Logical Necessity

Since Kit Fine presented his counter-examples to the standard versions of the modal view, many have been convinced that the standard versions of the modal view are not adequate. However, the scope of Fine's argument has not been fully appreciated. In this paper, I aim to carry Fine’s argument to its logical conclusion and argue that once we embrace the intuition underlying his counter-examples, we have to hold that properties obtained, totally or partially, by application of logical operations are not (...) essential to non-logical entities. I also demonstrate that most of the post-Finean versions of the modal view, which were developed to accommodate Fine’s counter-examples, entail that such properties are essential to the entities, and so fail to capture the notion of essence at issue in Fine's counter-examples. Additionally, I explore the consequence of my argument for Fine’s proposed logic of essence. The logic turns out to be inadequate in its present shape as it represents such properties to be essential to the entities. I conclude by developing a modification to the logic to overcome the shortcoming. (shrink)

I propose a comprehensive account of negation as a modal operator, vindicating a moderate logical pluralism. Negation is taken as a quantifier on worlds, restricted by an accessibility relation encoding the basic concept of compatibility. This latter captures the core meaning of the operator. While some candidate negations are then ruled out as violating plausible constraints on compatibility, different specifications of the notion of world support different logical conducts for negations. The approach unifies in a philosophically motivated picture the following (...) results: nothing can be called a negation properly if it does not satisfy Contraposition and Double Negation Introduction; the pair consisting of two split or Galois negations encodes a distinction without a difference; some paraconsistent negations also fail to count as real negations, but others may; intuitionistic negation qualifies as real negation, and classical Boolean negation does as well, to the extent that constructivist and paraconsistent doubts on it do not turn on the basic concept of compatibility but rather on the interpretation of worlds. (shrink)

In two early papers, Max Cresswell constructed two formal logics of propositional identity, pcr and fcr, which he observed to be respectively deductively equivalent to modal logics s4 and s5. Cresswell argued informally that these equivalences respectively “give . . . evidence” for the correctness of s4 and s5 as logics of broadly logical necessity. In this paper, I describe weaker propositional identity logics than pcr that accommodate core intuitions about identity and I argue that Cresswell’s informal arguments do not (...) firmly and without epistemic circularity justify accepting s4 or s5. I also describe how to formulate standard modal logics (k, s2, and their extensions) with strict equivalence as the only modal primitive. (shrink)

Some philosophers dispute the claim that there is a notion of logical necessity involved in the concept of logical consequence. They are sceptical about logical necessity. They argue that a proper characterisation of logical consequence - of what follows from what - need not and should not appeal to the notion of necessity at all. Quine is the most prominent philosopher holding such a view. In this doctoral dissertation, I argue that scepticism about logical necessity is not successful. Quine's scepticism (...) takes three forms. Firstly, he is often interpreted as undermining, in his classic paper 'Two Dogmas of Empiricism', the very intelligibility of notions such as meaning, necessity, and analyticity. If the notion of necessity is meaningless, it is clear that ascriptions of logical necessity are also meaningless. In the thesis, I defend Quine's criticism of these notions by situating it in its historical context and emphasising that the real target in those writings is not the intelligibility of these notions as such, but only their Platonistic interpretation. I agree with Quine that a good theory about meaning, necessity, or analyticity must avoid such an ontological commitment. Secondly, Quine advocates, in the same paper, a holistic picture of knowledge and claims that in this picture, ascriptions of logical necessity are superfluous. I then show that holism a la Quine is committed to admit the necessity of statements of logical consequence. Thirdly, there is Quine's substitutional account of logical consequence ). He contends that this theory makes no use of logical necessity, thus showing its superfluousness. I show that any plausible account of logical consequence needs to appeal to logical necessity, thus undercutting Quine's claim - and, more generally, undercutting scepticism about logical necessity. (shrink)

I show that intuitive and logical considerations do not justify introducing Leibniz’s Law of the Indiscernibility of Identicals in more than a limited form, as applying to atomic formulas. Once this is accepted, it follows that Leibniz’s Law generalises to all formulas of the first-order Predicate Calculus but not to modal formulas. Among other things, identity turns out to be logically contingent.

In this paper the logic of broad necessity is explored. Definitions of what it means for one modality to be broader than another are formulated, and it is proven, in the context of higher-order logic, that there is a broadest necessity, settling one of the central questions of this investigation. It is shown, moreover, that it is possible to give a reductive analysis of this necessity in extensional language. This relates more generally to a conjecture that it is not possible (...) to define intensional connectives from extensional notions. This conjecture is formulated precisely in higher-order logic, and concrete cases in which it fails are examined. The paper ends with a discussion of the logic of broad necessity. It is shown that the logic of broad necessity is a normal modal logic between S4 and Triv, and that it is consistent with a natural axiomatic system of higher-order logic that it is exactly S4. Some philosophical reasons to think that the logic of broad necessity does not include the S5 principle are given. (shrink)

Modal knowledge accounts like sensitivity or safety face a problem when it comes to knowing propositions that are necessarily true because the modal condition is always fulfilled no matter how random the belief forming method is. Pritchard models the anti-luck condition for knowledge in terms of the modal principle safety. Thus, his anti-luck epistemology faces the same problem when it comes to logical necessities. Any belief in a proposition that is necessarily true fulfills the anti-luck condition and, therefore, qualifies as (...) knowledge. Miščević shares Pritchard’s take on epistemic luck and acknowledges the resulting problem. In his intriguing article “Armchair Luck: Apriority, Intellection and Epistemic Luck” Miščević suggests solving the problem by supplementing safety with a virtue theoretic condition-“agent stability”-which he also spells out in modal terms. I will argue that Miščević is on the right track when he suggests adding a virtue-theoretic component to the safety condition. However, it should not be specified modally but rather in terms of performances that manifest competences. (shrink)

This paper examines the interaction between the philosophy and psychology of concepts and the modal characterization of the deductive concept of logical validity. The concept of logical consequence on which I focus is model-theoretic, where the concept records the property of necessary truth-preservation from the premise of an argument to its conclusion, as well as the condition that, in the class of all possible worlds in which a premise is true, a consequent formula or succedent class of formulas is true, (...) as well. Focusing on the case of logical necessity, I argue that the ability to account for the modal properties of concepts places a desideratum on the explanatory adequacy of theories of mental representation. I outline, then, five approaches to the nature of concepts pursued in philosophy and cognitive science, and argue that the above desideratum does not appear to be satisfiable by the candidate proposals. Given the limits of the foregoing, I appeal to a sixth approach -- namely, cognitivism about epistemic modality, which takes epistemic possibilities to be intensional functions in a language of thought -- and endeavor to demonstrate how the proposal can sufficiently account for the modal profile of the concept of logical consequence, and thus provide the resources necessary for a metamathematical analysis of the target notion in the setting of cognitive science. (shrink)

This paper explains Wittegnstein’s understanding of the ‘grammar’ of our language, tracing its origins in the Tractatus’s concept of logical syntax, and then examining the senses in which Wittegnstein, in his later work, viewed grammar as being ‘arbitrary’. Then, armed with this understanding, it moves on to the task of examining how, within the framework of a Wittegnsteinian view of language, we should understand the inescapable ‘compellingness’ of logical necessity – what Wittegnstein calls the “hardness of the logical must”. Whereas (...) it is often thought that Wittegnstein’s views on the nature of the ‘grammar’ of our concpets leads him towards a vitiatingly conventionalist or anti‐realist understanding of necessity, in which its logical ‘superhardness’ becomes problematic, this paper will argue that there is actually no such tension in Wittegnstein’s thought. In fact, it will be argued, an understanding of the ways in which our conceptual grammar is arbitrary casts a great deal of light on how it is that our concepts can nevertheless support a logically superhard, and normatively commanding, notion of necessity. In support of this view, I distinguish Wittegnstein’s views on necessity from the ‘classical’ conventionalism of the Vienna Circle, and from the radical conventionalism of Michael Dummett, and defend Wittegnstein’s view from a powerful recent attack from Quassim Cassam. (shrink)

It is thought that a valid inference to a logically necessary conclusion must proceed from entirely necessary premises. Counter-examples show this is false. Perhaps while the truth of a necessary proposition may follow from non-necessary premises, its necessity cannot so follow. Counter-examples show this to be mistaken. Must anyone who comes to know the non-necessary premises employed in the various counter-examples have prior knowledge of the necessity of the conclusions of the counter-examples? I argue against this. It is true that, (...) for any necessary proposition, there must be necessary premises from which it may validly be inferred; but no one need use these, or know these, or know how to use them, in order to know the necessity of any proposition. (shrink)

In this paper I advance an account of the necessity of logic in Wittgenstein’s Tractatus. I reject both the “metaphysical” reading of Peter Hacker, who takes Tractarian logical necessity to consist in the mode of truth of tautologies, and the “resolute” account of Cora Diamond, who argues that all Tractarian talk of necessity is to be thrown away. I urge an alternative conception based on remarks 3.342 and 6.124. Necessity consists in what is not arbitrary, and contingency in what is (...) up to our arbitrary choice, in the symbols we use, in how we picture or model the world. Necessity is not a mode of truth of propositions, but lies in the requirements of their intelligibility. I argue that this conception is implicit in certain “resolute” readings and in some of their critics. Both sides of the dispute are committed to certain logical features of language or thought, patterns of symbolizing constitutive of intelligibility that are not up to us to institute or alter. This conception of non-arbitrary patterns of symbolizing, I argue, is what logical syntax in the Tractatus consists in. I also argue that the well-known Tractarian view of propositions as truth-functions of elementary propositions can be understood in terms of patterns of norms governing our making sense with the affirmation and denial of propositions. (shrink)

To answer the question, we need first to consider the notion of necessity and the related notion of contingency. These are so-called "modal" notions. Other modal notions include those of possibility, impossibility, non-necessity, and noncontingency. All play a crucial role in philosophical thinking about matters to do with logic, metaphysics, morality, law, etc. This is because none of these modal notions is univocal in meaning. There are, so to speak, different "species" of the generic notions of necessity, contingency, possibility, and (...) the rest. (shrink)

Modal rationalism includes the thesis that ideal primary positive conceivability entails primary possibility. Modal monism is the thesis that the space of logically possible worlds is coextensive with the space of metaphysically possible worlds. In this paper I explore the relation between the two theses. My aim is to show that the former thesis implies the latter thesis, and that problems with the latter make the former implausible as a complete picture of the epistemology of modality. My argument explores the (...) relation between logical modality and metaphysical modality. (shrink)

This paper responds to criticism of the Kripkean account of logical truth in first-order modal logic. The criticism, largely ignored in the literature, claims that when the box and diamond are interpreted as the logical modality operators, the Kripkean account is extensionally incorrect because it fails to reflect the fact that all sentences stating truths about what is logically possible are themselves logically necessary. I defend the Kripkean account by arguing that some true sentences about logical possibility are not logically (...) necessary. (shrink)

This paper analyzes the logical truths as (very roughly) those truths that would still have been true under a certain range of counterfactual perturbations.What’s nice is that the relevant range is characterized without relying (overtly, at least) upon the notion of logical truth. This approach suggests a conception of necessity that explains what the different varieties of necessity (logical, physical, etc.) have in common, in virtue of which they are all varieties of necessity. However, this approach places the counterfactual conditionals (...) in an unfamiliar foundational role. (shrink)

Modal Platonism utilizes "weak" logical possibility, such that it is logically possible there are abstract entities, and logically possible there are none. Modal Platonism also utilizes a non-indexical actuality operator. Modal Platonism is the EASY WAY, neither reductionist nor eliminativist, but embracing the Platonistic language of abstract entities while eliminating ontological commitment to them. Statement of Modal Platonism. Any consistent statement B ontologically committed to abstract entities may be replaced by an empirically equivalent modalization, MOD(B), not so ontologically committed. This (...) equivalence is provable using Modal/Actuality Logic S5@. Let MAX be a strong set theory with individuals. Then the following Schematic Bombshell Result (SBR) can be shown: MAX logically yields [T is true if and only if MOD(T) is true], for scientific theories T. The proof utilizes Stephen Neale's clever model-theoretic interpretation of Quantified Lewis S5, which I extend to S5@. (shrink)

Which mathematical structures are possible, that is, instantiated by the concrete inhabitants of some possible world? Are there worlds with four-dimensional space? With infinite-dimensional space? Whence comes our knowledge of the possibility of structures? In this paper, I develop and defend a principle of plenitude according to which any mathematically natural generalization of possible structure is itself possible. I motivate the principle pragmatically by way of the role that logical possibility plays in our inquiry into the world.

An impossible world is a world which necessarily does not exist. Besides the paradigm of necessity, wich is logical necesslty, we must consider physical necessity and ethical necessity, both of wich can beexpressed in terms of logical necessity, in the way suggested by Montague. Accordingly, an impossible world can be logically impossible, physically impossible or ethically impossible, but in every case the impossibility can be reduced to logical impossibility, and in consequence an impossible world is irrational and cannot be understood (...) by us. An illustration is taken from the incongruities of Kafka’s story in Di Verwandlung. (shrink)

A study of various central and connected topics in philosophical logic and the theory of meaning. There are important sections on the relation between linguistic and abstract entities, on necessity and convention, on meaning, sense and reference, and on entailment. Dr Lewy proposes a number of original solutions to problems which have been widely discussed in literature, and there is in particular a sharp and sustained criticism of conventionalism and reductionism. These are among the most difficult and intricate issues in (...) contemporary philosophy, but Dr Lewy writes with great clarity and a minimum of technicality. Where his views are controversial they are explained and supported in a detail which makes it both possible and necessary for potential critics to state their disagreement precisely. The book should therefore be of value as an advanced textbook as well as an original contribution to philosophical logic. (shrink)

Interest in the metaphysics and logic of possible worlds goes back at least as far as Aristotle, but few books address the history of these important concepts. This volume offers new essays on the theories about the logical modalities held by leading philosophers from Aristotle in ancient Greece to Rudolf Carnap in the twentieth century. The story begins with an illuminating discussion of Aristotle's views on the connection between logic and metaphysics, continues through the Stoic and mediaeval traditions, and then (...) moves to the early modern period with particular attention to Locke and Leibniz. The views of Kant, Peirce, C. I. Lewis and Carnap complete the volume. Many of the essays illuminate the connection between the historical figures studied, and recent or current work in the philosophy of modality. The result is a rich and wide-ranging picture of the history of the logical modalities. (shrink)

This article deals with a question of a most general, comprehensive and profound content as it is the nature of mathematical–logical objects insofar as these are considered objects of knowledge and more specifically objects of formal mathematical theories. As objects of formal theories they are dealt with in the sense they have acquired primarily from the beginnings of the systematic study of mathematical foundations in connection with logic dating from the works of G. Cantor and G. Frege in the last (...) decades of the nineteenth century. Largely motivated by a phenomenologically founded view of mathematical objects/states-of-affairs, I try to consistently argue for their character as objects shaped to a certain extent by intentional forms exhibited by consciousness and the modes of constitution of inner temporality and at the same time as constrained in the form of immanent ‘appearances’ by what stands as their non-eliminable reference, that is, the world as primitive soil of experience and the mathematical intuitions developed in relations of reciprocity within-the-world. In this perspective and relative to my intentions I enter, in the last section, into a brief review of certain positions of G. Sher’s foundational holism and R. Tieszen’s constituted platonism, among others, respectively presented in Sher and Tieszen. (shrink)

Conventionalism about logical necessity has, since Quine, been criticised for falling into a vicious regress. The conventionalist model involves a base class of directly stipulated necessary truths, and all other logical necessities are taken to follow from this base. The essentialist model described by Hale is constructed in a broadly similar fashion: there is a class of those necessities which hold directly in virtue of the natures of things, and other necessities are from this class derivative. Hale expresses a concern (...) that the similarity of the two models might leave the essentialist position prey to a similar regress, but argues that such a situation will not arise. This should prompt a question: if the models are similar, could the solution not be also? In other words, can the conventionalist follow the essentialist to escape regress? This paper will offer a negative answer to this question. We will assume that Hale’s attempt to avert disaster for the essentialist was successful, but argue that the route taken by Hale is not open to the conventionalist, and that in attempting to take it the conventionalist fails to avoid regress. The paper will end with an attempt to generalise what we have seen to the conclusion that conventionalism about logical necessity cannot ever avoid regress. Keywords Modality, essentialism, conventionalism, regress, Hale. (shrink)

The starting point of this paper concerns the apparent difference between what we might call absolute truth and truth in a model, following Donald Davidson. The notion of absolute truth is the one familiar from Tarski’s T-schema: ‘Snow is white’ is true if and only if snow is white. Instead of being a property of sentences as absolute truth appears to be, truth in a model, that is relative truth, is evaluated in terms of the relation between sentences and models. (...) I wish to examine the apparent dual nature of logical truth (without dwelling on Davidson), and suggest that we are dealing with a distinction between a metaphysical and a linguistic interpretation of truth. I take my cue from John Etchemendy, who suggests that absolute truth could be considered as being equivalent to truth in the ‘right model’, i.e., the model that corresponds with the world. However, the notion of ‘model’ is not entirely appropriate here as it is closely associated with relative truth. Instead, I propose that the metaphysical interpretation of truth may be illustrated in modal terms, by metaphysical modality in particular. One of the tasks that I will undertake in this paper is to develop this modal interpretation, partly building on my previous work on the metaphysical interpretation of the law of non-contradiction (Tahko 2009). After an explication of the metaphysical interpretation of logical truth, a brief study of how this interpretation connects with some recent important themes in philosophical logic follows. In particular, I discuss logical pluralism and propose an understanding of pluralism from the point of view of the metaphysical interpretation. (shrink)

This essay investigates the nature of necessary truth in logic and mathematics through a discussion of Wittgenstein's writings on logic and mathematics. My principal scholarly interest in the dissertation is Wittgenstein' s later philosophy, but I also examine his philosophy in the Tractatus and the transitional texts of the 1930s. ;In chapter one I present the Tractatus' general theory of representation through an examination of Wittgenstein's work on the ancient problem of the unity of a proposition. I argue that Wittgenstein's (...) solution draws upon Frege's philosophy of logic, despite the fact that Wittgenstein's interest in the problem is part of Russell's patrimony. In chapter two I present my interpretation of Wittgenstein's early treatment of logic and arithmetic. A major achievement of Wittgenstein's early work is the progress he makes towards showing that the logic of language is implicit in the concept of a proposition and that arithmetic is implicit in the concept of iteration. In chapter two I also discuss Wittgenstein's criticisms of Frege and Russell, on the one hand, and his relation to conventionalism of the logical empiricists, on the other. ;In the second half of the thesis I argue that the Tractarian idea that logic and mathematics belong to an ahistorical logico-metaphysical scaffolding surrounding language, thought, and the world evolves into the view that logical and mathematical truths are constitutive rules of the multifarious "language-games" that make up our human lives. Logical and mathematical truths are necessary because they are implicit in our concept of thinking or speaking about anything at all, as Wittgenstein had held in the Tractatus; but they enjoy this status on account of their position relative to our practices and not because they exhibit the prior constitution of the world. This treatment of these issues and my reading of Wittgenstein avoids both Kripke's interpretation of Wittgenstein, which collapses truth into assertability based on the agreement of a community and makes mathematical necessity a function of the affective side of human nature; and conventionalism, which is another standard reading of Wittgenstein and so-called "Wittgensteinian" account of mathematics. (shrink)

It is usually held that what distinguishes a good inference from a bad one is that the former is but the latter is not truth-preserving. What is behind this view is the basic assumption that whether a certain inference is truth-preserving or not is a genuine issue, an issue the answer to which is determined objectively. This view is called the realist view of logic. ;In this dissertation the realist view of logic is criticized and an alternative view is presented. (...) The alternative is called the fictionalist view of logic. On this view, what makes a logical system good is not that its inferential rules are truth-preserving, but rather that it has a certain proof-theoretic property, called deductive conservativeness. More specifically, it is argued that if the inferential rules for the logical operators involved in the system are conservative over the deductive system which is devoid of logical operators, then the conservative system will have all the virtues we expect in a good logical system. It is shown however that insofar as the system is conservative, there is no way to determine whether its inferential rules are truth-preserving or not; thus the latter is simply irrelevant to the goodness of the logical system. To support this claim, the view that the meaning of a logical operator is determined not by its semantic properties such as truth but by its inferential role, is advocated. A significant implication of this view is that there is no answer to the question which of the conservative logical systems is the correct logic for the world. ;Furthermore, it is maintained that metalogical claims of implication and consistency, namely claims of the forms 'A implies B', and 'A is consistent', can be taken as modal claims which involve modal operators as logical operators, and thus the above approach is applicable to such metalogical claims, too. This idea is used to avoid our ontological commitment to mathematical entities such as models and proofs employed in metalogic. (shrink)

This paper defends logical nominalism, the thesis that logically necessary truth belongs primarily to sentences and depends solely on the conventions of human language. A sentence is logically necessary iff its negation entails a contradiction. A sentence is a posteriori metaphysically necessary iff it reduces to a logical necessity when we substitute for rigid designators of objects or properties canonical descriptions of the essential properties of those objects or properties. The truth-conditions of necessary sentences are not to be found in (...) any transcendent reality, such as God's thoughts. ‘There is a God’ is neither a priori nor a posteriori metaphysically necessary; God is necessary in the sense that His existence is not causally contingent on anything else. (shrink)

This dissertation is divided into three parts. The first part consists of a detailed descriptive summary of the interpretive commentaries following the publication of Wittgenstein's Remarks on the Foundation of Mathematics. The material, thus assembled and presented as a whole for the first time, reveals unities directly traceable to assumptions and issues introduced by one man, Michael Dummett, who gave the interpretive literature its impetus, form and substance. ;The second part of the dissertation is of a critical nature. In this (...) part it will be shown that Dummett's interpretation and assessment of Wittgenstein's philosophical activities is inadequate. Dummett's requirements for solving the problem of logical necessity require contradictory things of Wittgenstein. Further, the criteria which Dummett uses to judge the success of Wittgenstein's efforts are the very criteria which Wittgenstein attacks successfully in the Remarks. ;Having shown that the literature depends heavily on Dummett, and that Dummett is fundamentally mistaken, the third and remaining part of the dissertation attempts an expository sketch in which Wittgenstein's thoughts on logical necessity are provided. This sketch attempts to avoid the pitfalls of Dummett's approach and preserve Wittgenstein's avoidance of philosophical theorizing. (shrink)

In these few pages I shall try to demonstrate the emptiness of the most cumbersome piece of unexamined intellectual baggage at present being hauled about by English philosophers. I here cite one example to be going on with, at the end of the paper I shall give a handful more, and it would be easy to multiply the number by ten from the writings of reputable philosophers. The outstanding philosophical achievement of the ha1f-century which has just drawn to a close (...) [i.e. the period 1900-1950] has been an appreciation of the peculiar status of a priori judgments and of logically necessary or formally true propositions. . . . Though many problems remain unsolved, the main outline is now clear: formally true statements assert nothing about the world; instead, their function is to state principles according to which empirical propositions are deduced from other empirical propositions . . . (R. B. Braithwaite, ‘Moral Principles and Inductive Policies’, Proceedings of the British Academy 1950). What is wrong with this passage and with the myth of which it is an expression is its assumption that we have clear notions of what it is for a proposition to be logically necessary and of what it is for a proposition to assert something about the world, these notions being such that it is plausible to say that it has recently been discovered that every proposition having the first of these properties lacks the second. This assumption is wrong: there is no body of published theory giving a clear account of such notions, and despite fairly diligent searching I have so far failed to find, among the many philosophers who accept the myth, one who is able when challenged to supplement the literature on this vital point. Let me make it clear at once that I am not going to defend synthetic a priori truths - I am going to attack a popular mishandling of the truth that all necessary truths are analytic, and through this attack to draw right-wing conclusions from left-wing premisses.. (shrink)

Hale has argued that logical necessities are absolute in the sense that there is no competing kind of modality under which they may be false. In this paper, I argue that there are competing kinds of modality, which I call “essentialist modalities,” under which logical necessities may be false. Since it is counter-intuitive to say that logical necessities are not absolute, my argument, if correct, shows that Hale’s characterization of absolute necessity does not adequately capture the intuitive notion of absolute (...) necessity. Then, a qualified version of Hale’s characterization of absolute necessity is proposed. On the qualified version, the absoluteness of logical necessities is no longer defeated by essentialist possibilities. (shrink)