version of what is now called a conservative quantifier in general quantification theory. Contrary to the interpretation of Jean-Claude Parienté and others, the truth-conditions do (...) not require the introduction of a new concept of ?indefinite? term restriction because the notion of conservative quantifier is formulated in terms of the standard notion of term intersection. The discussion shows the following. Distributive supposition could not be used in an analysis of truth because it is explained in terms of entailment, and entailment in terms of truth. By abstracting from semantic identities that underlie distribution, the new concept of distributive term is definitionally prior to truth and can, therefore, be used in a non-circular way to state truth-conditions. Using only standard restriction, the Logic?s truth-conditions for the categorical propositions are stated solely in terms of (1) universal (distributive) term, (2) conservative quantifier, and (3) affirmative and negative proposition. It is explained why the Cartesian notion of extension as a set of ideas is in this context equivalent to medieval and modern notions of extension. (shrink)
The paper explores the existential import of universal affirmative in Descartes, Arnauld and Malebranche. Descartes holds, inconsistently, that eternal truths are true even if the subject term is empty but that a proposition with a false idea as subject is false. Malebranche extends Descartes? truth-conditions for eternal truths, which lack existential import, to all knowledge, allowing only for non-propositional knowledge of contingent existence. Malebranche's rather implausible Neoplatonic semantics is detailed as consisting of three key semantic relations: illumination by which God's (...) ideas cause mental terms, creation by which God's ideas cause material substances by a kind of ?ontic privation?, and sensation in which brain events occasion states of mental awareness. In contrast, Arnauld distinguishes two types of propositions ? necessary and contingent ? with distinct truth-conditions, one with and one without existential import. Arnauld's more modern semantics is laid out as a theory of reference that substitutes earlier causal accounts with one that adapts the medieval notion of objective being. His version anticipates modern notions of intentional content and appeals in its ontology only to substances and their modes. (shrink)
Though acknowledged by scholars, Plato’s identification of the Beautiful and the Good has generated little interest, even in aesthetics where the moral concepts are a current topic. The view is suspect because, e.g., it is easy to find examples of ugly saints and beautiful sinners. In this paper the thesis is defended using ideas from Plato’s ancient commentators, the Neoplatonists. Most interesting is Proclus, who applied to value theory a battery of linguistic tools with fixed semantic properties—comparative adjectives, associated gradable (...) adjectives, mass nouns, and predicate negations—all with a semantics that demand a privative scale of value. It is shown how it is perfectly possible to interpret value terms Platonically over privative Boolean algebras so that beautifuland good diverge while at higher levels other value terms are coextensional. Considerations are offered that this structure conforms to actual usage. (shrink)
The mediaeval logic of Aristotelian privation, represented by Ockham's expositionof All A is non-P as All S is of a type T that is naturally P and no S is P, iscritically evaluated as an account of privative negation. It is argued that there aretwo senses of privative negation: (1) an intensifier (as in subhuman), the dualof Neoplatonic hypernegation (superhuman), which is studied in linguistics asan operator on scalar adjectives, and (2) a (often lexicalized) Boolean complementrelative to the extension of (...) a privative negation in sense (1) (e.g., Brute). Thissecond sense, which is the privative negation discussed in modern linguistics, isshown to be Aristotle's. It is argued that Ockham's exposition fails to capture muchof the logic of Aristotelian privation due to limitations in the expressive power of thesyllogistic. (shrink)
This paper explores the modal interpretation of ?ukasiewicz's n -truth-values, his conditional and the puzzles they generate by exploring his suggestion that by ?necessity? he intends the concept used in traditional philosophy. Scalar adjectives form families with nested extensions over the left and right fields of an ordering relation described by an associated comparative adjective. Associated is a privative negation that reverses the ?rank? of a predicate within the field. If the scalar semantics is interpreted over a totally ordered domain (...) of cardinality n and metric ?, an n-valued Lukasiewicz algebra is definable. Privation is analysed in terms of non-scalar adjectives. Any Boolean algebra of 2 n ?properties? determines an n + 1 valued Lukasiewicz algebra. The Neoplatonic ?hierarchy of Being? is essentially the order presupposed by natural language modal scalars. ?ukasiewicz's ≈ is privative negation, and ? proves to stand for the extensional (antitonic) dual if ? then for scalar adjectives, especially modals. Relations to product logics and frequency interpretations of probability are sketched. (shrink)
An investigation of Proclus' logic of the syllogistic and of negations in the Elements of Theology, On the Parmenides, and Platonic Theology. It is shown that Proclus employs interpretations over a linear semantic structure with operators for scalar negations (hypernegation/alpha-intensivum and privative negation). A natural deduction system for scalar negations and the classical syllogistic (as reconstructed by Corcoran and Smiley) is shown to be sound and complete for the non-Boolean linear structures. It is explained how Proclus' syllogistic presupposes converting the (...) tree of genera and species from Plato's diairesis into the Neoplatonic linear hierarchy of Being by use of scalar hyper and privative negations. (shrink)
The paper is a study of the logic of existence, negation, and order in the Neoplatonic tradition. The central idea is that Neoplatonists assume a logic in which the existence predicate is a comparative adjective and in which monadic predicates function as scalar adjectives that nest the background order. Various scalar predicate negations are then identifiable with various Neoplatonic negations, including a privative negation appropriate for the lower orders of reality and a hyper-negation appropriate for the higher. Reversion to the (...) One can then be explained as the logical inference of hyper-negations from mundane knowledge. Part I develops the relevant linguistic and logical theory, and Part II defends Wolfson and the scalar interpretation against the more traditional Aristotelian understanding of Whittaker and others of reversion as intensional abstraction. (shrink)
Using concepts from abstract algebra and type theory, I analyze the structural presuppositions of any holistic ethical theory. This study is motivated by such recent holistic theories in environmental ethics as Aldo Leopold’s land ethic, James E. Lovelock’s Gaia hypothesis, Arne Naess’ deep ecology, and various aesthetic ethics of the sublime. I also discuss the holistic and type theoretic assumptions of suchstandard ethical theories as hedonism, natural rights theory, utilitarianism, Rawls’ difference principle, and fascism. I argue that although there are (...) several common senses of part-whole in ethical theory, the central sense of holism in ethics is that of a theory that defines its key moral idea as an emergent group property grounded in the relational properties of its individual constituents. Hedonism and Kantianism do not count as holistic in this sense. Natural rights theory does in adegenerate way. Utilitarianism and various environmental ethics are paradigm examples. I point out as a general structural weakness of environmental holistic theories that their first-order grounding in nonmoral vocabulary seems to preclude an explanation of many moral intuitions about human ethics. (shrink)
An interpretation in modal and tense logic is proposed for Boethius's reconciliation of God's foreknowledge with human freedom from The consolation of philosophy, Book V. The interpretation incorporates a suggestion by Paul Spade that God's special status in time be explained as a restriction of God's knowledge to eternal sentences. The argument proves valid, and the seeming restriction on omnipotence is mitigated by the very strong expressive power of eternal sentences.
Quantifiers in Frege's Grundgesetze like are not well-defined because the part Fx & Gx stands for a concept but the yoking conjunction is horizontalised and must stand for a truth-value. This standard interpretation is rejected in favor of a substitutional reading that, it is argued, both conforms better to the text and is well-defined. The theory of the horizontal is investigated in detail and the composite reading of Frege's connectives as made up of horizontals is rejected. The sense in which (...) the Grundgesetze has a many-valued but classical logic is explained by proving that its propositional fragment (under the standard interpretation) and Bochvar's 3-valued logic are instances of the same metatheoretic methods. Historically, the paper argues for a more naive but well-defined reading of the text. Theoretically, it provides a formally adequate statement of that semantics, as well as developing the abstract metatheory which embraces Bochvar's language and the Fregean fragment. (shrink)
An analysis is proposed for the common argument that something should be preserved because it is irreplaceable. The argument is shown to depend on modal elements in irreplaceable, existence assumptions of preserve, and the logic of obligation. In terms of this theory it is argued that utilitarianism can account for most, but not all instances of persuasive appeals to irreplaceability. Beingessentially backwards looking, utilitarianism cannot in principle justify preservation of objects irreplaceable because of their history or genesis.