Spinoza on Having a False Idea

Naturalism pervades Spinoza’s doctrines of The Ethics, but the contours of it often bewilder us. In this light, I consider the account of falsity, or having a false idea, as presented by Spinoza in Proposition thirty_five of the Second Part, its demonstration, and the subsequent note. Based on my interpretation I argue for the claim that his account has coherence and makes sense. Further, I examine the significance of what Spinoza says about falsity for comprehension of his philosophy overall, especially as regards its contrasts with the philosophy of Descartes.     

Towards a More General Concept of Inference

The main objective of this paper is to sketch unifying conceptual and formal framework for inference that is able to explain various proof techniques without implicitly changing the underlying notion of inference rules. We base this framework upon the so-called two-dimensional, i.e., deduction to deduction, account of inference introduced by Tichý in his seminal work The Foundation’s of Frege’s Logic (1988). Consequently, it will be argued that sequent calculus provides suitable basis for such general concept of inference and therefore should not be seen just as technical tool, but philosophically well-founded system that can rival natural deduction in terms of its “naturalness”.     

Russell on Spinoza’s Substance Monism

Russell??s critique of substance monism is an ideal starting point from which to understand some main concepts in Spinoza??s difficult metaphysics. This paper provides an in-depth examination of Spinoza??s proof that only one substance exists. On this basis, it rejects Russell??s interpretation of Spinoza??s theory of reality as founded upon the logical doctrine that all propositions consist of a predicate and a subject. An alternative interpretation is offered: Spinoza??s substance is not a bearer of properties, as Russell implied, but an eternally active, self-actualizing creative power. Eventually, Spinoza the Monist and Russell the Pluralist are at one in holding that process and activity rather than enduring things are the most fundamental realities.     

Hume’s Argument for the Ontological Independence of Simple Properties

In this paper, I will reconstruct Hume's argument for the ontological (in the sense of rigid existential) independence of simple properties in A Treatise of Human Nature, Book 1 (1739). According to my reconstruction, the main premises of the argument are the real distinctness of every perception of a simple property, Hume's Separability Principle and his Conceivability Principle. In my view, Hume grounds the real distinctness of every perception of a simple property in his atomistic theory of sense perception and his Copy Principle. I will also show why Hume's argument should be seen as relevant nowadays. David Lewis and his followers in metaphysics continue Hume's line of thinking in this respect, which is opposed by power ontologists (Brian Ellis, Stephen Mumford), for example.     

Bolzano’s Argument for the Existence of Substances: a Formalization with Two Types of Predication

The topic of our analysis is the argument for the existence of substances given by Bernard Bolzano in Athanasia (1827), where he essentially employs two ontological categories: substance and adherence. Bolzano considers the real and conditioned Inbegriff of all adherences, which are wirklich and nicht selbst bestehen. He claims that the formed collection is dependent on something external and non-adherential, which therefore is a substance. Bolzano’s argumentation turns out to be structurally similar to his argument for the existence of God from Lehrbuch der Religionswissenschaft (1834), but in each of these reasonings, we find different plausible interpretations of the key concept “Inbegriff”. The latter argumentation refers to the mereological totality of existentially conditioned objects. We propose the explication of the Bolzanian Inbegriff of all adherences using two types of predication: we consider its extension as composed of certain intensional counterparts of adherences. In our approach, we use a fragment of the theory of abstract objects formulated by E. Zalta (1983), describing two different relations between individuals and properties: extensional exemplification and intensional encoding. We put our reconstruction in a wider context of Bolzano’s ontology, formulating the needed axioms with two primitive predicates of second order ... is an adherence, ... is conditioned by something real as well as the conditionally introduced first order predicate constant \(\mathfrak {In}\) for Inbegriff of all adherential ideas. Finally, we sketch a model for our theory.     

A Logical Analysis of the Anselm’s Unum Argumentum (from <Emphasis Type="BoldItalic">Proslogion</Emphasis>)

Anselm of Cantorbery wrote Proslogion (1077–1078), where is formulated the famous ‘Unum argumentum’ about the existence of God. This argument was been disputed and criticized by numerous logicians from an extensional view point. The classical predicate logic is not able to give a formal frame to develop an adequate analysis of this argument. According to us, this argument is not an ontological proof; it analyses the meaning of the “quo nihil maius cogitari posit”, a characterization of God, and establish, by absurd, that “quod non posit cogitare non esse” by using the hypothesis: “to think in re” is taller than “to think in solo intelectu”. We discuss this relation and the difference between the meanings of the elementary predicates ‘to be in re’, ‘to be in intellectu’ and ‘to be in solo intellectu’. We propose a new logical approach of this ‘Unum argumentum’ of Anselm by using Curry’s Combinatory Logic (1958, 1973). Indeed, Combinatory Logic is an abstract applicative formalism of operators applied to operands; in this formalism, the predicates, viewed as specific operators, can be composed and can be transformed, by an intrinsic way, into more complex predicates, by means of abstract operators, called “combinators”, studied by Combinatory Logic. We show that this formalism is a logical frame where it becomes possible to discuss and to formulate cognitive representations of the meanings of predicates used inside of the ‘Unum argumentum’ and to explain how the argument runs.     

Immense Multiple Realization

In his latest book Physicalism, or Something near Enough, Jaegwon Kim argues that his version of functional reductionism is the most promising way for saving mental causation. I argue, on the other hand, that there is an internal tension in his position: Functional reductionism does not save mental causation if Kim’s own supervenience argument is sound. My line of reasoning has the following steps: (1) I discuss the supervenience argument and I explain how it motivates Kim’s functional reductionism; (2) I present what I call immense multiple realization, which says that macro-properties are immensely multiply realized in determinate micro-based properties; (3) on that background I argue that functional reductionism leads to a specified kind of irrealism for mental properties. Assuming that such irrealism is part of Kim’s view, which Kim himself seems to acknowledge, I argue that Kim’s position gets the counterfactual dependencies between macro-causal relata wrong. Consequently, his position does not give a conservative account of mental causation. I end the paper by discussing some alternative moves that Kim seems to find viable in his latest book. I argue on the assumption that the supervenience argument is sound, so the discussion provides further reasons to critically reevaluate that argument because it generalizes in deeply problematic ways.     

Leibniz’s Ontological Proof of the Existence of God and the Problem of »Impossible Objects«

The core idea of the ontological proof is to show that the concept of existence is somehow contained in the concept of God, and that therefore God’s existence can be logically derived—without any further assumptions about the external world—from the very idea, or definition, of God. Now, G.W. Leibniz has argued repeatedly that the traditional versions of the ontological proof are not fully conclusive, because they rest on the tacit assumption that the concept of God is possible, i.e. free from contradiction. A complete proof will rather have to consist of two parts. First, a proof of premise

1. (1)
   God is possible.
    

Second, a demonstration of the “remarkable proposition”

1. (2)
   If God is possible, then God exists.
    

The present contribution investigates an interesting paper in which Leibniz tries to prove proposition (2). It will be argued that the underlying idea of God as a necessary being has to be interpreted with the help of a distinguished predicate letter ‘E’ (denoting the concept of existence) as follows:

1. (3)
   \(g=_{\mathrm{df}} \,\upiota x\square E(x)\).
    

Proposition (2) which Leibniz considered as “the best fruit of the entire logic” can then be formalized as follows:

1. (4)
   \(\diamondsuit E(\upiota x\square E(x)) \rightarrow \, E(\upiota x\square E(x))\).
    

At first sight, Leibniz’s proof appears to be formally correct; but a closer examination reveals an ambiguity in his use of the modal notions. According to (4), the possibility of the necessary being has to be understood in the sense of something which possibly exists. However, in other places of his proof, Leibniz interprets the assumption that the necessary being is impossible in the diverging sense of something which involves a contradiction. Furthermore, Leibniz believes that an »impossible thing«, y, is such that contradictory propositions like \(\hbox {F}(y)\) and \(\lnot F(y)\) might both be true of y. It will be argued that the latter assumption is incompatible with Leibniz’s general views about logic and that the crucial proof is better reinterpreted as dealing with the necessity, possibility, and impossibility of concepts rather than of objects. In this case, the counterpart of (2) turns out to be a theorem of Leibniz’s second order logic of concepts; but in order to obtain a full demonstration of the existence of God, the counterpart of (1), i.e. the self-consistency of the concept of a necessary being, remains to be proven.

     

Dispositions, Laws, and Categories

After a short sketch of Lowe’s account of his four basic categories, I discuss his theory of formal ontological relations and how Lowe wants to account for dispositional predications. I argue that on the ontic level Lowe is a pan-categoricalist, while he is a language dualist and an exemplification dualist with regard to the dispositional/categorical distinction. I argue that Lowe does not present an adequate account of disposition. From an Aristotelian point of view, Lowe conflates dispositional predication with hôs epi to poly statements about what is normally or mostly the case.     

Analyticity and the Deviant Logician: Williamson’s Argument from Disagreement

One way to discredit the suggestion that a statement is true just in virtue of its meaning is to observe that its truth is the subject of genuine disagreement. By appealing to the case of the unorthodox philosopher, Timothy Williamson has recast this response as an argument foreclosing any appeal to analyticity. Reconciling Quine’s epistemological holism with his treatment of the ‘deviant logician’, I show that we may discharge the demands of charitable interpretation even while attributing trivial semantic error to Williamson’s philosophers. Williamson’s effort to generalize the argument from disagreement therefore fails.     

Confirmation,Increase in Probability,and the Likelihood Ratio Measure: a Reply to Glass and McCartney

Bayesian confirmation theory is rife with confirmation measures. Zalabardo (2009) focuses on the probability difference measure, the probability ratio measure, the likelihood difference measure, and the likelihood ratio measure. He argues that the likelihood ratio measure is adequate, but each of the other three measures is not. He argues for this by setting out three adequacy conditions on confirmation measures and arguing in effect that all of them are met by the likelihood ratio measure but not by any of the other three measures. Glass and McCartney (2015), hereafter “G&M,” accept the conclusion of Zalabardo’s argument along with each of the premises in it. They nonetheless try to improve on Zalabardo’s argument by replacing his third adequacy condition with a weaker condition. They do this because of a worry to the effect that Zalabardo’s third adequacy condition runs counter to the idea behind his first adequacy condition. G&M have in mind confirmation in the sense of increase in probability: the degree to which E confirms H is a matter of the degree to which E increases H’s probability. I call this sense of confirmation “IP.” I set out four ways of precisifying IP. I call them “IP1,” “IP2,” “IP3,” and “IP4.” Each of them is based on the assumption that the degree to which E increases H’s probability is a matter of the distance between p(H | E) and a certain other probability involving H. I then evaluate G&M’s argument (with a minor fix) in light of them.     

Bloom’s inequality: commutators in a two-weight setting

In 1985, Bloom characterized the boundedness of the commutator [b, H] as a map between a pair of weighted L^p spaces, where both weights are in A_p. The characterization is in terms of a novel BMO condition. We give a ‘modern’ proof of this result, in the case of p = 2. In a subsequent paper, this argument will be used to generalize Bloom’s result to all Calderón–Zygmund operators and dimensions.     

Sharp upper diameter bounds for compact shrinking Ricci solitons

We give a sharp upper diameter bound for a compact shrinking Ricci soliton in terms of its scalar curvature integral and the Perelman’s entropy functional. The sharp cases could occur at round spheres. The proof mainly relies on a sharp logarithmic Sobolev inequality of gradient shrinking Ricci solitons and a Vitali-type covering argument.

     

Birational p-adic Galois sections in higher dimensions

This note explores the consequences of Koenigsmann’s model theoretic argument from the proof of the birational p-adic section conjecture for curves in the context of higher dimensional varieties over p-adic local fields.     

On a Theorem of Barker–Puig that Refines Alperin’s Weight Conjecture for Blocks in p-Solvable Finite Groups via Clifford Theory

We present a much shorter Clifford Theoretic Proof of an important Theorem of Barker and Puig on a refinement of Alperin’s Weight Conjecture for Blocks in p-solvable finite groups. Our proof employs a standard Clifford Theoretic approach. We also demonstrate a relationship between the Green correspondence and Sibley’s concept of a vertex pair. Consequently the main theorem can be stated in terms of Sibley’s vertex pairs.     

Brouwer and Euclid

We explore the relationship between Brouwer’s intuitionistic mathematics and Euclidean geometry. Brouwer wrote a paper in 1949 called The contradictority of elementary geometry. In that paper, he showed that a certain classical consequence of the parallel postulate implies Markov’s principle, which he found intuitionistically unacceptable. But Euclid’s geometry, having served as a beacon of clear and correct reasoning for two millennia, is not so easily discarded.Brouwer started from a “theorem” that is not in Euclid, and requires Markov’s principle for its proof. That means that Brouwer’s paper did not address the question whether Euclid’s Elements really requires Markov’s principle. In this paper we show that there is a coherent theory of “non-Markovian Euclidean geometry”. We show in some detail that our theory is an adequate formal rendering of (at least) Euclid’s Book I, and suffices to define geometric arithmetic, thus refining the author’s previous investigations (which include Markov’s principle as an axiom).Philosophically, Brouwer’s proof that his version of the parallel postulate implies Markov’s principle could be read just as well as geometric evidence for the truth of Markov’s principle, if one thinks the geometrical “intersection theorem” with which Brouwer started is geometrically evident.     

Triangles, Schemes and Worlds: Reply to Nulty

Nulty proposes a “Davidsonian” argument for metaphysical pluralism, the thesis that there are (or could be) many actual worlds, which appeals to the possibility of alien forms of triangulation. I dispute Nulty’s reading of Davidson on two important points: Davidson’s attack on the notion of a conceptual scheme is not, as Nulty thinks, directed at pluralism, and his understanding of the notions of objective truth and reality is at odds with the conception needed for Nulty’s argument. I also show that the pluralist argument fails on its own terms as it requires an assimilation of worlds to worldviews. But there is much of value in Nulty’s paper despite these important flaws. When the confusions are cleared up, we are left with an intriguing and novel line of argument for conceptual relativism.     

An Axiomatisation of the Conditionals of Post's Many Valued Logics

The paper provides a method for a uniform complete Hilbert-style axiomatisation of Post's (m, u)-conditionals and Post's negation, where m is the number of truth values and u is the number of designated truth values (cf. [5]). The main feature of the technique which we employ in this proof generalises the well-known Kalmár Lemma which was used by its author in his completeness argument for the ordinary, two-valued logic (cf. [2]).     

Smale’s Fundamental Theorem of Algebra Reconsidered

In his 1981 Fundamental Theorem of Algebra paper Steve Smale initiated the complexity theory of finding a solution of polynomial equations of one complex variable by a variant of Newton’s method. In this paper we reconsider his algorithm in the light of work done in the intervening years. Smale’s upper bound estimate was infinite average cost. Ours is polynomial in the Bézout number and the dimension of the input. Hence it is polynomial for any range of dimensions where the Bézout number is polynomial in the input size. In particular it is not just for the case that Smale considered but for a range of dimensions as considered by Bürgisser–Cucker, where the max of the degrees is greater than or equal to n ^1+? for some fixed ?. It is possible that Smale’s algorithm is polynomial cost in all dimensions and our main theorem raises some problems that might lead to a proof of such a theorem.