Can Only Human Lives Be Meaningful?

Duncan Purves and Nicolas Delon have argued that one’s life will be meaningful to the extent that one contributes to valuable states of affairs and this contribution is a result of one’s intentional actions. They then argue, contrary to some theorists’ intuitions, that non-human animals are capable of fulfilling these requirements, and that this finding might entail important things for the animal ethics movement. In this paper, I also argue that things besides human beings can have meaningful existences, but I disagree with Purves and Delon’s theory of meaning, and some of the practical implications they suggest arise from their conclusion. Specifically, I argue that Purves and Delon are wrong to suggest that intentional agency is necessary for one’s life to be meaningful; contributing to valuable states of affairs can be sufficient by itself. Purves and Delon’s objection to such a claim is that it would allow even inanimate objects’ existences to count as meaningful. However, while I accept this consequence, I argue that it only seems counterintuitive because of two false beliefs they appear to hold: that some X cannot have a meaningful existence without that meaning (a) making X better off or (b) giving X reasons for pride.     

Continuum,name and paradox

The article deals with Cantor’s argument for the non-denumerability of reals somewhat in the spirit of Lakatos’ logic of mathematical discovery. At the outset Cantor’s proof is compared with some other famous proofs such as Dedekind’s recursion theorem, showing that rather than usual proofs they are resolutions to do things differently. Based on this I argue that there are “ontologically” safer ways of developing the diagonal argument into a full-fledged theory of continuum, concluding eventually that famous semantic paradoxes based on diagonal construction are caused by superficial understanding of what a name is.     

The Consistency of The Naïve Theory of Properties

If properties are to play a useful role in semantics, it is hard to avoid assuming the naïve theory of properties: for any predicate Θ(x), there is a property such that an object o has it if and only if Θ(o). Yet this appears to lead to various paradoxes. I show that no paradoxes arise as long as the logic is weakened appropriately; the main difficulty is finding a semantics that can handle a conditional obeying reasonable laws without engendering paradox. I employ a semantics which is infinite-valued, with the values only partially ordered. Can the solution be adapted to naïve set theory? Probably not, but limiting naïve comprehension in set theory is perfectly satisfactory, whereas this is not so in a property theory used for semantics.     

Non‐Classical Knowledge

The Knower paradox purports to place surprising a priori limitations on what we can know. According to orthodoxy, it shows that we need to abandon one of three plausible and widely‐held ideas: that knowledge is factive, that we can know that knowledge is factive, and that we can use logical/mathematical reasoning to extend our knowledge via very weak single‐premise closure principles. I argue that classical logic, not any of these epistemic principles, is the culprit. I develop a consistent theory validating all these principles by combining Hartry Field's theory of truth with a modal enrichment developed for a different purpose by Michael Caie. The only casualty is classical logic: the theory avoids paradox by using a weaker‐than‐classical K₃ logic. I then assess the philosophical merits of this approach. I argue that, unlike the traditional semantic paradoxes involving extensional notions like truth, its plausibility depends on the way in which sentences are referred to—whether in natural languages via direct sentential reference, or in mathematical theories via indirect sentential reference by Gödel coding. In particular, I argue that from the perspective of natural language, my non‐classical treatment of knowledge as a predicate is plausible, while from the perspective of mathematical theories, its plausibility depends on unresolved questions about the limits of our idealized deductive capacities.     

PARADOXICAL PROPOSITIONS

This paper concerns two paradoxes involving propositions. The first is Russell's paradox from Appendix B of The Principles of Mathematics, a version of which was later given by Myhill. The second is a paradox in the framework of possible worlds, given by Kaplan. This paper shows a number of things about these paradoxes. First, we will see that, though the Russell/Myhill paradox and the Kaplan paradox might appear somewhat different, they are really just variants of the same phenomenon. Though they do this in different ways, the core of each paradox is to use the notion of a proposition to construct a function, f, from the power set of some set into the set itself. Next we will see how this paradox fits into the Inclosure Schema. Finally, I will provide a model of the paradox in question, showing its results to be non‐trivial, though inconsistent.     

Composition as a Kind of Identity

Composition as identity, as I understand it, is a theory of the composite structure of reality. The theory’s underlying logic is irreducibly plural; its fundamental primitive is a generalized identity relation that takes either plural or singular arguments. Strong versions of the theory that incorporate a generalized version of the indiscernibility of identicals are incompatible with the framework of plural logic, and should be rejected. Weak versions of the theory that are based on the idea that composition is merely analogous to (one-one) identity are too weak to be interesting, lacking in metaphysical consequence. I defend a moderate version according to which composition is a kind of identity, and argue that the difference is metaphysically substantial, not merely terminological. I then consider whether the notion of generalized identity, though fundamental, can be elucidated in modal terms by reverse engineering Hume’s Dictum. Unfortunately, for realists about possible worlds, such as myself, who understand Hume’s Dictum in terms of duplicates, the elucidation never gets off the ground; for modal primitivists, the elucidation may succeed in capturing some general notion of identity, but it’s characterization is too general to target the particular notion of identity, tied to classical mereology, that is the object of my theorizing. In the end, I have little to offer the skeptic who claims not to understand the relevant notion of generalized identity, or not to take it to be compulsory in providing an account of the structure of reality.     

Modalising Plurals

There has been very little discussion of the appropriate principles to govern a modal logic of plurals. What debate there has been has accepted a principle I call (NecInc); informally if this is one of those then, necessarily: this is one of those. On this basis Williamson has criticised the Boolosian plural interpretation of monadic second-order logic. I argue against (NecInc), noting that it isn’t a theorem of any logic resulting from adding modal axioms to the plural logic PFO+, and showing that the most obvious formal argument in its favour is question begging. I go on to discuss the behaviour of natural language plurals, motivating a case against (NecInc) by developing a case that natural language plural terms are not de jure rigid designators. The paper concludes by developing a model theory for modal PFO+ which does not validate (NecInc). An Appendix discusses (NecInc) in relation to counterpart theory.     

Necessity and Triviality

In this paper I argue that there are some sentences whose truth makes no demands on the world, being trivially true in that their truth-conditions are trivially met. I argue that this does not amount to their truth-conditions being met necessarily: we need a non-modal understanding of the notion of the demands the truth of a sentence makes, lest we be blinded to certain conceptual possibilities. I defend the claim that the truths of pure mathematics and set theory are trivially true, and hence accepting their truth brings no ontological commitment; I further defend the claim that the truths of applied mathematics and set theory do not demand the existence of numbers or sets. While the notion of a demand must not be reduced to anything modal, I nonetheless argue that sentences that are trivially true must also be necessary, lest we violate a very weak version of the principle that truth depends on the world. I further argue that all necessary truths are trivially true, lest we admit unexplained necessities. I end by showing one important consequence of this: I argue that if there are truthmakers for intrinsic predications, they must be states of affairs rather than tropes.     

Monism, Emergence, and Plural Logic

In this paper I argue that we need to take irreducibly plural logic more seriously in metaphysical debates due to the fact that the verdict of many metaphysical debates hangs on it. I give two examples. The main example I focus on is the debate recently revived by Jonathan Schaffer over the fundamental cardinality of the world. I show how the three main arguments provided by Schaffer are unsound in virtue of an employment of plural logic. The second example I give is a more general issue about the possibility of emergent properties of mereological wholes. Employing plural logic there is a new way to understand such cases. The upshot is that plural logic greatly matters to metaphysics and hence can no longer be ignored the way it has in this area.     

Georg cantor's influence on bertrand russell

This paper is concerned with the influence that the set theory of Georg Cantor (1845–1918) bore upon the mathematical logic of Bertrand Russell (1872–1970). In some respects the influence is positive, and stems directly from Cantor's writings or through intermediary figures such as Peano; but in various ways negative influence is evident, for Russell adopted alternative views about the form and foundations of set theory. After an opening biographical section, six sections compare and contrast their views on matters of common interest; irrational numbers, infinitesimals, cardinal and ordinal numbers, the axiom of infinity, the paradoxes, and the axioms of choice. Two further sections compare the two men over more general questions: the role of logic and the philosophy of mathematics. In a final section I draw some conclusions.     

On the Innocence and Determinacy of Plural Quantification

Plural logic is widely assumed to have two important virtues: ontological innocence and determinacy. It is claimed to be innocent in the sense that it incurs no ontological commitments beyond those already incurred by the first‐order quantifiers. It is claimed to be determinate in the sense that it is immune to the threat of non‐standard (Henkin) interpretations that confronts higher‐order logics on their more traditional, set‐based semantics. We challenge both claims. Our challenge is based on a Henkin‐style semantics for plural logic that does not resort to sets or set‐like objects to interpret plural variables, but adopts the view that a plural variable has many objects as its values. Using this semantics, we also articulate a generalized notion of ontological commitment which enables us to develop some ideas of earlier critics of the alleged ontological innocence of plural logic.     

The Logic and Meaning of Plurals. Part I

Contemporary accounts of logic and language cannot give proper treatments of plural constructions of natural languages. They assume that plural constructions are redundant devices used to abbreviate singular constructions. This paper and its sequel, “The logic and meaning of plurals, II”, aim to develop an account of logic and language that acknowledges limitations of singular constructions and recognizes plural constructions as their peers. To do so, the papers present natural accounts of the logic and meaning of plural constructions that result from the view that plural constructions are, by and large, devices for talking about many things (as such). The account of logic presented in the papers surpasses contemporary Fregean accounts in its scope. This extension of the scope of logic results from extending the range of languages that logic can directly relate to. Underlying the view of language that makes room for this is a perspective on reality that locates in the world what plural constructions can relate to. The papers suggest that reflections on plural constructions point to a broader framework for understanding logic, language, and reality that can replace the contemporary Fregean framework as this has replaced its Aristotelian ancestor.     

The Logics of Strict-Tolerant Logic

Adding a transparent truth predicate to a language completely governed by classical logic is not possible. The trouble, as is well-known, comes from paradoxes such as the Liar and Curry. Recently, Cobreros, Egré, Ripley and van Rooij have put forward an approach based on a non-transitive notion of consequence which is suitable to deal with semantic paradoxes while having a transparent truth predicate together with classical logic. Nevertheless, there are some interesting issues concerning the set of metainferences validated by this logic. In this paper, we show that this logic, once it is adequately understood, is weaker than classical logic. Moreover, the logic is in a way similar to the paraconsistent logic LP.     

Mental models and probabilistic thinking

This paper outlines the theory of reasoning based on mental models, and then shows how this theory might be extended to deal with probabilistic thinking. The same explanatory framework accommodates deduction and induction: there are both deductive and inductive inferences that yield probabilistic conclusions. The framework yields a theoretical conception of strength of inference, that is, a theory of what the strength of an inference is objectively: it equals the proportion of possible states of affairs consistent with the premises in which the conclusion is true, that is, the probability that the conclusion is true given that the premises are true. Since there are infinitely many possible states of affairs consistent with any set of premises, the paper then characterizes how individuals estimate the strength of an argument. They construct mental models, which each correspond to an infinite set of possibilities (or, in some cases, a finite set of infinite sets of possibilities). The construction of models is guided by knowledge and beliefs, including lay conceptions of such matters as the “law of large numbers”. The paper illustrates how this theory can account for phenomena of probabilistic reasoning.     

Liar-type Paradoxes and the Incompleteness Phenomena

We define a liar-type paradox as a consistent proposition in propositional modal logic which is obtained by attaching boxes to several subformulas of an inconsistent proposition in classical propositional logic, and show several famous paradoxes are liar-type. Then we show that we can generate a liar-type paradox from any inconsistent proposition in classical propositional logic and that undecidable sentences in arithmetic can be obtained from the existence of a liar-type paradox. We extend these results to predicate logic and discuss Yablo’s Paradox in this framework. Furthermore, we define explicit and implicit self-reference in paradoxes in the incompleteness phenomena.     

Implicational Paradoxes and the Meaning of Logical Constants

I discuss paradoxes of implication in the setting of a proof-conditional theory of meaning for logical constants. I argue that a proper logic of implication should be not only relevant, but also constructive and nonmonotonic. This leads me to select as a plausible candidate LL, a fragment of linear logic that differs from R in that it rejects both contraction and distribution.     

FINDING AN APPROPRIATE ETHIC IN A WORLD OF MORAL ACQUAINTANCES

This paper discusses the possibility of finding an ethic of at least partial and perhaps ever-growing content in a world not that of moral strangers (where we have nothing except our desire to live freely to unite us) and one of moral friends (in which values, goals and ways of doing things are held in common). I argue that both the world of moral strangers which Engelhardt's world view would support, as the world of moral friends which is the one Pellegrino seeks both are untenable and that furthermore both can lead to a similar state of affairs. I suggest a dynamic world of moral acquaintances in which different belief systems and ways of doing things can come to some broad agreements about some essential thing. This is made possible because although we do not share the intimate framework Pellegrino might suggest, yet we are united by a much broader framework than the one moral strangers share.     

The paradoxes

Conclusion The paradoxes act as restraints on undisciplined or erroneous reasoning and so perform a valuable role. The fact that they have resisted solution for so long suggests that the current systems of logic are defective. The paradoxes of set theory, in my opinion, completely condemn all the current forms of set theory and their associated definitions of the natural numbers. Nothing short of a complete review of the whole area seems capable of remedying the situation.