# BARRY LOEWER LEIBNIZ AND THE ONTOLOGICAL ARGUMENT* (Received 2 February, 1977)

Save this PDF as:

Size: px
Start display at page:

## Transcription

2 106 BARRY LOEWER correct then a proof of 'It is possible that God exists' would quite literally be required to complete the ontological argument. In this paper I will formalize a version of the ontological argument, which seems to be Leibniz's version, in a system of modal logic. It turns out that in this formal system the paraphrase of 'If it is possible that a necessary being exists then a necessary being exists' is valid. Furthermore, the paraphrase of 'A necessary being exists' is consistent in this system. If we follow Leibniz and conclude from this that 'It is possible that a necessary being exists' is true, we arrive at what seems to be a valid and sound version of the ontological argument. I will call the line of argumentation just sketched 'Leibniz's strategy'. However, I do not claim that Leibniz clearly viewed himself as following this strategy. Be that as it may, the strategy is suggested by Leibniz's discussion of the ontological argument and is worth pursuing for its own sake. I will formalize the argument in a system of modal logic closely related to the system devised by Kripke in 'Semantical Considerations on Modal Logic'. s Let L be a first order language with modal operators. A K-structure for L is a triple, (I4/, D, f) where W is a non-empty set of possible worlds, D is a nonempty set of possible individuals, and f is a function which assigns to each member w of W a subset of D. The value of f(w) is the set of individuals which exist at w. The definition of an interpretation of L on (W, D, f) proceeds is the usual manner. For our purposes it will suffice to note the clauses for necessity and quantification. NA is true at w iffa is true at every u E IV. 3 xa is true at w iff Ab/x is true at w for some individual b which exists at w. A formula B of L is K-valid if and only if for every K-structure (W, D, f) and every interpretation on (W, D, f) B is true at each member of W. According to Leibniz, it is part of the definition of God that existence belongs to God's essence. This is expressed in L by (1) N((x)(Gx DN 3y(y =x 9 Gx). 6 Formula (1) says that it is necessary that if anything is God then it necessarily is God and it exists. Although (1) does not imply 3xGx it does imply (2) P 3 xgx ~ 3 xgx. The proof goes as follows: Suppose that P 3 xgx is true at w. Then there is a u E W at which 3 xgx is true. So there is an individual b which exists at u and Gb holds at u. Notice that (1) implies that (x)(gx D N 3 y O' = x " Gx))

3 LEIBNIZ AND THE ONTOLOGICAL ARGUMENT 107 is true at u, so N( 3y)y = b 9 Gb) holds at u. But this implies that 3 xgx holds at w. A similar proof shows that (3) P3xN3y(.y=x) D 3xN3y(y=x) is K-valid. 7 Leibniz's strategy for demonstrating that God is possible is to show that the concept of God is consistent; that 3 xgx does not imply a contradiction. His argument that 3 xgx is consistent is based on the idea that God is the being which possesses every perfection. I call every simple quality which is positive and absolute, or expresses whatever it expresses without any limits, a perfection. But a quality of this sort, because it is simple, is therefore irresolvable or indefinable, for otherwise either it will not be a simple quality but an aggregate of many, or, if it is one, it will be circumscribed by limits and so be known through negations of further progress contrary to the hypothesis, for a purely positive quality was assumed. From these considerations it is not difficult to show that all perfections are compatible with each other or can exist in the same subject. ~ Leibniz's argument that 'there is a perfect being' is consistent is proof- theoretic. He reasons that for this statement to be contradictory two of the perfections A, B must be incompatible. But this could happen, he claims, only if either A, B were negations of one another or one limited the other. But since A, B are simple, positive, and absolute neither of these situations can arise. Leibniz claims that the preceding argument demonstrates that it is possible that a perfect being or God exists. Since necessary existence is, according to Leibniz, a perfection, it follows by the ontological argument that a perfect being exists. Most commentators have found Leibniz's argument that 3 xgx is con- sistent unpersuasive. For example, Norman Malcolm comments For another thing, it assumes that some qualities are intrinsically simple. I believe that Wittgenstein has shown in the Investigations that nothing is intrinsically simple, but that whatever has the status of a simple, an indefinable, in one system of concepts, may have the status of a complex thing, a definable thing, in another system of concepts.9 Although proving that the idea of a perfect being is consistent may face insurmountable difficulties it is an easy matter to show that 3xN 3y(y = x) is K-consistent. If we follow Leibniz we will then conclude the truth of (4) P3xN3y(y =x). Formulas (4) and (3) together imply 3 xn 3yO' = x). So it seems that by

4 108 BARRY LOEWER pursuing 'Leibniz's strategy' we have produced a valid and sound ontological argument. II The preceding version of the ontological argument implicitly appealed to the following principle: (5) For all A if A is K-consistent then PA is true (at the actual world). Clearly Leibniz also assumed that possibility is identifiable with consistency although, of course, he did not relativize consistency to a particular formal system. However, it turns out that (5) is false. This can be easily seen by noting that both 3 xn 3 y (y = x) and - 3 xn 3 y (y = x) are K-consistent but there can be no K-structure and interpretation which contains a world at which both P3xN3y(y =x) and P(-3xN3y(y =x) are true. (Nor can there be a K-structure which contains a world at which P 3 xn 3 y (V = x) is true and a world at which P- ~ xn 3y(y = x) is true.) This shows that our Leibnizian argument for (4) fails since it is based on an appeal to (5). The question naturally arises whether there are systems of modal logic K* in which a principle analagous to (5) (but for K*) holds. The answer is affirmative. Define a K* structure (W, D, f) and interpretation I* on it as before with the added restriction (R) For each atomic predicate R of L and each subset E of D and each subset U of D there is a w~ W such that f(w) =E and I* (R, w) = U. K* structures are also K structures so system K is a subsystem of system K*. Although the analogue for principle (5) holds for K* it turns out that 3 xn3 y(y = x) is not K*-consistent. This is so simply because (R) requires that for each subset of D there is a world whose existants are precisely that subset. This rules out there being any necessary existants. Are there any systems K', at which the analogue of(5)holds and in which 3 xn3y(y =x) is K'-consistent? The answer here is also affirmative. We can produce such a system by dropping the requirement in (R) that to each subset of E of D there is a w E W such that f(w) = E and replace it by

5 LEIBNIZ AND THE ONTOLOGICAL ARGUMENT 109 the requirement that for each w, f(w)= D. Notice that in this system 3 xn 3y(y = x) is not merely consistent it is valid. One can more easily see what is going on here by noting that the contrapositive of (5) for K' is (6) IfNA holds at w then A is K'-valid. This means that the only systems in which an analogue to (5) holds and in which 3 xn3 y(y = x) is consistent are systems in which 3 xn3 y(y = x) is already valid. So an analogue to (5) can be used to establish P 3 xn3 y(y = x) only for systems in which 3 xn3 y(y = x) is valid. The attempted demonstration of P3xN3y(y =x) via the consistency of 3xN3y(y =x) turns out to be completely redundant. The preceding discussion shows that 'Leibniz's strategy', though it is tempting to follow, is ultimately unsuccessful. One may, of course, still find a demonstration that the concept of God contains no hidden contradictions desirable, since if the concept were contradictory no sound argument, ontological or otherwise, could establish that God exists. University of South Carolina NOTES * I would like to thank Professors Robert Mulvaney and Ferdinand Schoeman for helpful comments on previous versions of this paper. i Gottfried Wilhelm Leibniz: Philosophical Papers and Letters, 2nd ed., ed. and transl. by L. E. Loemker (Dordrecht, Holland 1969), p. 386, hereafter cited as Loemker. 2 Loemker, p Loemker, p Loemker, p. 386,293. s This system is practically identical to the system of quantified modal logic devised by Saul Kripke in 'Semantical Considerations on Modal Logic'. In Reference and Modality, ed. by L. Linsky, Oxford University Press, Glasgow, 1971, pp NEy (V = x) is true of b at w, just in case b exists at every possible world; i.e., existence is part of b's essence. 7 Charles Hartshorne, Norman Malcolm, and Alvin Plantinga each formulate versions of the ontological argument whose conclusions are expressed by (2) or (3). The relevant articles are reprinted in The Ontological Argument, ed. by A. Plantinga, Anchor Books Plantinga also formulates a version of the ontological argument in terms of possible world semantics in The Nature of Necessity, Clarendon Press, Oxford, Leibniz; 'Two Notations for Discussion with Spinoza' in The Ontological Argument, ed. by A. Plantinga, Anchor Books, 1965, pp Norman Malcolm, ibid., p. 157.

### ST ANSELM S VERSION OF THE ONTOLOGICAL ARGUMENT Anselm s argument relies on conceivability :

Michael Lacewing The ontological argument St Anselm and Descartes both famously presented an ontological argument for the existence of God. (The word ontological comes from ontology, the study of (-ology)

### Gödel s Ontological Proof of the Existence of God

Prof. Dr. Elke Brendel Institut für Philosophie Lehrstuhl für Logik und Grundlagenforschung g Rheinische Friedrich-Wilhelms-Universität Bonn ebrendel@uni-bonn.de Gödel s Ontological Proof of the Existence

### PHILOSOPHY 4360/5360 METAPHYSICS

PHILOSOPHY 4360/5360 METAPHYSICS Topic IV: The Nature of the Mind Arguments for (Persisting) Substance Dualism Argument 1: The Modal Argument from Personal Identity This first type of argument is advanced

### Th e ontological argument distinguishes itself from the cosmological

Aporia vol. 18 no. 1 2008 Charles Hartshorne and the Ontological Argument Joshua Ernst Th e ontological argument distinguishes itself from the cosmological and teleological arguments for God s existence

### I WISH A VALID ONTOLOGICAL ARGUMENT?

A VALID ONTOLOGICAL ARGUMENT? I WISH to discuss Professor Malcolm's absorbingly powerful defense of a version of Anselm's ontological proof for the existence of God.' Professor Malcolm believes "that in

### Aquinas on Essence, Existence, and Divine Simplicity Strange but Consistent. In the third question of the Summa Theologiae, Aquinas is concerned with

Aquinas on Essence, Existence, and Divine Simplicity Strange but Consistent In the third question of the Summa Theologiae, Aquinas is concerned with divine simplicity. This is important for him both theologically

### 1/8. Descartes 4: The Fifth Meditation

1/8 Descartes 4: The Fifth Meditation Recap: last time we found that Descartes in the 3 rd Meditation set out to provide some grounds for thinking that God exists, grounds that would answer the charge

### This is because the quality of extension is part of the essence of material objects.

UNIT 1: RATIONALISM HANDOUT 5: DESCARTES MEDITATIONS, MEDITATION FIVE 1: CONCEPTS AND ESSENCES In the Second Meditation Descartes found that what we know most clearly and distinctly about material objects

### INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28

### DIVINE CONTINGENCY Einar Duenger Bohn IFIKK, University of Oslo

DIVINE CONTINGENCY Einar Duenger Bohn IFIKK, University of Oslo Brian Leftow s God and Necessity is interesting, full of details, bold and ambitious. Roughly, the main question at hand is: assuming there

### The Slate Is Not Empty: Descartes and Locke on Innate Ideas

The Slate Is Not Empty: Descartes and Locke on Innate Ideas René Descartes and John Locke, two of the principal philosophers who shaped modern philosophy, disagree on several topics; one of them concerns

### Perfect being theology and modal truth

Perfect being theology and modal truth Jeff Speaks February 9, 2016 Perfect being theology is the attempt to use the principle that God is the greatest possible being to derive claims about the divine

### Anselm s Ontological Argument for the Existence of God

Anselm s Ontological Argument for the Existence of God Anselm s argument is an a priori argument; that is, it is an argument that is independent of experience and based solely on concepts and logical relations,

### Handout #1: Mathematical Reasoning

Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or

### 3.3 Proofs Involving Quantifiers

3.3 Proofs Involving Quantifiers 1. In exercise 6 of Section 2.2 you use logical equivalences to show that x(p (x) Q(x)) is equivalent to xp (x) xq(x). Now use the methods of this section to prove that

### Descartes rationalism

Michael Lacewing Descartes rationalism Descartes Meditations provide an extended study in establishing knowledge through rational intuition and deduction. We focus in this handout on three central claims:

### THE KNOWLEDGE ARGUMENT

Michael Lacewing Descartes arguments for distinguishing mind and body THE KNOWLEDGE ARGUMENT In Meditation II, having argued that he knows he thinks, Descartes then asks what kind of thing he is. Discussions

### CONCEPTUAL CONTINGENCY AND ABSTRACT EXISTENCE

87 CONCEPTUAL CONTINGENCY AND ABSTRACT EXISTENCE BY MARK COLYVAN Mathematical statements such as There are infinitely many prime numbers and 2 ℵ 0 > ℵ 0 are usually thought to be necessarily true. Not

### DISCRETE MATHEMATICS W W L CHEN

DISCRETE MATHEMATICS W W L CHEN c W W L Chen, 1982, 2008. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It is available free

### We now explore a third method of proof: proof by contradiction.

CHAPTER 6 Proof by Contradiction We now explore a third method of proof: proof by contradiction. This method is not limited to proving just conditional statements it can be used to prove any kind of statement

### Existence Is Not a Predicate by Immanuel Kant

Existence Is Not a Predicate by Immanuel Kant Immanuel Kant, Thoemmes About the author.... Immanuel Kant (1724-1804) studied in Königsberg, East Prussia. Before he fully developed an interest in philosophy,

### PROOF AND PROBABILITY(teaching notes)

PROOF AND PROBABILITY(teaching notes) In arguing for God s existence, it is important to distinguish between proof and probability. Different rules of reasoning apply depending on whether we are testing

### Descartes Fourth Meditation On human error

Descartes Fourth Meditation On human error Descartes begins the fourth Meditation with a review of what he has learned so far. He began his search for certainty by questioning the veracity of his own senses.

### Must God Be Perfectly Good?

Must God Be Perfectly Good? Jay Michael Arnold Introduction Richard Swinburne, in his seminal book, The Coherence of Theism, claims I propose to argue that not merely is perfect goodness compatible with

### On Cuneo s defence of the parity premise

1 On Cuneo s defence of the parity premise G.J.E. Rutten 1. Introduction In his book The Normative Web Terence Cuneo provides a core argument for his moral realism of a paradigmatic sort. Paradigmatic

### Kant s Possibility Proof

History of Philosophy Quarterly Volume 27, Number 3, July 2010 Kant s Possibility Proof Nicholas F. Stang 1. In t r o d u c t i o n In his 1763 work The Only Possible Ground of Proof [Beweisgrund] for

### CHAPTER 7 GENERAL PROOF SYSTEMS

CHAPTER 7 GENERAL PROOF SYSTEMS 1 Introduction Proof systems are built to prove statements. They can be thought as an inference machine with special statements, called provable statements, or sometimes

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### Kant on Time. Diana Mertz Hsieh (diana@dianahsieh.com) Kant (Phil 5010, Hanna) 28 September 2004

Kant on Time Diana Mertz Hsieh (diana@dianahsieh.com) Kant (Phil 5010, Hanna) 28 September 2004 In the Transcendental Aesthetic of his Critique of Pure Reason, Immanuel Kant offers a series of dense arguments

### Things That Might Not Have Been Michael Nelson University of California at Riverside mnelson@ucr.edu

Things That Might Not Have Been Michael Nelson University of California at Riverside mnelson@ucr.edu Quantified Modal Logic (QML), to echo Arthur Prior, is haunted by the myth of necessary existence. Features

### Predicate Logic Review

Predicate Logic Review UC Berkeley, Philosophy 142, Spring 2016 John MacFarlane 1 Grammar A term is an individual constant or a variable. An individual constant is a lowercase letter from the beginning

### Math 317 HW #7 Solutions

Math 17 HW #7 Solutions 1. Exercise..5. Decide which of the following sets are compact. For those that are not compact, show how Definition..1 breaks down. In other words, give an example of a sequence

### Plantinga and the Problem of Evil

Plantinga and the Problem of Evil Heimir Geirsson, Iowa State University Michael Losonsky, Colorado State University The logical problem of evil centers on the apparent inconsistency of the following two

### 5. ABSOLUTE EXTREMA. Definition, Existence & Calculation

5. ABSOLUTE EXTREMA Definition, Existence & Calculation We assume that the definition of function is known and proceed to define absolute minimum. We also assume that the student is familiar with the terms

### 2.1.1 Examples of Sets and their Elements

Chapter 2 Set Theory 2.1 Sets The most basic object in Mathematics is called a set. As rudimentary as it is, the exact, formal definition of a set is highly complex. For our purposes, we will simply define

### MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take

### Possible Worlds. Curtis Brown Metaphysics Spring, 2005

Possible Worlds Curtis Brown Metaphysics Spring, 2005 Possible Worlds - Uses 1. Understanding claims about possibility and necessity it is possible that P means there is a possible world in which P is

### Metaphysical preliminaries

Metaphysical preliminaries Ted Sider Properties seminar Our questions about properties affect and are affected by the choice of metaphysical tools certain key concepts that frame the debate. 1. Ontology

### The Ontological Argument by St. Anselm

The Ontological Argument by St. Anselm Canterbury Cathedral, Library of Congress, Detroit Publishing About the author.... St. Anselm (1033-1109), a member of the Benedictine Order and Bishop of Canterbury,

### Gödel s Theorem: An Incomplete Guide to Its Use and Abuse

Gödel s Theorem: An Incomplete Guide to Its Use and Abuse by Torkel Franzén Summary by Xavier Noria August 2006 fxn@hashref.com This unique exposition of Kurt Gödel s stunning incompleteness theorems for

### Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing

### (Refer Slide Time: 1:41)

Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture # 10 Sets Today we shall learn about sets. You must

### Propositional Logic and Methods of Inference SEEM

Propositional Logic and Methods of Inference SEEM 5750 1 Logic Knowledge can also be represented by the symbols of logic, which is the study of the rules of exact reasoning. Logic is also of primary importance

### Logic, Sets, and Proofs

Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Statements. A logical statement is a mathematical statement that is either true or false. Here we denote logical

### Grounded abstraction

Grounded abstraction Øystein Linnebo Mini-seminar on groundedness. Winter term 2012. Session 3 1 The bad company problem (Σ) α = β α β (HP) (V) (H 2 P) #F = #G F G ˆx.F x = ˆx.Gx x(f x Gx) R = S R S 2

### Discrete Mathematics, Chapter : Propositional Logic

Discrete Mathematics, Chapter 1.1.-1.3: Propositional Logic Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.1-1.3 1 / 21 Outline 1 Propositions

### Proof: A logical argument establishing the truth of the theorem given the truth of the axioms and any previously proven theorems.

Math 232 - Discrete Math 2.1 Direct Proofs and Counterexamples Notes Axiom: Proposition that is assumed to be true. Proof: A logical argument establishing the truth of the theorem given the truth of the

### The Philosophical Importance of Mathematical Logic Bertrand Russell

The Philosophical Importance of Mathematical Logic Bertrand Russell IN SPEAKING OF "Mathematical logic", I use this word in a very broad sense. By it I understand the works of Cantor on transfinite numbers

### THE WORD TO GOD Philosophy as Prayer in St. Anselm s Proslogion (Chapters 1-3)

THE WORD TO GOD Philosophy as Prayer in St. Anselm s Proslogion (Chapters 1-3) I believe this also, that unless I believe, I shall not understand [Isa. 7: 9]. 1 So ends the first chapter of St. Anselm

### RSR Episode #11 Ontological Argument

RSR Episode #11 Ontological Argument The Ontological Argument for God s existence was developed in the Middle Ages by St. Anselm. It claims that reflection on God s nature as the greatest conceivable being

### If f is a 1-1 correspondence between A and B then it has an inverse, and f 1 isa 1-1 correspondence between B and A.

Chapter 5 Cardinality of sets 51 1-1 Correspondences A 1-1 correspondence between sets A and B is another name for a function f : A B that is 1-1 and onto If f is a 1-1 correspondence between A and B,

### ON THE NECESSARY EXISTENCE OF GOD-MINUS

Faraci and Linford 1 ON THE NECESSARY EXISTENCE OF GOD-MINUS David Faraci and Daniel Linford Abstract In this paper, we offer a novel reductio of Anselm s (in)famous Ontological Argument for the existence

### Introduction. I. Proof of the Minor Premise ( All reality is completely intelligible )

Philosophical Proof of God: Derived from Principles in Bernard Lonergan s Insight May 2014 Robert J. Spitzer, S.J., Ph.D. Magis Center of Reason and Faith Lonergan s proof may be stated as follows: Introduction

### CHAPTER 1. Logic, Proofs Propositions

CHAPTER 1 Logic, Proofs 1.1. Propositions A proposition is a declarative sentence that is either true or false (but not both). For instance, the following are propositions: Paris is in France (true), London

### 1.3 Induction and Other Proof Techniques

4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.

### Logic and Proofs. Chapter 1

Section 1.0 1.0.1 Chapter 1 Logic and Proofs 1.1 Propositional Logic 1.2 Propositional Equivalences 1.3 Predicates and Quantifiers 1.4 Nested Quantifiers 1.5 Rules of Inference 1.6 Introduction to Proofs

### 2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try.

2. METHODS OF PROOF 69 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. Trivial Proof: If we know q is true then

### Notes on Modal Logic

Notes on Modal Logic Notes for Philosophy 151 Eric Pacuit January 28, 2009 These short notes are intended to supplement the lectures and text ntroduce some of the basic concepts of Modal Logic. The primary

### Chapter 4. Descartes, Third Meditation. 4.1 Homework

Chapter 4 Descartes, Third Meditation 4.1 Homework Readings : - Descartes, Meditation III - Objections and Replies: a) Third O and R: CSM II, 132; 127-8. b) Fifth O and R: CSM II, 195-97, 251. c) First

### The Predicate Calculus in AI

Last time, we: The Predicate Calculus in AI Motivated the use of Logic as a representational language for AI (Can derive new facts syntactically - simply by pushing symbols around) Described propositional

### More Mathematical Induction. October 27, 2016

More Mathematical Induction October 7, 016 In these slides... Review of ordinary induction. Remark about exponential and polynomial growth. Example a second proof that P(A) = A. Strong induction. Least

### A Hajós type result on factoring finite abelian groups by subsets II

Comment.Math.Univ.Carolin. 51,1(2010) 1 8 1 A Hajós type result on factoring finite abelian groups by subsets II Keresztély Corrádi, Sándor Szabó Abstract. It is proved that if a finite abelian group is

### Section 1. Statements and Truth Tables. Definition 1.1: A mathematical statement is a declarative sentence that is true or false, but not both.

M3210 Supplemental Notes: Basic Logic Concepts In this course we will examine statements about mathematical concepts and relationships between these concepts (definitions, theorems). We will also consider

### Phil 420: Metaphysics Spring 2008. [Handout 4] Hilary Putnam: Why There Isn t A Ready-Made World

1 Putnam s Main Theses: 1. There is no ready-made world. Phil 420: Metaphysics Spring 2008 [Handout 4] Hilary Putnam: Why There Isn t A Ready-Made World * [A ready-made world]: The world itself has to

### IS HUME S ACCOUNT SCEPTICAL?

Michael Lacewing Hume on knowledge HUME S FORK Empiricism denies that we can know anything about how the world outside our own minds is without relying on sense experience. In IV, Hume argues that we can

### MATH 55: HOMEWORK #2 SOLUTIONS

MATH 55: HOMEWORK # SOLUTIONS ERIC PETERSON * 1. SECTION 1.5: NESTED QUANTIFIERS 1.1. Problem 1.5.8. Determine the truth value of each of these statements if the domain of each variable consists of all

### Time and Causation in Gödel s Universe.

Time and Causation in Gödel s Universe. John L. Bell In 1949 the great logician Kurt Gödel constructed the first mathematical models of the universe in which travel into the past is, in theory at least,

### 2. Propositional Equivalences

2. PROPOSITIONAL EQUIVALENCES 33 2. Propositional Equivalences 2.1. Tautology/Contradiction/Contingency. Definition 2.1.1. A tautology is a proposition that is always true. Example 2.1.1. p p Definition

### Quine on truth by convention

Quine on truth by convention March 8, 2005 1 Linguistic explanations of necessity and the a priori.............. 1 2 Relative and absolute truth by definition.................... 2 3 Is logic true by convention?...........................

### Last time we had arrived at the following provisional interpretation of Aquinas second way:

Aquinas Third Way Last time we had arrived at the following provisional interpretation of Aquinas second way: 1. 2. 3. 4. At least one thing has an efficient cause. Every causal chain must either be circular,

### Chapter I Logic and Proofs

MATH 1130 1 Discrete Structures Chapter I Logic and Proofs Propositions A proposition is a statement that is either true (T) or false (F), but or both. s Propositions: 1. I am a man.. I am taller than

### 4 Domain Relational Calculus

4 Domain Relational Calculus We now present two relational calculi that we will compare to RA. First, what is the difference between an algebra and a calculus? The usual story is that the algebra RA is

### Mathematical Logic. Tableaux Reasoning for Propositional Logic. Chiara Ghidini. FBK-IRST, Trento, Italy

Tableaux Reasoning for Propositional Logic FBK-IRST, Trento, Italy Outline of this lecture An introduction to Automated Reasoning with Analytic Tableaux; Today we will be looking into tableau methods for

### Quine on Empiricism. G. J. Mattey. Fall, 2005 / Philosophy 156. Logical empiricism was not widely embraced in Britain or the United States.

Quine on Empiricism G. J. Mattey Fall, 2005 / Philosophy 156 Reaction to Logical Empiricism Logical empiricism was not widely embraced in Britain or the United States. British philosophers were more attracted

### 13 Infinite Sets. 13.1 Injections, Surjections, and Bijections. mcs-ftl 2010/9/8 0:40 page 379 #385

mcs-ftl 2010/9/8 0:40 page 379 #385 13 Infinite Sets So you might be wondering how much is there to say about an infinite set other than, well, it has an infinite number of elements. Of course, an infinite

### Frege s theory of sense

Frege s theory of sense Jeff Speaks August 25, 2011 1. Three arguments that there must be more to meaning than reference... 1 1.1. Frege s puzzle about identity sentences 1.2. Understanding and knowledge

### Semantics for the Predicate Calculus: Part I

Semantics for the Predicate Calculus: Part I (Version 0.3, revised 6:15pm, April 14, 2005. Please report typos to hhalvors@princeton.edu.) The study of formal logic is based on the fact that the validity

### Partitioning edge-coloured complete graphs into monochromatic cycles and paths

arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edge-coloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political

### Section 3 Sequences and Limits

Section 3 Sequences and Limits Definition A sequence of real numbers is an infinite ordered list a, a 2, a 3, a 4,... where, for each n N, a n is a real number. We call a n the n-th term of the sequence.

### 1/9. Locke 1: Critique of Innate Ideas

1/9 Locke 1: Critique of Innate Ideas This week we are going to begin looking at a new area by turning our attention to the work of John Locke, who is probably the most famous English philosopher of all

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us

### Descriptive Names vs. Descriptive Anaphora

Descriptive Names vs. Descriptive Anaphora By Scott Soames School of Philosophy USC To Appear in a Symposium on Alan Berger s Terms and Truth In PHILOSOPHY AND PHENOMENOLOGICAL RESEARCH Descriptive Names

### Modality & Other Matters: An Interview with Timothy Williamson

Perspectives: International Postgraduate Journal of Philosophy Modality & Other Matters: An Interview with Timothy Williamson Timothy Williamson (TW) has been the Wykeham Professor of Logic at Oxford since

### God and Reality. Arman Hovhannisyan

God and Reality Arman Hovhannisyan Metaphysics has done everything to involve God in the world of being. However, in case of considering Reality as being and nothingness, naturally, the metaphysical approach

### Title: Duty Derives from Telos: The Teleology behind Kant s Categorical Imperative. Author: Micah Tillman

Title: Duty Derives from Telos: The Teleology behind Kant s Categorical Imperative Author: Micah Tillman Word Count: 3,358 (3,448, including content notes) Abstract: This paper argues that Kant s view

### 1/10. Descartes 2: The Cogito and the Mind

1/10 Descartes 2: The Cogito and the Mind Recap: last week we undertook to follow Descartes path of radical doubt in order to attempt to discover what, if anything, can be known for certain. This path

### This chapter is all about cardinality of sets. At first this looks like a

CHAPTER Cardinality of Sets This chapter is all about cardinality of sets At first this looks like a very simple concept To find the cardinality of a set, just count its elements If A = { a, b, c, d },

### A Devilish Twist on Anselm s Ontological Argument. Emily Thomas (Birmingham)

1 A Devilish Twist on Anselm s Ontological Argument Emily Thomas (Birmingham) AET400@bham.ac.uk St Anselm formulated his ontological argument to prove the existence of God. This paper is not intended to

### Midterm Examination 1 with Solutions - Math 574, Frank Thorne Thursday, February 9, 2012

Midterm Examination 1 with Solutions - Math 574, Frank Thorne (thorne@math.sc.edu) Thursday, February 9, 2012 1. (3 points each) For each sentence below, say whether it is logically equivalent to the sentence

### ON WHITCOMB S GROUNDING ARGUMENT FOR ATHEISM Joshua Rasmussen Andrew Cullison Daniel Howard-Snyder

ON WHITCOMB S GROUNDING ARGUMENT FOR ATHEISM Joshua Rasmussen Andrew Cullison Daniel Howard-Snyder Abstract: Dennis Whitcomb argues that there is no God on the grounds that (i) God is omniscient, yet (ii)

### 1 Proposition, Logical connectives and compound statements

Discrete Mathematics: Lecture 4 Introduction to Logic Instructor: Arijit Bishnu Date: July 27, 2009 1 Proposition, Logical connectives and compound statements Logic is the discipline that deals with the

### mywbut.com Propositional Logic inference rules

Propositional Logic inference rules 1 5.5 Rules of Inference Here are some examples of sound rules of inference. Each can be shown to be sound once and for all using a truth table. The left column contains

### Sound and Complete Inference Rules for SE-Consequence

Journal of Artificial Intelligence Research 31 (2008) 205 216 Submitted 10/07; published 01/08 Sound and Complete Inference Rules for SE-Consequence Ka-Shu Wong University of New South Wales and National

### Quine s critique of conventionalism

Quine s critique of conventionalism PHIL 83104 September 26, 2011 One of the puzzling aspects of Ayer s discussion is that although he seems to lay great weight on the notion of truth in virtue of definitions

### Chapter Three. Functions. In this section, we study what is undoubtedly the most fundamental type of relation used in mathematics.

Chapter Three Functions 3.1 INTRODUCTION In this section, we study what is undoubtedly the most fundamental type of relation used in mathematics. Definition 3.1: Given sets X and Y, a function from X to

### Introduction to Proofs

Chapter 1 Introduction to Proofs 1.1 Preview of Proof This section previews many of the key ideas of proof and cites [in brackets] the sections where they are discussed thoroughly. All of these ideas are

### Theorem 2. If x Q and y R \ Q, then. (a) x + y R \ Q, and. (b) xy Q.

Math 305 Fall 011 The Density of Q in R The following two theorems tell us what happens when we add and multiply by rational numbers. For the first one, we see that if we add or multiply two rational numbers

### (LMCS, p. 317) V.1. First Order Logic. This is the most powerful, most expressive logic that we will examine.

(LMCS, p. 317) V.1 First Order Logic This is the most powerful, most expressive logic that we will examine. Our version of first-order logic will use the following symbols: variables connectives (,,,,

### Basic Proof Techniques

Basic Proof Techniques David Ferry dsf43@truman.edu September 13, 010 1 Four Fundamental Proof Techniques When one wishes to prove the statement P Q there are four fundamental approaches. This document

### ESSAY BA course year 1 no. 5 Philosophy

ESSAY BA course year 1 no. 5 Philosophy Assess THREE of the proofs of the existence of God in the Summa Theologiae of St. Thomas Aquinas. In what sense are they described as converging and convincing arguments