Instead of the half-century old foundational feud between set theory and category theory, this paper argues that they are theories about two different complementary types of universals. The set-theoretic antinomies forced naïve set theory to be reformulated using some iterative notion of a set so that a set would always have higher type or rank than its members. Then the universal u_{F}={x|F(x)} for a property F() could never be self-predicative in the sense of u_{F}∈u_{F}. But the mathematical theory of categories, (...) dating from the mid-twentieth century, includes a theory of always-self-predicative universals--which can be seen as forming the "other bookend" to the never-self-predicative universals of set theory. The self-predicative universals of category theory show that the problem in the antinomies was not self-predication per se, but negated self-predication. They also provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato's Theory of Forms as well as for the idea of a "concrete universal" in Hegel and similar ideas of paradigmatic exemplars in ordinary thought. (shrink)

The principle of alternate possibilities (PAP), making the ability to do otherwise a necessary condition for moral responsibility, is supposed by Harry Frankfurt, John Fischer, and others to succumb to a peculiar kind of counterexample. The paper reviews the main problems with the counterexample that have surfaced over the years, and shows how most can be addressed within the terms of the current debate. But one problem seems ineliminable: because Frankfurt''s example relies on a counterfactual intervener to preclude alternatives to (...) the person''s action, it is not possible for it to preclude all alternatives (intervention that is contingent upon a trigger cannot bring it about that the trigger never occurred). This makes it possible for the determined PAPist to maintain that some pre-intervention deviation is always available to ground moral responsibility. In reply, the critic of PAP can examine all the candidate deviations and argue their irrelevance to moral responsibility (a daunting prospect); or the critic can dispense with counterfactual intervention altogether. The paper pursues the second of these strategies, developing three examples of noncounterfactual intervention in which (i) the agent has no alternatives (and a fortiori no morally relevant alternatives), yet (ii) there is just as much reason to think that the agent is morally responsible as there was in Frankfurt''s original example. The new counterexamples do suffer from one liability, but this is insufficient in the end to repair PAP''s conceptual connection between moral responsibility and alternate possibilities. (shrink)

One strategy in recent discussions of theological fatalism is to draw on Harry Frankfurt’s famous counterexamples to the principle of alternate possibilities (PAP) to defend human freedom from divine foreknowledge. For those who endorse this line, “Frankfurt counterexamples” are supposed to show that PAP is false, and this conclusion is then extended to the foreknowledge case. This makes it critical to determine whether Frankfurt counterexamples perform as advertised, an issue recently debated in this journal via a pair of articles by (...) David Widerker and John Martin Fischer. I suggest that this debate can be avoided: divine foreknowledge is itself aparadigmatic counterexample to PAP, requiring no support from suspect Frankfurt counterexamples. (shrink)

The computational genomics community has come increasingly to rely on the methodology of creating annotations of scientific literature using terms from controlled structured vocabularies such as the Gene Ontology (GO). We here address the question of what such annotations signify and of how they are created by working biologists. Our goal is to promote a better understanding of how the results of experiments are captured in annotations in the hope that this will lead to better representations of biological reality through (...) both the annotation process and ontology development, and in more informed use of the GO resources by experimental scientists. (shrink)

This paper seeks to rehabilitate St. Augustine’s widely dismissed response to the alleged incompatibility of divine foreknowledge and free will. This requires taking a fresh look at his analysis in On Free Choice of the Will, and arguing its relevance to the current debate. Along the way, mistaken interpretations of Augustine are rebutted, his real solution is developed and defended, a reason for his not anticipating Boethius’s a temporalist solution is suggested, a favorable comparison with Ockham is made, rival solutions (...) are rejected, and the aporetic nature of the problem is explained. (shrink)

This is a contribution to an Author Meets Critics special issue on John Martin Fischer's _Our Fate: Essays on God and Free Will_.

The principle that one cannot deliberate over what one already knows is going to happen, when suitably qualified, has seemed to many philosophers to be about as secure a truth as one is likely to find in this life. Fortunately, it poses little restriction on human deliberation, since the conditions which would trigger its prohibition seldom arise for us: our knowledge of the future is intermittent at best, and those things of which we do have advance knowledge are not the (...) sorts of things over which we would deliberate in any case. But matters appear to stand otherwise with an all-knowing agent such as God is traditionally conceived to be; for what an omniprescient deity ‘already knows is going to happen’ is everything that is going to happen; and if He cannot deliberate over such things, there is nothing over which He can deliberate. (shrink)

The paper that follows continues a discussion with Tomis Kapitan in the pages of this journal over the compatibility of divine agency with divine foreknowledge. I had earlier argued against two premises in Kapitan's case for omniscient impotence: (i) that intentionally A-ing presupposes prior acquisition of the intention to A, and (ii) that acquiring the intention to A presupposes prior ignorance whether one will A. In response to my criticisms, Kapitan has recently offered new defences for these two premises. I (...) show in reply why neither defence succeeds in rehabilitating the case against omniscient agency. (Note that the very first sentence of the published version has a misprint: instead of "To claim that divine foreknowledge is inconsistent with divine agency has been pressed with increasing frequency of late . . .", it should read, "_The_ claim . . ."!). (shrink)

I distinguish between a _metaphysical_ problem generated by the argument for theological fatalism, and a _theological_ problem posed by the argument. Some responses to the argument, including ones associated with Boethius, Aquinas and Ockham, address only the theological problem. Even if such responses succeed in showing that God's foreknowledge doesn't threaten human freedom, they fail to take the full measure of the argument for theological fatalism, since the metaphysical problem remains to be solved.

I first argue that the Matrix trilogy is a Gnostic cyber-epic; I then use this interpretive lens to review the films' treatment of fundamental questions in epistemology, metaphysics, and value theory.

The paper explores possible resources available to Plotinus for responding to Plato's famous "Third Man Argument" in the _Parmenides_.

Can free agency exist within a Minkowskian "block universe"? A negative answer to this question has been labeled 'chronogeometrical fatalism'. I look at five theses associated with Minkowskian space-time which have been thought to entail chronogeometrical fatalism, and argue that none of them delivers the goods.

According to the thesis of divine ‘middle knowledge’, first propounded by the Jesuit theologian Luis de Molina in the sixteenth century, subjunctive conditionals stating how free agents would freely respond under counter-factual conditions may be straightforwardly true, and thus serve as the objects of divine knowledge. This thesis has provoked considerable controversy, and the recent revival of interest in middle knowledge, initiated by Anthony Kenny, Robert Adams and Alvin Plantinga in the 1970s, has led to two ongoing debates. One is (...) a theoretical debate over the very intelligibility of middle knowledge;1 the other is a practical debate over its philosophical and theological utility.2. (shrink)

The traditional doctrine of divine omniscience ascribes to God the fully exercised power to know all truths. but why is God’s excellence with respect to knowing not treated on a par with his excellence with respect to doing, where the latter requires only that God have the power to do all things? The prima facie problem with divine ”omni-knowledgeability’ -- roughly, being able to know whatever one wants to know whenever one wants to know it -- is that knowledge requires (...) an internal representation, whereas mere ”knowledgeability’ does not. I argue to the contrary that knowledge does not require an internal representation, and that even if it did, an omni-knowledgeable God would satisfy this requirement. omni-knowledgeability therefore represents a distinct understanding of God’s cognitive excellence while satisfying the traditional insistence on full omniscience. (shrink)

The most serious challenge to Frankfurt-type counterexamples to the Principle of Alternate Possibilities (PAP) comes in the form of a dilemma: either the counterexample presupposes determinism, in which case it begs the question; or it does not presuppose determinism, in which case it fails to deliver on its promise to eliminate all alternatives that might plausibly be thought to satisfy PAP. I respond to this challenge with a counterexample in which considering an alternative course of action is a necessary condition (...) for deciding to act otherwise, and the agent does not in fact consider the alternative. I call this a “buffer case,” because the morally relevant alternative is “buffered” by the requirement that the agent first consider the alternative. Suppose further that the agent’s considering an alternative action—entering the buffer zone—is what would trigger the counterfactual intervener. Then it would appear that PAP-relevant alternatives are out of reach. I defend this counterexample to PAP against three objections: that considering an alternative is itself a morally relevant alternative; that buffer cases can be shown to contain other alternatives that arguably satisfy PAP; and that even if the agent’s present access to PAP-relevant alternatives were eliminated, PAP could still be satisfied in virtue of earlier alternatives. I conclude that alternative possibilities are a normal symptom, but not an essential constituent, of moral agency. (shrink)

Open theists deny that God knows future contingents. Most open theists justify this denial by adopting the position that there are no future contingent truths to be known. In this paper we examine some of the arguments put forward for this position in two recent articles in this journal, one by Dale Tuggy and one by Alan Rhoda, Gregory Boyd, and Thomas Belt. The arguments concern time, modality, and the semantics of ‘will’ statements. We explain why we find none of (...) these arguments persuasive. This wide road leads only to destruction. (shrink)

Classical logic is usually interpreted as the logic of propositions. But from Boole's original development up to modern categorical logic, there has always been the alternative interpretation of classical logic as the logic of subsets of any given (nonempty) universe set. Partitions on a universe set are dual to subsets of a universe set in the sense of the reverse-the-arrows category-theoretic duality--which is reflected in the duality between quotient objects and subobjects throughout algebra. Hence the idea arises of a dual (...) logic of partitions. That dual logic is described here. Partition logic is at the same mathematical level as subset logic since models for both are constructed from (partitions on or subsets of) arbitrary unstructured sets with no ordering relations, compatibility or accessibility relations, or topologies on the sets. Just as Boole developed logical finite probability theory as a quantitative treatment of subset logic, applying the analogous mathematical steps to partition logic yields a logical notion of entropy so that information theory can be refounded on partition logic. But the biggest application is that when partition logic and the accompanying logical information theory are "lifted" to complex vector spaces, then the mathematical framework of quantum mechanics is obtained. Partition logic models indefiniteness (i.e., numerical attributes on a set become more definite as the inverse-image partition becomes more refined) while subset logic models the definiteness of classical physics (an entity either definitely has a property or definitely does not). Hence partition logic provides the backstory so the old idea of "objective indefiniteness" in QM can be fleshed out to a full interpretation of quantum mechanics. (shrink)

Categorical logic has shown that modern logic is essentially the logic of subsets (or "subobjects"). Partitions are dual to subsets so there is a dual logic of partitions where a "distinction" [an ordered pair of distinct elements (u,u′) from the universe U ] is dual to an "element". An element being in a subset is analogous to a partition π on U making a distinction, i.e., if u and u′ were in different blocks of π. Subset logic leads to finite (...) probability theory by taking the (Laplacian) probability as the normalized size of each subset-event of a finite universe. The analogous step in the logic of partitions is to assign to a partition the number of distinctions made by a partition normalized by the total number of ordered pairs |U|² from the finite universe. That yields a notion of "logical entropy" for partitions and a "logical information theory." The logical theory directly counts the (normalized) number of distinctions in a partition while Shannon's theory gives the average number of binary partitions needed to make those same distinctions. Thus the logical theory is seen as providing a conceptual underpinning for Shannon's theory based on the logical notion of "distinctions.". (shrink)

Liberal-contractarian philosophies of justice see the unjust systems of slavery and autocracy in the past as being based on coercion—whereas the social order in modern democratic market societies is based on consent and contract. However, the ‘best’ case for slavery and autocracy in the past were consent-based contractarian arguments. Hence, our first task is to recover those ‘forgotten’ apologia for slavery and autocracy. To counter those consent-based arguments, the historical anti-slavery and democratic movements developed a theory of inalienable rights. Our (...) second task is to recover that theory and to consider several other applications of the theory. Finally, the liberal theories of justice expounded by John Rawls and by Robert Nozick are briefly examined from this perspective. (shrink)

After Marx, dissenting economics almost always used 'the labour theory' as a theory of value. This paper develops a modern treatment of the alternative labour theory of property that is essentially the property theoretic application of the juridical principle of responsibility: impute legal responsibility in accordance with who was in fact responsible. To understand descriptively how assets and liabilities are appropriated in normal production, a 'fundamental myth' needs to be cleared away, and then the market mechanism of appropriation can be (...) understood. On the normative side, neoclassical theory represents marginal productivity theory as showing that (a metaphorical version of) the imputation principle is satisfied ('people get what they produce') in competitive enterprises. Since that shows the moral commitment of neoclassical economics to the imputation principle, the labour theory of property is presented here as the actual non-metaphorical application of the imputation principle to property appropriation. The property-theoretic analysis at the firm level shows how the neoclassical (and much heterodox) analysis in terms of 'distributive shares' wholly misframed the basic questions. Finally, the paper shows how the imputation principle (modernised labour theory of property) is systematically violated in the present wage labour system of renting persons. The paper can be seen as taking up the recent challenge posed by Donald Katzner for a dialogue between neoclassical and heterodox microeconomics. (shrink)

The purpose of this paper is to show that the mathematics of quantum mechanics is the mathematics of set partitions linearized to vector spaces, particularly in Hilbert spaces. That is, the math of QM is the Hilbert space version of the math to describe objective indefiniteness that at the set level is the math of partitions. The key analytical concepts are definiteness versus indefiniteness, distinctions versus indistinctions, and distinguishability versus indistinguishability. The key machinery to go from indefinite to more definite (...) states is the partition join operation at the set level that prefigures at the quantum level projective measurement as well as the formation of maximally-definite state descriptions by Dirac’s Complete Sets of Commuting Operators. This development is measured quantitatively by logical entropy at the set level and by quantum logical entropy at the quantum level. This follow-the-math approach supports the Literal Interpretation of QM—as advocated by Abner Shimony among others which sees a reality of objective indefiniteness that is quite different from the common sense and classical view of reality as being “definite all the way down”. (shrink)

Logical information theory is the quantitative version of the logic of partitions just as logical probability theory is the quantitative version of the dual Boolean logic of subsets. The resulting notion of information is about distinctions, differences and distinguishability and is formalized using the distinctions of a partition. All the definitions of simple, joint, conditional and mutual entropy of Shannon information theory are derived by a uniform transformation from the corresponding definitions at the logical level. The purpose of this paper (...) is to give the direct generalization to quantum logical information theory that similarly focuses on the pairs of eigenstates distinguished by an observable, i.e., qudits of an observable. The fundamental theorem for quantum logical entropy and measurement establishes a direct quantitative connection between the increase in quantum logical entropy due to a projective measurement and the eigenstates that are distinguished by the measurement. Both the classical and quantum versions of logical entropy have simple interpretations as “two-draw” probabilities for distinctions. The conclusion is that quantum logical entropy is the simple and natural notion of information for quantum information theory focusing on the distinguishing of quantum states. (shrink)

The new logic of partitions is dual to the usual Boolean logic of subsets (usually presented only in the special case of the logic of propositions) in the sense that partitions and subsets are category-theoretic duals. The new information measure of logical entropy is the normalized quantitative version of partitions. The new approach to interpreting quantum mechanics (QM) is showing that the mathematics (not the physics) of QM is the linearized Hilbert space version of the mathematics of partitions. Or, putting (...) it the other way around, the math of partitions is a skeletal version of the math of QM. The key concepts throughout this progression from logic, to logical information, to quantum theory are distinctions versus indistinctions, definiteness versus indefiniteness, or distinguishability versus indistinguishability. The distinctions of a partition are the ordered pairs of elements from the underlying set that are in different blocks of the partition and logical entropy is defined (initially) as the normalized number of distinctions. The cognate notions of definiteness and distinguishability run throughout the math of QM, e.g., in the key non-classical notion of superposition (= ontic indefiniteness) and in the Feynman rules for adding amplitudes (indistinguishable alternatives) versus adding probabilities (distinguishable alternatives). (shrink)

Modern categorical logic as well as the Kripke and topological models of intuitionistic logic suggest that the interpretation of ordinary “propositional” logic should in general be the logic of subsets of a given universe set. Partitions on a set are dual to subsets of a set in the sense of the category-theoretic duality of epimorphisms and monomorphisms—which is reflected in the duality between quotient objects and subobjects throughout algebra. If “propositional” logic is thus seen as the logic of subsets of (...) a universe set, then the question naturally arises of a dual logic of partitions on a universe set. This paper is an introduction to that logic of partitions dual to classical subset logic. The paper goes from basic concepts up through the correctness and completeness theorems for a tableau system of partition logic. (shrink)

The logical basis for information theory is the newly developed logic of partitions that is dual to the usual Boolean logic of subsets. The key concept is a "distinction" of a partition, an ordered pair of elements in distinct blocks of the partition. The logical concept of entropy based on partition logic is the normalized counting measure of the set of distinctions of a partition on a finite set--just as the usual logical notion of probability based on the Boolean logic (...) of subsets is the normalized counting measure of the subsets (events). Thus logical entropy is a measure on the set of ordered pairs, and all the compound notions of entropy (join entropy, conditional entropy, and mutual information) arise in the usual way from the measure (e.g., the inclusion-exclusion principle)--just like the corresponding notions of probability. The usual Shannon entropy of a partition is developed by replacing the normalized count of distinctions (dits) by the average number of binary partitions (bits) necessary to make all the distinctions of the partition. (shrink)

A theory of property needs to give an account of the whole life-cycle of a property right: how it is initiated, transferred, and terminated. Economics has focused on the transfers in the market and has almost completely neglected the question of the initiation and termination of property in normal production and consumption (not in some original state or in the transition from common to private property). The institutional mechanism for the normal initiation and termination of property is an invisible-hand function (...) of the market, the market mechanism of appropriation. Does this mechanism satisfy an appropriate normative principle? The standard normative juridical principle is to assign or impute legal responsibility according to de facto responsibility. It is given a historical tag of being "Lockean" but the basis is contemporary jurisprudence, not historical exegesis. Then the fundamental theorem of the property mechanism is proven which shows that if "Hume's conditions" (no transfers without consent and all contracts fulfilled) are satisfied, then the market automatically satisfies the Lockean responsibility principle, i.e., "Hume implies Locke." As a major application, the results in their contrapositive form, "Not Locke implies Not Hume," are applied to a market economy based on the employment contract. It is shown the production based on the employment contract violates the Lockean principle (all who work in an employment enterprise are de facto responsible for the positive and negative results) and thus Hume's conditions must also be violated in the marketplace (de facto responsible human action cannot be transferred from one person to another—as is readily recognized when and employer and employee together commit a crime). (shrink)