Modal Propositional Logic
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Logic in Action: Wittgenstein's Logical Pragmatism and the Impotence of Scepticism
This is the final, pre-publication draft. Please cite only from published paper in Philosophical Investigations 26:2 (April 2003), 125-48. LOGIC IN ACTION: WITTGENSTEIN'S LOGICAL PRAGMATISM AND THE IMPOTENCE OF SCEPTICISM DANIÈLE MOYAL-SHARROCK UNIVERSITY OF GENEVA 1. The Many Faces of Certainty: Wittgenstein's Logical Pragmatism So I am trying to say something that sounds like pragmatism. (OC 422) In his struggle to uncover the nature of our basic beliefs, Wittgenstein depicts them variously in On Certainty: he thinks of them in propositional terms, in pictorial terms and in terms of acting. As propositions, they would be of a peculiar sort – a hybrid between a logical and an empirical proposition (OC 136, 309). These are the so-called 'hinge propositions' of On Certainty (OC 341). Wittgenstein also thinks of these beliefs as forming a picture, a World-picture – or Weltbild (OC 167). This is a step in the right (nonpropositional) direction, but not the ultimate step. Wittgenstein's ultimate and crucial depiction of our basic beliefs is in terms of a know-how, an attitude, a way of acting (OC 204). Here, he treads on pragmatist ground. But can Wittgenstein be labelled a pragmatist, having himself rejected the affiliation because of its utility implication? But you aren't a pragmatist? No. For I am not saying that a proposition is true if it is useful. (RPP I, 266) Wittgenstein resists affiliation with pragmatism because he does not want his use of use to be confused with the utility use of use. For him, it is not that a proposition is true if it is useful, but that use gives the proposition its sense. -
The Modal Logic of Potential Infinity, with an Application to Free Choice
The Modal Logic of Potential Infinity, With an Application to Free Choice Sequences Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Ethan Brauer, B.A. ∼6 6 Graduate Program in Philosophy The Ohio State University 2020 Dissertation Committee: Professor Stewart Shapiro, Co-adviser Professor Neil Tennant, Co-adviser Professor Chris Miller Professor Chris Pincock c Ethan Brauer, 2020 Abstract This dissertation is a study of potential infinity in mathematics and its contrast with actual infinity. Roughly, an actual infinity is a completed infinite totality. By contrast, a collection is potentially infinite when it is possible to expand it beyond any finite limit, despite not being a completed, actual infinite totality. The concept of potential infinity thus involves a notion of possibility. On this basis, recent progress has been made in giving an account of potential infinity using the resources of modal logic. Part I of this dissertation studies what the right modal logic is for reasoning about potential infinity. I begin Part I by rehearsing an argument|which is due to Linnebo and which I partially endorse|that the right modal logic is S4.2. Under this assumption, Linnebo has shown that a natural translation of non-modal first-order logic into modal first- order logic is sound and faithful. I argue that for the philosophical purposes at stake, the modal logic in question should be free and extend Linnebo's result to this setting. I then identify a limitation to the argument for S4.2 being the right modal logic for potential infinity. -
7.1 Rules of Implication I
Natural Deduction is a method for deriving the conclusion of valid arguments expressed in the symbolism of propositional logic. The method consists of using sets of Rules of Inference (valid argument forms) to derive either a conclusion or a series of intermediate conclusions that link the premises of an argument with the stated conclusion. The First Four Rules of Inference: ◦ Modus Ponens (MP): p q p q ◦ Modus Tollens (MT): p q ~q ~p ◦ Pure Hypothetical Syllogism (HS): p q q r p r ◦ Disjunctive Syllogism (DS): p v q ~p q Common strategies for constructing a proof involving the first four rules: ◦ Always begin by attempting to find the conclusion in the premises. If the conclusion is not present in its entirely in the premises, look at the main operator of the conclusion. This will provide a clue as to how the conclusion should be derived. ◦ If the conclusion contains a letter that appears in the consequent of a conditional statement in the premises, consider obtaining that letter via modus ponens. ◦ If the conclusion contains a negated letter and that letter appears in the antecedent of a conditional statement in the premises, consider obtaining the negated letter via modus tollens. ◦ If the conclusion is a conditional statement, consider obtaining it via pure hypothetical syllogism. ◦ If the conclusion contains a letter that appears in a disjunctive statement in the premises, consider obtaining that letter via disjunctive syllogism. Four Additional Rules of Inference: ◦ Constructive Dilemma (CD): (p q) • (r s) p v r q v s ◦ Simplification (Simp): p • q p ◦ Conjunction (Conj): p q p • q ◦ Addition (Add): p p v q Common Misapplications Common strategies involving the additional rules of inference: ◦ If the conclusion contains a letter that appears in a conjunctive statement in the premises, consider obtaining that letter via simplification. -
Branching Time
24.244 Modal Logic, Fall 2009 Prof. Robert Stalnaker Lecture Notes 16: Branching Time Ordinary tense logic is a 'multi-modal logic', in which there are two (pairs of) modal operators: P/H and F/G. In the semantics, the accessibility relations, for past and future tenses, are interdefinable, so in effect there is only one accessibility relation in the semantics. However, P/H cannot be defined in terms of F/G. The expressive power of the language does not match the semantics in this case. But this is just a minimal multi-modal theory. The frame is a pair consisting of an accessibility relation and a set of points, representing moments of time. The two modal operators are defined on this one set. In the branching time theory, we have two different W's, in a way. We put together the basic tense logic, where the points are moments of time, with the modal logic, where they are possible worlds. But they're not independent of one another. In particular, unlike abstract modal semantics, where possible worlds are primitives, the possible worlds in the branching time theory are defined entities, and the accessibility relation is defined with respect to the structure of the possible worlds. In the pure, abstract theory, where possible worlds are primitives, to the extent that there is structure, the structure is defined in terms of the relation on these worlds. So the worlds themselves are points, the structure of the overall frame is given by the accessibility relation. In the branching time theory, the possible worlds are possible histories, which have a structure defined by a more basic frame, and the accessibility relation is defined in terms of that structure. -
29 Alethic Modal Logics and Semantics
29 Alethic Modal Logics and Semantics GERHARD SCHURZ 1 Introduction The first axiomatic development of modal logic was untertaken by C. I. Lewis in 1912. Being anticipated by H. McCall in 1880, Lewis tried to cure logic from the ‘paradoxes’ of extensional (i.e. truthfunctional) implication … (cf. Hughes and Cresswell 1968: 215). He introduced the stronger notion of strict implication <, which can be defined with help of a necessity operator ᮀ (for ‘it is neessary that:’) as follows: A < B iff ᮀ(A … B); in words, A strictly implies B iff A necessarily implies B (A, B, . for arbitrary sen- tences). The new primitive sentential operator ᮀ is intensional (non-truthfunctional): the truth value of A does not determine the truth-value of ᮀA. To demonstrate this it suf- fices to find two particular sentences p, q which agree in their truth value without that ᮀp and ᮀq agree in their truth-value. For example, it p = ‘the sun is identical with itself,’ and q = ‘the sun has nine planets,’ then p and q are both true, ᮀp is true, but ᮀq is false. The dual of the necessity-operator is the possibility operator ‡ (for ‘it is possible that:’) defined as follows: ‡A iff ÿᮀÿA; in words, A is possible iff A’s negation is not necessary. Alternatively, one might introduce ‡ as new primitive operator (this was Lewis’ choice in 1918) and define ᮀA as ÿ‡ÿA and A < B as ÿ‡(A ŸÿB). Lewis’ work cumulated in Lewis and Langford (1932), where the five axiomatic systems S1–S5 were introduced. -
Contradiction Or Non-Contradiction? Hegel’S Dialectic Between Brandom and Priest
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Padua@research CONTRADICTION OR NON-CONTRADICTION? HEGEL’S DIALECTIC BETWEEN BRANDOM AND PRIEST by Michela Bordignon Abstract. The aim of the paper is to analyse Brandom’s account of Hegel’s conception of determinate negation and the role this structure plays in the dialectical process with respect to the problem of contradiction. After having shown both the merits and the limits of Brandom’s account, I will refer to Priest’s dialetheistic approach to contradiction as an alternative contemporary perspective from which it is possible to capture essential features of Hegel’s notion of contradiction, and I will test the equation of Hegel’s dialectic with Priest dialetheism. 1. Introduction According to Horstmann, «Hegel thinks of his new logic as being in part incompatible with traditional logic»1. The strongest expression of this new conception of logic is the first thesis of the work Hegel wrote in 1801 in order to earn his teaching habilitation: «contradictio est regula veri, non contradictio falsi»2. Hegel seems to claim that contradictions are true. The Hegelian thesis of the truth of contradiction is highly problematic. This is shown by Popper’s critique based on the principle of ex falso quodlibet: «if a theory contains a contradiction, then it entails everything, and therefore, indeed, nothing […]. A theory which involves a contradiction is therefore entirely useless I thank Graham Wetherall for kindly correcting a previous English translation of this paper and for his suggestions and helpful remarks. Of course, all remaining errors are mine. -
Absolute Contradiction, Dialetheism, and Revenge
THE REVIEW OF SYMBOLIC LOGIC,Page1of15 ABSOLUTE CONTRADICTION, DIALETHEISM, AND REVENGE FRANCESCO BERTO Department of Philosophy, University of Amsterdam and Northern Institute of Philosophy, University of Aberdeen Abstract. Is there a notion of contradiction—let us call it, for dramatic effect, “absolute”— making all contradictions, so understood, unacceptable also for dialetheists? It is argued in this paper that there is, and that spelling it out brings some theoretical benefits. First it gives us a foothold on undisputed ground in the methodologically difficult debate on dialetheism. Second, we can use it to express, without begging questions, the disagreement between dialetheists and their rivals on the nature of truth. Third, dialetheism has an operator allowing it, against the opinion of many critics, to rule things out and manifest disagreement: for unlike other proposed exclusion-expressing-devices (for instance, the entailment of triviality), the operator used to formulate the notion of absolute contradiction appears to be immune both from crippling expressive limitations and from revenge paradoxes—pending a rigorous nontriviality proof for a formal dialetheic theory including it. Nothing is, and nothing could be, literally both true and false. [. ] That may seem dogmatic. And it is: I am affirming the very thesis that [the dialetheists] have called into question and—contrary to the rules of debate—I decline to defend it. Further, I concede that it is indefensible against their challenge. They have called so much into question that I have no foothold on undisputed ground. So much the worse for the demand that philosophers always must be ready to defend their theses under the rules of debate. -
Probabilistic Semantics for Modal Logic
Probabilistic Semantics for Modal Logic By Tamar Ariela Lando A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Philosophy in the Graduate Division of the University of California, Berkeley Committee in Charge: Paolo Mancosu (Co-Chair) Barry Stroud (Co-Chair) Christos Papadimitriou Spring, 2012 Abstract Probabilistic Semantics for Modal Logic by Tamar Ariela Lando Doctor of Philosophy in Philosophy University of California, Berkeley Professor Paolo Mancosu & Professor Barry Stroud, Co-Chairs We develop a probabilistic semantics for modal logic, which was introduced in recent years by Dana Scott. This semantics is intimately related to an older, topological semantics for modal logic developed by Tarski in the 1940’s. Instead of interpreting modal languages in topological spaces, as Tarski did, we interpret them in the Lebesgue measure algebra, or algebra of measurable subsets of the real interval, [0, 1], modulo sets of measure zero. In the probabilistic semantics, each formula is assigned to some element of the algebra, and acquires a corresponding probability (or measure) value. A formula is satisfed in a model over the algebra if it is assigned to the top element in the algebra—or, equivalently, has probability 1. The dissertation focuses on questions of completeness. We show that the propo- sitional modal logic, S4, is sound and complete for the probabilistic semantics (formally, S4 is sound and complete for the Lebesgue measure algebra). We then show that we can extend this semantics to more complex, multi-modal languages. In particular, we prove that the dynamic topological logic, S4C, is sound and com- plete for the probabilistic semantics (formally, S4C is sound and complete for the Lebesgue measure algebra with O-operators). -
Chapter 10: Symbolic Trails and Formal Proofs of Validity, Part 2
Essential Logic Ronald C. Pine CHAPTER 10: SYMBOLIC TRAILS AND FORMAL PROOFS OF VALIDITY, PART 2 Introduction In the previous chapter there were many frustrating signs that something was wrong with our formal proof method that relied on only nine elementary rules of validity. Very simple, intuitive valid arguments could not be shown to be valid. For instance, the following intuitively valid arguments cannot be shown to be valid using only the nine rules. Somalia and Iran are both foreign policy risks. Therefore, Iran is a foreign policy risk. S I / I Either Obama or McCain was President of the United States in 2009.1 McCain was not President in 2010. So, Obama was President of the United States in 2010. (O v C) ~(O C) ~C / O If the computer networking system works, then Johnson and Kaneshiro will both be connected to the home office. Therefore, if the networking system works, Johnson will be connected to the home office. N (J K) / N J Either the Start II treaty is ratified or this landmark treaty will not be worth the paper it is written on. Therefore, if the Start II treaty is not ratified, this landmark treaty will not be worth the paper it is written on. R v ~W / ~R ~W 1 This or statement is obviously exclusive, so note the translation. 427 If the light is on, then the light switch must be on. So, if the light switch in not on, then the light is not on. L S / ~S ~L Thus, the nine elementary rules of validity covered in the previous chapter must be only part of a complete system for constructing formal proofs of validity. -
Generalized Molecular Formulas in Logical Atomism
Journal for the History of Analytical Philosophy An Argument for Completely General Facts: Volume 9, Number 7 Generalized Molecular Formulas in Logical Editor in Chief Atomism Audrey Yap, University of Victoria Landon D. C. Elkind Editorial Board Annalisa Coliva, UC Irvine In his 1918 logical atomism lectures, Russell argued that there are Vera Flocke, Indiana University, Bloomington no molecular facts. But he posed a problem for anyone wanting Henry Jackman, York University to avoid molecular facts: we need truth-makers for generaliza- Frederique Janssen-Lauret, University of Manchester tions of molecular formulas, but such truth-makers seem to be Kevin C. Klement, University of Massachusetts both unavoidable and to have an abominably molecular charac- Consuelo Preti, The College of New Jersey ter. Call this the problem of generalized molecular formulas. I clarify Marcus Rossberg, University of Connecticut the problem here by distinguishing two kinds of generalized Anthony Skelton, Western University molecular formula: incompletely generalized molecular formulas Mark Textor, King’s College London and completely generalized molecular formulas. I next argue that, if empty worlds are logically possible, then the model-theoretic Editors for Special Issues and truth-functional considerations that are usually given ad- Sandra Lapointe, McMaster University dress the problem posed by the first kind of formula, but not the Alexander Klein, McMaster University problem posed by the second kind. I then show that Russell’s Review Editors commitments in 1918 provide an answer to the problem of com- Sean Morris, Metropolitan State University of Denver pletely generalized molecular formulas: some truth-makers will Sanford Shieh, Wesleyan University be non-atomic facts that have no constituents. -
31 Deontic, Epistemic, and Temporal Modal Logics
31 Deontic, Epistemic, and Temporal Modal Logics RISTO HILPINEN 1 Modal Concepts Modal logic is the logic of modal concepts and modal statements. Modal concepts (modalities) include the concepts of necessity, possibility, and related concepts. Modalities can be interpreted in different ways: for example, the possibility of a propo- sition or a state of affairs can be taken to mean that it is not ruled out by what is known (an epistemic interpretation) or believed (a doxastic interpretation), or that it is not ruled out by the accepted legal or moral requirements (a deontic interpretation), or that it has not always been or will not always be false (a temporal interpretation). These interpre- tations are sometimes contrasted with alethic modalities, which are thought to express the ways (‘modes’) in which a proposition can be true or false. For example, logical pos- sibility and physical (real or substantive) possibility are alethic modalities. The basic modal concepts are represented in systems of modal logic as propositional operators; thus they are regarded as syntactically analogous to the concept of negation and other propositional connectives. The main difference between modal operators and other connectives is that the former are not truth-functional; the truth-value (truth or falsity) of a modal sentence is not determined by the truth-values of its subsentences. The concept of possibility (‘it is possible that’ or ‘possibly’) is usually symbolized by ‡ and the concept of necessity (‘it is necessary that’ or ‘necessarily’) by ᮀ; thus the modal formula ‡p represents the sentence form ‘it is possible that p’ or ‘possibly p,’ and ᮀp should be read ‘it is necessary that p.’ Modal operators can be defined in terms of each other: ‘it is possible that p’ means the same as ‘it is not necessary that not-p’; thus ‡p can be regarded as an abbreviation of ÿᮀÿp, where ÿ is the sign of negation, and ᮀp is logically equivalent to ÿ‡ÿp. -
The Other Side of Peirce's Phaneroscopy: Questioning And
The Other Side of Peirce’s Phaneroscopy: Questioning and Analogising Phaneron... 1 without ‘Being’ Iraklis Ioannidis (University of Glasgow) Abstract Research on Peirce’s phaneroscopy has been done with and through the paradigm or the conceptual schema of “Being” — what has been cri- tiqued by post-structuralist philosophers as the metaphysics of Being. Thus, such research is either limited to attempts to define “phaneron,” or to identify whether there is a particular and consistent meaning intention behind Peirce’s use of this term. Another problematic characteristic with such a way of engaging with phaneroscopy is the very anonymity of the schema of “Being.” While all scholars admit to the universality of “phaneron,” rarely, if ever, do we see an account of how such universal- ity can be instantiated. In this paper, I attempt to engage with phan- eroscopy differently. Instead of presenting a better version of what phaneroscopy is, or making arguments about what is the case with phan- eroscopy, both of which are ways of philosophising with “being,” I at- tempt to enact phaneroscopy. This would mean to undertake to follow Peirce’s instructions for the phaneroscopist and report the findings. Based on the latter, I shall analogise phaneron with the possibility of understanding. Finally, instead of having a conclusion which would im- ply an intention of making a case, and thus closure, I shall open up the 1 The first version of this paper was presented at the “Pragmatism and the Ana- lytic — Continental Split” conference held at the University of Sheffield in Au- gust 2017. I would like to thank Professor Shannon Dea, Professor James Wil- liams, and Dr.