Is the law of excluded middle a law of logic?


 Vanessa Cross
 5 years ago
 Views:
Transcription
1 Is the law of excluded middle a law of logic? Introduction I will conclude that the intuitionist s attempt to rule out the law of excluded middle as a law of logic fails. They do so by appealing to harmony as a constraint on the deductive rules we have in our logic, but I shall show that such an appeal throws the baby out with the bathwater. This is because an appeal to harmony precludes us from incorporating identity into our logic. Additionally, a failure to give a plausible and precise definition of harmony suggests that harmonytype approaches to logic fail. Conceptual Clarification The law of excluded middle is the syntactic law that: P P In other words, it is provable from no premises that; it is a theorem that P P. A syntactic law, unlike a semantic law, introduces a claim that is provable from the deductive rules alone no premises are needed. A semantic law is a claim that is true on all possible interpretations. This is important because the syntactic law of excluded middle is sometimes conflated with the semantic principle of bivalence, x(px (Tx Fx) Which operates on the universal domain, and where Px: x is a proposition, Tx: x is true, and Fx: x is false. In English, it claims that every proposition is either true or false. They are not the same law; I shall focus exclusively on the syntactic law of excluded middle LEM. A logic is constituted of a language, deductive system and semantics. The language is the vocabulary of primitive symbols (,, ) and grammar (a definition of a sentence). The deductive system is the rules that tell us what follows from what, and thus which particular theorems are possible in that logic. The semantics is a specification of what, in the world, the symbols refer to. Implications of losing LEM The law of excluded middle is a law of logic, if and only if, our logic is (1) correct and (2) contains a deductive system such that LEM is a theorem. To deny (2), our deductive system shouldn t be able to prove LEM from no premises. Then, our logic would have to exclude all those rules that enable LEM to be a theorem. This includes tertium non datur (TND), which states 1 : 1 I have directly copied these proofs from O Griffiths Part IB lecture handouts on Intuitionism.
2 Since TND can prove LEM: We d also have to ditch double negation elimination (DNE), since it, too, can prove LEM: As well as Pierce s law, which states that (A B) A) A: Then, we d also have to get rid of any derived rule, which is provable only from DNE. This includes one of the CQ rules that takes us from xfx to x Fx:
3 And we d also have to get rid of any derived rule, which is provable only from TND. This includes a De Morgan rule that takes us from (A B) to A B: So in summary, ditching LEM actually has wider implications. We also have to get rid of TND, DNE, Pierce s Law, a CQ rule and a DeM rule. Keeping all other deductive rules of classical logic, we have created a system equivalent to the deductive system of Intuitionistic logic. Intuitionistic Logic Intuitionistic logic has the same language as classical logic. It deviates from classical logic insofar as it gets rid of the above deductive rules. Having lost TND, intuitionists use contradiction introduction as a surrogate for a negation elimination rule. Intuitionistic semantics also deviates from classical logic; the standard semantics is called the BHK interpretation, after Brouwer, Heyting and Kolmogorov, but my focus is on the syntactical deductive laws. Intuitionistic logic is not a threevalued logic, nor does it claim (P P). It merely doesn t assert LEM as a theorem. It s important that it doesn t claim (P P), because then we could, via De Morgan, derive P P, which is a contradiction that even intuitionistic logicians recognise 2. Having articulated a logic which denies that the law of excluded middle is a law of logic, I shall examine the motivations for embracing Intuitionistic logic in the first place. Tonk and Harmony Which deductive rules should we have in our logic? There seem to be some deductive rules which we definitely shouldn t have. The classic example is of tonk 3. The introduction and elimination rules for tonk are: Introduction: If you have A on line m, you can show A tonk B on some further line, citing the rule TI m. Elimination: If you have A tonk B on line m, you can show A on some further line, or (inclusive or) you can show B on some other further line, citing the rule TE m. 2 O Griffiths, Lecture Series 3 A N Prior, The Runabout InferenceTicket
4 As such, tonkintroduction is similar to disjunction introduction, and tonkelimination is similar to conjunction elimination. However, tonk is problematic, insofar as it lets us move via deductive rules from any proposition to any proposition, e.g. (for any A and B): 1 A 2 A tonk B TI 1 3 B TE 2 And indeed, it can even result in contradiction from any premise A: 1 A 2 A tonk A TI 1 3 A TE 2 4 A A I 1, 3 5 Contradiction Contradiction introduction Tonk is problematic because it is clearly a terrible connective  in some way, its introduction and elimination rules are faulty. One can conceive of the broader issue as a demarcation problem: how do we demarcate the good deductive rules from the bad? What makes tonk rubbish, and the proper rules good? One answer is the quality of harmony 4. Roughly, an elimination rule and introduction rule are said to be in harmony with one another iff they tell us no more and no less than they ought. For example, tonk is not harmonious, because we can move from any premise, to any conclusion, so tonk can tell us far more than it ought. There is another motivation for implementing harmony as a constraint on the possible deductive rules we should accept. Assuming we already accept the following principle, harmony is a good constraint because it fulfills the principle of innocence 5, which states: it should not be possible, solely by engaging in deductive logical reasoning, to discover hitherto unknown (atomic) truths that we would have been incapable of discovering independently of logic Harmony fulfills the principle of innocence because it guarantees that our deductive rules won t tell us too much. (I am being deliberately vague about the notion of harmony because  as I shall argue  I doubt whether there is any way to precisely define harmony without the concept losing its intended force) Now that we have our answer to the demarcation problem, we can implement it. We can get rid of all the rules that are not harmonious, keeping only those that are harmonious. Application of Harmony to Classical Negation Harmony definitely excludes the tonkrules from our deductive system. But some have even argued that it excludes a classical elimination rule. In particular, it excludes DNE. This is because we can use DNE to arrive at a conclusion that we couldn t have otherwise established, as in the following proof: 4 N Belnap, Tonk, Plonk and Plink 5 F Steinberger, What Harmony Could and Could Not Be
5 In the above schematic proof, we arrive at A, which we couldn t have otherwise arrived at. Therefore, DNE isn t harmonious. Therefore, DNE shouldn t be one of our deductive rules. Therefore, we should accept a logic that doesn t include DNE as a deductive rule. One candidate (albeit not the only possible candidate) for this is intuitionistic logic. So, in the absence of any feasible alternatives, we should embrace intuitionistic logic. Intuitionistic logic doesn t include LEM as a law of logic. So LEM is not a law of logic. Criticism 1: Harmony excludes Identity I shall focus on two criticisms of the above argument for intuitionistic logic (and thus indirectly for the LEM). First, harmony is too strict a constraint on our deductive rules, because it excludes the deductive rules around identity, and so throws the baby out with the bathwater. If we embrace harmony, we have to get rid of the identity rules. We need the identity rules. So we shouldn t embrace harmony. Identity elimination is plausibly not a harmonious rule, because we can start from the premises a=b and Fa, to arrive at Fb, which we could not have otherwise arrived at: But we need some kind of identity elimination rule. So harmony is too strong. Although it successfully gets rid of tonk, in doing so, it gets rid of too much. So we should reject harmony as a constraint. We thus lose any reason to reject DNE as a deductive rule, and so too lose any motivation for intuitionistic logic. Read s harmonisation of identity Read 6 attempts to defend harmony, by giving a new identity elimination rule that is harmonious. The rule, he suggests, could operate: If (1) F does not appear in any undischarged assumptions other than i 6 S Read, Identity and Harmony
6 (2) On some line i we assume Fa, and within the same subproof conclude Fb we can conclude a=b on some further line, citing the rule =I However, even if Read s new rule is harmonious, it must be false. This is because, as Griffiths 7 points out, Read s identity rule assumes the identity of indiscernibles. Read s rule, in English, can be paraphrased as For all of a s properties, if b has those same properties, then a is identical to b. This is equivalent to the identity of indiscernibles, which states that: F x y((fx Fy) x=y) However, even if two objects have exactly the same properties, this is insufficient to show that they are one and the same object. For example, consider a universe in which the only existing objects are two black spheres of equal size, orbiting around each other. They share the exact same properties. And yet they are not identical. So no matter how many properties we show two objects to share, this is insufficient to show that they are identical. So Read s new rule must be wrong even if it is harmonious. So his defence of harmony fails, and so too does his indirect defence of intuitionistic logic. Criticism 2: Putting pressure on the very concept of harmony Just what is harmony anyway? My rough definition of harmony claimed that harmony obtains iff the introduction and elimination rules of a connective don t tell us anything new. What constitutes something new? It can t mean not obvious, because obviousness is a psychological quality, and harmony is a logical relation as Frege repeatedly maintained in Begriffsschrift, logic is not psychology. Also, if new meant not obvious, then, given that different claims have different degrees of obviousness for different people, we d be forced to say that different rules are harmonious for different people, and so different people should accept different logics dependent on what it obvious to them and not obvious to them. This seems wrong. We can redefine it to be a logical relationship. For example we can define harmony as that relation that obtains iff the introduction and elimination rules of a connective don t tell us anything we couldn t deduce from logical laws. But this, too, is problematic. Given that all logical laws are proved from, or are identical to, the introduction and elimination rules of the connectives, we can rephrase the definition, as the introduction and elimination rules of a connective don t tell us anything we couldn t deduce from the introduction and elimination rules of the connectives. And if we embrace this definition of harmony, then all connectives pass the test of harmony. All connectives will pass the test of harmony, because they will always tell you something you can deduce by applying that very same connective. For example, even tonk passes this test of harmony, because tonk will never tell you something that you can t deduce from the connectives (including tonk itself!). In other words, tonk will never surprise you, if you already accept tonk. So this definition is too liberal it lets in too many connectives. So perhaps we can restrict the latter clause of the above definition to only include other connectives than the one we re currently testing. On this definition, the tonk rules can t be used to show that tonk is harmonious. Our definition is now: Harmony is that relation that obtains iff the introduction and elimination rules of a connective don t tell us anything we couldn t deduce from the introduction and elimination rules of the other connectives. 7 O Griffiths, Harmonious Rules for Identity
7 However, this, too, is problematic, because it doesn t determine what the other connectives are. The upshot is that this definition seems to let in too many rules as a test for the particular connective we re testing. For example, if we re testing tonk for harmony on this definition, we just check and see if tonk can tell us anything that, for example,,, and any other nontonk connective can t tell us. This isn t so problematic. But then what happens when we test for harmony? We just check and see if can tell us anything that, for example,,, tonk can t tell us. But it seems wrong to include the tonkrules (or any other bizarre tonktype rules) as a test to see if is harmonious. In other words, the phrase the other connectives in the above definition is still too indeterminate, because it lets us use tonktyperules to assess other connectives. Alternatively, we could just stipulate that only the connectives we already accept should be included in the domain of other connectives. This excludes tonktyperules, because presumably nobody actually accepts tonktype rules. The problem with this approach is that it makes harmony relative to the deductive rules that one already accepts, which seems wrong. On this account of harmony, different rules may be harmonious for different people. So it runs into the same sort of trouble that defining harmony in psychological terms ran into. All this trouble in defining harmony suggests that there is no coherent formulation of the concept. As such, we d be better ditching the concept. As before argued, this leaves us with no motivation to get rid of DNE, and so, no motivation to embrace intuitionistic logic, and so, no motivation to exclude the law of excluded middle from our logic. Conclusion I have shown that one attempt to argue for intuitionistic logic, which excludes the law of excluded middle, fails. If we use harmony as a constraint on our deductive rules, we throw the baby out with the bathwater by removing identity also. But an appeal to harmony has even deeper problems: there is doubt as to whether harmony can even be precisely defined in the first place. As such, one attempt to show that the law of excluded middle is not a law of logic, fails.
UC Berkeley, Philosophy 142, Spring 2016
Logical Consequence UC Berkeley, Philosophy 142, Spring 2016 John MacFarlane 1 Intuitive characterizations of consequence Modal: It is necessary (or apriori) that, if the premises are true, the conclusion
More informationConstructive Logic, Truth and Warranted Assertibility
Constructive Logic, Truth and Warranted Assertibility Greg Restall Department of Philosophy Macquarie University Version of May 20, 2000....................................................................
More informationAn Introduction to. Formal Logic. Second edition. Peter Smith, February 27, 2019
An Introduction to Formal Logic Second edition Peter Smith February 27, 2019 Peter Smith 2018. Not for reposting or recirculation. Comments and corrections please to ps218 at cam dot ac dot uk 1 What
More informationSemantics and the Justification of Deductive Inference
Semantics and the Justification of Deductive Inference Ebba Gullberg ebba.gullberg@philos.umu.se Sten Lindström sten.lindstrom@philos.umu.se Umeå University Abstract Is it possible to give a justification
More informationA Liar Paradox. Richard G. Heck, Jr. Brown University
A Liar Paradox Richard G. Heck, Jr. Brown University It is widely supposed nowadays that, whatever the right theory of truth may be, it needs to satisfy a principle sometimes known as transparency : Any
More informationPotentialism about set theory
Potentialism about set theory Øystein Linnebo University of Oslo SotFoM III, 21 23 September 2015 Øystein Linnebo (University of Oslo) Potentialism about set theory 21 23 September 2015 1 / 23 Openendedness
More informationWRIGHT ON BORDERLINE CASES AND BIVALENCE 1
WRIGHT ON BORDERLINE CASES AND BIVALENCE 1 HAMIDREZA MOHAMMADI Abstract. The aim of this paper is, firstly to explain Crispin Wright s quandary view of vagueness, his intuitionistic response to sorites
More informationSemantic Entailment and Natural Deduction
Semantic Entailment and Natural Deduction Alice Gao Lecture 6, September 26, 2017 Entailment 1/55 Learning goals Semantic entailment Define semantic entailment. Explain subtleties of semantic entailment.
More informationLecture Notes on Classical Logic
Lecture Notes on Classical Logic 15317: Constructive Logic William Lovas Lecture 7 September 15, 2009 1 Introduction In this lecture, we design a judgmental formulation of classical logic To gain an intuition,
More informationAppeared in: AlMukhatabat. A Trilingual Journal For Logic, Epistemology and Analytical Philosophy, Issue 6: April 2013.
Appeared in: AlMukhatabat. A Trilingual Journal For Logic, Epistemology and Analytical Philosophy, Issue 6: April 2013. Panu Raatikainen Intuitionistic Logic and Its Philosophy Formally, intuitionistic
More informationModule 5. Knowledge Representation and Logic (Propositional Logic) Version 2 CSE IIT, Kharagpur
Module 5 Knowledge Representation and Logic (Propositional Logic) Lesson 12 Propositional Logic inference rules 5.5 Rules of Inference Here are some examples of sound rules of inference. Each can be shown
More informationVagueness and supervaluations
Vagueness and supervaluations UC Berkeley, Philosophy 142, Spring 2016 John MacFarlane 1 Supervaluations We saw two problems with the threevalued approach: 1. sharp boundaries 2. counterintuitive consequences
More informationScott Soames: Understanding Truth
Philosophy and Phenomenological Research Vol. LXV, No. 2, September 2002 Scott Soames: Understanding Truth MAlTHEW MCGRATH Texas A & M University Scott Soames has written a valuable book. It is unmatched
More informationStudy Guides. Chapter 1  Basic Training
Study Guides Chapter 1  Basic Training Argument: A group of propositions is an argument when one or more of the propositions in the group is/are used to give evidence (or if you like, reasons, or grounds)
More informationA Guide to FOL Proof Rules ( for Worksheet 6)
A Guide to FOL Proof Rules ( for Worksheet 6) This lesson sheet will be a good deal like last class s. This time, I ll be running through the proof rules relevant to FOL. Of course, when you re doing any
More informationWhat would count as Ibn Sīnā (11th century Persia) having first order logic?
1 2 What would count as Ibn Sīnā (11th century Persia) having first order logic? Wilfrid Hodges Herons Brook, Sticklepath, Okehampton March 2012 http://wilfridhodges.co.uk Ibn Sina, 980 1037 3 4 Ibn Sīnā
More informationPHILOSOPHY OF LOGIC AND LANGUAGE OVERVIEW FREGE JONNY MCINTOSH 1. FREGE'S CONCEPTION OF LOGIC
PHILOSOPHY OF LOGIC AND LANGUAGE JONNY MCINTOSH 1. FREGE'S CONCEPTION OF LOGIC OVERVIEW These lectures cover material for paper 108, Philosophy of Logic and Language. They will focus on issues in philosophy
More informationAyer on the criterion of verifiability
Ayer on the criterion of verifiability November 19, 2004 1 The critique of metaphysics............................. 1 2 Observation statements............................... 2 3 In principle verifiability...............................
More informationParadox of Deniability
1 Paradox of Deniability Massimiliano Carrara FISPPA Department, University of Padua, Italy Peking University, Beijing  6 November 2018 Introduction. The starting elements Suppose two speakers disagree
More informationExercise Sets. KS Philosophical Logic: Modality, Conditionals Vagueness. Dirk Kindermann University of Graz July 2014
Exercise Sets KS Philosophical Logic: Modality, Conditionals Vagueness Dirk Kindermann University of Graz July 2014 1 Exercise Set 1 Propositional and Predicate Logic 1. Use Definition 1.1 (Handout I Propositional
More informationCan Negation be Defined in Terms of Incompatibility?
Can Negation be Defined in Terms of Incompatibility? Nils Kurbis 1 Introduction Every theory needs primitives. A primitive is a term that is not defined any further, but is used to define others. Thus
More informationChapter 9 Sentential Proofs
Logic: A Brief Introduction Ronald L. Hall, Stetson University Chapter 9 Sentential roofs 9.1 Introduction So far we have introduced three ways of assessing the validity of truthfunctional arguments.
More informationWhat is the Nature of Logic? Judy Pelham Philosophy, York University, Canada July 16, 2013 PanHellenic Logic Symposium Athens, Greece
What is the Nature of Logic? Judy Pelham Philosophy, York University, Canada July 16, 2013 PanHellenic Logic Symposium Athens, Greece Outline of this Talk 1. What is the nature of logic? Some history
More informationCan Negation be Defined in Terms of Incompatibility?
Can Negation be Defined in Terms of Incompatibility? Nils Kurbis 1 Abstract Every theory needs primitives. A primitive is a term that is not defined any further, but is used to define others. Thus primitives
More informationVerificationism. PHIL September 27, 2011
Verificationism PHIL 83104 September 27, 2011 1. The critique of metaphysics... 1 2. Observation statements... 2 3. In principle verifiability... 3 4. Strong verifiability... 3 4.1. Conclusive verifiability
More informationA Puzzle about Knowing Conditionals i. (final draft) Daniel Rothschild University College London. and. Levi Spectre The Open University of Israel
A Puzzle about Knowing Conditionals i (final draft) Daniel Rothschild University College London and Levi Spectre The Open University of Israel Abstract: We present a puzzle about knowledge, probability
More information16. Universal derivation
16. Universal derivation 16.1 An example: the Meno In one of Plato s dialogues, the Meno, Socrates uses questions and prompts to direct a young slave boy to see that if we want to make a square that has
More informationWilliams on Supervaluationism and Logical Revisionism
Williams on Supervaluationism and Logical Revisionism Nicholas K. Jones Noncitable draft: 26 02 2010. Final version appeared in: The Journal of Philosophy (2011) 108: 11: 633641 Central to discussion
More informationRussell: On Denoting
Russell: On Denoting DENOTING PHRASES Russell includes all kinds of quantified subject phrases ( a man, every man, some man etc.) but his main interest is in definite descriptions: the present King of
More informationArtificial Intelligence: Valid Arguments and Proof Systems. Prof. Deepak Khemani. Department of Computer Science and Engineering
Artificial Intelligence: Valid Arguments and Proof Systems Prof. Deepak Khemani Department of Computer Science and Engineering Indian Institute of Technology, Madras Module 02 Lecture  03 So in the last
More informationThe distinction between truthfunctional and nontruthfunctional logical and linguistic
FORMAL CRITERIA OF NONTRUTHFUNCTIONALITY Dale Jacquette The Pennsylvania State University 1. TruthFunctional Meaning The distinction between truthfunctional and nontruthfunctional logical and linguistic
More informationLeibniz, Principles, and Truth 1
Leibniz, Principles, and Truth 1 Leibniz was a man of principles. 2 Throughout his writings, one finds repeated assertions that his view is developed according to certain fundamental principles. Attempting
More informationTruth and Modality  can they be reconciled?
Truth and Modality  can they be reconciled? by Eileen Walker 1) The central question What makes modal statements statements about what might be or what might have been the case true or false? Normally
More informationTOWARDS A PHILOSOPHICAL UNDERSTANDING OF THE LOGICS OF FORMAL INCONSISTENCY
CDD: 160 http://dx.doi.org/10.1590/01006045.2015.v38n2.wcear TOWARDS A PHILOSOPHICAL UNDERSTANDING OF THE LOGICS OF FORMAL INCONSISTENCY WALTER CARNIELLI 1, ABÍLIO RODRIGUES 2 1 CLE and Department of
More informationDay 3. Wednesday May 23, Learn the basic building blocks of proofs (specifically, direct proofs)
Day 3 Wednesday May 23, 2012 Objectives: Learn the basics of Propositional Logic Learn the basic building blocks of proofs (specifically, direct proofs) 1 Propositional Logic Today we introduce the concepts
More informationA Judgmental Formulation of Modal Logic
A Judgmental Formulation of Modal Logic Sungwoo Park Pohang University of Science and Technology South Korea Estonian Theory Days Jan 30, 2009 Outline Study of logic Model theory vs Proof theory Classical
More informationA Logical Approach to Metametaphysics
A Logical Approach to Metametaphysics Daniel Durante Departamento de Filosofia UFRN durante10@gmail.com 3º Filomena  2017 What we take as true commits us. Quine took advantage of this fact to introduce
More informationRemarks on a Foundationalist Theory of Truth. Anil Gupta University of Pittsburgh
For Philosophy and Phenomenological Research Remarks on a Foundationalist Theory of Truth Anil Gupta University of Pittsburgh I Tim Maudlin s Truth and Paradox offers a theory of truth that arises from
More informationSemantic Foundations for Deductive Methods
Semantic Foundations for Deductive Methods delineating the scope of deductive reason Roger Bishop Jones Abstract. The scope of deductive reason is considered. First a connection is discussed between the
More information2.3. Failed proofs and counterexamples
2.3. Failed proofs and counterexamples 2.3.0. Overview Derivations can also be used to tell when a claim of entailment does not follow from the principles for conjunction. 2.3.1. When enough is enough
More informationIntuitive evidence and formal evidence in proofformation
Intuitive evidence and formal evidence in proofformation Okada Mitsuhiro Section I. Introduction. I would like to discuss proof formation 1 as a general methodology of sciences and philosophy, with a
More informationFrom Necessary Truth to Necessary Existence
Prequel for Section 4.2 of Defending the Correspondence Theory Published by PJP VII, 1 From Necessary Truth to Necessary Existence Abstract I introduce new details in an argument for necessarily existing
More informationComments on Truth at A World for Modal Propositions
Comments on Truth at A World for Modal Propositions Christopher Menzel Texas A&M University March 16, 2008 Since Arthur Prior first made us aware of the issue, a lot of philosophical thought has gone into
More informationA Note on a Remark of Evans *
Penultimate draft of a paper published in the Polish Journal of Philosophy 10 (2016), 715. DOI: 10.5840/pjphil20161028 A Note on a Remark of Evans * Wolfgang Barz Johann Wolfgang GoetheUniversität Frankfurt
More informationWhat is Logical Validity?
What is Logical Validity? Whatever other merits prooftheoretic and modeltheoretic accounts of validity may have, they are not remotely plausible as accounts of the meaning of valid. And not just because
More informationA. Problem set #3 it has been posted and is due Tuesday, 15 November
Lecture 9: Propositional Logic I Philosophy 130 1 & 3 November 2016 O Rourke & Gibson I. Administrative A. Problem set #3 it has been posted and is due Tuesday, 15 November B. I am working on the group
More informationInternational Phenomenological Society
International Phenomenological Society The Semantic Conception of Truth: and the Foundations of Semantics Author(s): Alfred Tarski Source: Philosophy and Phenomenological Research, Vol. 4, No. 3 (Mar.,
More informationThe Greatest Mistake: A Case for the Failure of Hegel s Idealism
The Greatest Mistake: A Case for the Failure of Hegel s Idealism What is a great mistake? Nietzsche once said that a great error is worth more than a multitude of trivial truths. A truly great mistake
More informationPHI 1500: Major Issues in Philosophy
PHI 1500: Major Issues in Philosophy Session 3 September 9 th, 2015 All About Arguments (Part II) 1 A common theme linking many fallacies is that they make unwarranted assumptions. An assumption is a claim
More informationMaudlin s Truth and Paradox Hartry Field
Maudlin s Truth and Paradox Hartry Field Tim Maudlin s Truth and Paradox is terrific. In some sense its solution to the paradoxes is familiar the book advocates an extension of what s called the KripkeFeferman
More informationPhilosophy of Mathematics Nominalism
Philosophy of Mathematics Nominalism Owen Griffiths oeg21@cam.ac.uk Churchill and Newnham, Cambridge 8/11/18 Last week Ante rem structuralism accepts mathematical structures as Platonic universals. We
More information3.3. Negations as premises Overview
3.3. Negations as premises 3.3.0. Overview A second group of rules for negation interchanges the roles of an affirmative sentence and its negation. 3.3.1. Indirect proof The basic principles for negation
More informationLing 98a: The Meaning of Negation (Week 1)
Yimei Xiang yxiang@fas.harvard.edu 17 September 2013 1 What is negation? Negation in twovalued propositional logic Based on your understanding, select out the metaphors that best describe the meaning
More informationIntersubstitutivity Principles and the Generalization Function of Truth. Anil Gupta University of Pittsburgh. Shawn Standefer University of Melbourne
Intersubstitutivity Principles and the Generalization Function of Truth Anil Gupta University of Pittsburgh Shawn Standefer University of Melbourne Abstract We offer a defense of one aspect of Paul Horwich
More informationTruth At a World for Modal Propositions
Truth At a World for Modal Propositions 1 Introduction Existentialism is a thesis that concerns the ontological status of individual essences and singular propositions. Let us define an individual essence
More informationDoes Deduction really rest on a more secure epistemological footing than Induction?
Does Deduction really rest on a more secure epistemological footing than Induction? We argue that, if deduction is taken to at least include classical logic (CL, henceforth), justifying CL  and thus deduction
More information1. Introduction. Against GMR: The Incredulous Stare (Lewis 1986: 133 5).
Lecture 3 Modal Realism II James Openshaw 1. Introduction Against GMR: The Incredulous Stare (Lewis 1986: 133 5). Whatever else is true of them, today s views aim not to provoke the incredulous stare.
More informationReductio ad Absurdum, Modulation, and Logical Forms. Miguel LópezAstorga 1
International Journal of Philosophy and Theology June 25, Vol. 3, No., pp. 5965 ISSN: 2333575 (Print), 23335769 (Online) Copyright The Author(s). All Rights Reserved. Published by American Research
More informationEntailment, with nods to Lewy and Smiley
Entailment, with nods to Lewy and Smiley Peter Smith November 20, 2009 Last week, we talked a bit about the AndersonBelnap logic of entailment, as discussed in Priest s Introduction to NonClassical Logic.
More informationSqueezing arguments. Peter Smith. May 9, 2010
Squeezing arguments Peter Smith May 9, 2010 Many of our concepts are introduced to us via, and seem only to be constrained by, roughandready explanations and some sample paradigm positive and negative
More informationINTUITION AND CONSCIOUS REASONING
The Philosophical Quarterly Vol. 63, No. 253 October 2013 ISSN 00318094 doi: 10.1111/14679213.12071 INTUITION AND CONSCIOUS REASONING BY OLE KOKSVIK This paper argues that, contrary to common opinion,
More informationLGCS 199DR: Independent Study in Pragmatics
LGCS 99DR: Independent Study in Pragmatics Jesse Harris & Meredith Landman September 0, 203 Last class, we discussed the difference between semantics and pragmatics: Semantics The study of the literal
More informationSubjective Logic: Logic as Rational Belief Dynamics. Richard Johns Department of Philosophy, UBC
Subjective Logic: Logic as Rational Belief Dynamics Richard Johns Department of Philosophy, UBC johns@interchange.ubc.ca May 8, 2004 What I m calling Subjective Logic is a new approach to logic. Fundamentally
More informationUnderstanding Truth Scott Soames Précis Philosophy and Phenomenological Research Volume LXV, No. 2, 2002
1 Symposium on Understanding Truth By Scott Soames Précis Philosophy and Phenomenological Research Volume LXV, No. 2, 2002 2 Precis of Understanding Truth Scott Soames Understanding Truth aims to illuminate
More informationTHESES SIS/LIBRARY TELEPHONE:
THESES SIS/LIBRARY TELEPHONE: +61 2 6125 4631 R.G. MENZIES LIBRARY BUILDING NO:2 FACSIMILE: +61 2 6125 4063 THE AUSTRALIAN NATIONAL UNIVERSITY EMAIL: library.theses@anu.edu.au CANBERRA ACT 0200 AUSTRALIA
More information1. Lukasiewicz s Logic
Bulletin of the Section of Logic Volume 29/3 (2000), pp. 115 124 Dale Jacquette AN INTERNAL DETERMINACY METATHEOREM FOR LUKASIEWICZ S AUSSAGENKALKÜLS Abstract An internal determinacy metatheorem is proved
More informationThe Relationship between the Truth Value of Premises and the Truth Value of Conclusions in Deductive Arguments
The Relationship between the Truth Value of Premises and the Truth Value of Conclusions in Deductive Arguments I. The Issue in Question This document addresses one single question: What are the relationships,
More informationA Solution to the Gettier Problem Keota Fields. the three traditional conditions for knowledge, have been discussed extensively in the
A Solution to the Gettier Problem Keota Fields Problem cases by Edmund Gettier 1 and others 2, intended to undermine the sufficiency of the three traditional conditions for knowledge, have been discussed
More informationpart one MACROSTRUCTURE Cambridge University Press X  A Theory of Argument Mark Vorobej Excerpt More information
part one MACROSTRUCTURE 1 Arguments 1.1 Authors and Audiences An argument is a social activity, the goal of which is interpersonal rational persuasion. More precisely, we ll say that an argument occurs
More informationPhilosophy of Mathematics Kant
Philosophy of Mathematics Kant Owen Griffiths oeg21@cam.ac.uk St John s College, Cambridge 20/10/15 Immanuel Kant Born in 1724 in Königsberg, Prussia. Enrolled at the University of Königsberg in 1740 and
More informationReply to Kit Fine. Theodore Sider July 19, 2013
Reply to Kit Fine Theodore Sider July 19, 2013 Kit Fine s paper raises important and difficult issues about my approach to the metaphysics of fundamentality. In chapters 7 and 8 I examined certain subtle
More informationLogic & Proofs. Chapter 3 Content. Sentential Logic Semantics. Contents: Studying this chapter will enable you to:
Sentential Logic Semantics Contents: TruthValue Assignments and TruthFunctions TruthValue Assignments TruthFunctions Introduction to the TruthLab TruthDefinition Logical Notions TruthTrees Studying
More informationThe way we convince people is generally to refer to sufficiently many things that they already know are correct.
Theorem A Theorem is a valid deduction. One of the key activities in higher mathematics is identifying whether or not a deduction is actually a theorem and then trying to convince other people that you
More informationBeyond Symbolic Logic
Beyond Symbolic Logic 1. The Problem of Incompleteness: Many believe that mathematics can explain *everything*. Gottlob Frege proposed that ALL truths can be captured in terms of mathematical entities;
More informationA BRIEF INTRODUCTION TO LOGIC FOR METAPHYSICIANS
A BRIEF INTRODUCTION TO LOGIC FOR METAPHYSICIANS 0. Logic, Probability, and Formal Structure Logic is often divided into two distinct areas, inductive logic and deductive logic. Inductive logic is concerned
More informationLogic and Pragmatics: linear logic for inferential practice
Logic and Pragmatics: linear logic for inferential practice Daniele Porello danieleporello@gmail.com Institute for Logic, Language & Computation (ILLC) University of Amsterdam, Plantage Muidergracht 24
More informationComments on Saul Kripke s Philosophical Troubles
Comments on Saul Kripke s Philosophical Troubles Theodore Sider Disputatio 5 (2015): 67 80 1. Introduction My comments will focus on some loosely connected issues from The First Person and Frege s Theory
More informationPrimitive Thisness and Primitive Identity Robert Merrihew Adams
Robert Merrihew Adams Let us begin at the end, where Adams states simply the view that, he says, he has defended in his paper: Thisnesses and transworld identities are primitive but logically connected
More informationA Defense of Contingent Logical Truths
Michael Nelson and Edward N. Zalta 2 A Defense of Contingent Logical Truths Michael Nelson University of California/Riverside and Edward N. Zalta Stanford University Abstract A formula is a contingent
More informationConditionals II: no truth conditions?
Conditionals II: no truth conditions? UC Berkeley, Philosophy 142, Spring 2016 John MacFarlane 1 Arguments for the material conditional analysis As Edgington [1] notes, there are some powerful reasons
More informationINTERMEDIATE LOGIC Glossary of key terms
1 GLOSSARY INTERMEDIATE LOGIC BY JAMES B. NANCE INTERMEDIATE LOGIC Glossary of key terms This glossary includes terms that are defined in the text in the lesson and on the page noted. It does not include
More informationILLOCUTIONARY ORIGINS OF FAMILIAR LOGICAL OPERATORS
ILLOCUTIONARY ORIGINS OF FAMILIAR LOGICAL OPERATORS 1. ACTS OF USING LANGUAGE Illocutionary logic is the logic of speech acts, or language acts. Systems of illocutionary logic have both an ontological,
More informationLogic I, Fall 2009 Final Exam
24.241 Logic I, Fall 2009 Final Exam You may not use any notes, handouts, or other material during the exam. All cell phones must be turned off. Please read all instructions carefully. Good luck with the
More information(Some More) Vagueness
(Some More) Vagueness Otávio Bueno Department of Philosophy University of Miami Coral Gables, FL 33124 Email: otaviobueno@mac.com Three features of vague predicates: (a) borderline cases It is common
More informationLogical Mistakes, Logical Aliens, and the Laws of Kant's Pure General Logic Chicago February 21 st 2018 Tyke Nunez
Logical Mistakes, Logical Aliens, and the Laws of Kant's Pure General Logic Chicago February 21 st 2018 Tyke Nunez 1 Introduction (1) Normativists: logic's laws are unconditional norms for how we ought
More informationLOGIC ANTHONY KAPOLKA FYF 1019/3/2010
LOGIC ANTHONY KAPOLKA FYF 1019/3/2010 LIBERALLY EDUCATED PEOPLE......RESPECT RIGOR NOT SO MUCH FOR ITS OWN SAKE BUT AS A WAY OF SEEKING TRUTH. LOGIC PUZZLE COOPER IS MURDERED. 3 SUSPECTS: SMITH, JONES,
More informationIntuitionistic Epistemic Logic
Intuitionistic Epistemic Logic arxiv:1406.1582v4 [math.lo] 16 Jan 2016 Sergei Artemov & Tudor Protopopescu The CUNY Graduate Center 365 Fifth Avenue, rm. 4329 New York City, NY 10016, USA January 19, 2016
More informationROBERT STALNAKER PRESUPPOSITIONS
ROBERT STALNAKER PRESUPPOSITIONS My aim is to sketch a general abstract account of the notion of presupposition, and to argue that the presupposition relation which linguists talk about should be explained
More informationQuandary and Intuitionism: Crispin Wright on Vagueness
Forthcoming in A. Miller (ed), Essays for Crispin Wright: Logic, Language and Mathematics (OUP) Quandary and Intuitionism: Crispin Wright on Vagueness Stephen Schiffer New York University I 1. The philosophical
More informationMoral Psychology
MIT OpenCourseWare http://ocw.mit.edu 24.120 Moral Psychology Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 24.120 MORAL PSYCHOLOGY RICHARD
More informationAm I free? Freedom vs. Fate
Am I free? Freedom vs. Fate We ve been discussing the free will defense as a response to the argument from evil. This response assumes something about us: that we have free will. But what does this mean?
More informationChapter Six. Putnam's AntiRealism
119 Chapter Six Putnam's AntiRealism So far, our discussion has been guided by the assumption that there is a world and that sentences are true or false by virtue of the way it is. But this assumption
More informationPhilosophical Perspectives, 16, Language and Mind, 2002 THE AIM OF BELIEF 1. Ralph Wedgwood Merton College, Oxford
Philosophical Perspectives, 16, Language and Mind, 2002 THE AIM OF BELIEF 1 Ralph Wedgwood Merton College, Oxford 0. Introduction It is often claimed that beliefs aim at the truth. Indeed, this claim has
More informationAn Inferentialist Conception of the A Priori. Ralph Wedgwood
An Inferentialist Conception of the A Priori Ralph Wedgwood When philosophers explain the distinction between the a priori and the a posteriori, they usually characterize the a priori negatively, as involving
More informationHåkan Salwén. Hume s Law: An Essay on Moral Reasoning Lorraine BesserJones Volume 31, Number 1, (2005) 177180. Your use of the HUME STUDIES archive indicates your acceptance of HUME STUDIES Terms and
More informationNatural Deduction for Sentence Logic
Natural Deduction for Sentence Logic Derived Rules and Derivations without Premises We will pursue the obvious strategy of getting the conclusion by constructing a subderivation from the assumption of
More informationConstructive Logic for All
Constructive Logic for All Greg Restall Philosophy Department Macquarie University June 14, 2000 Abstract It is a commonplace in recent metaphysics that one s logical commitments go hand in hand with one
More informationSome remarks on verificationism, constructivism and the Principle of Excluded Middle in the context of Colour Exclusion Problem
URRJ 5 th June, 2017 Some remarks on verificationism, constructivism and the Principle of Excluded Middle in the context of Colour Exclusion Problem Marcos Silva marcossilvarj@gmail.com https://sites.google.com/site/marcossilvarj/
More information2.1 Review. 2.2 Inference and justifications
Applied Logic Lecture 2: Evidence Semantics for Intuitionistic Propositional Logic Formal logic and evidence CS 4860 Fall 2012 Tuesday, August 28, 2012 2.1 Review The purpose of logic is to make reasoning
More informationLecture 3. I argued in the previous lecture for a relationist solution to Frege's puzzle, one which
1 Lecture 3 I argued in the previous lecture for a relationist solution to Frege's puzzle, one which posits a semantic difference between the pairs of names 'Cicero', 'Cicero' and 'Cicero', 'Tully' even
More information