About the Class: Some General Course Information Who Should Take This Class? Course Mechanics What You Need to Do About Logic: Why Is It Important?
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Rules of Replacement II, §7.4
Philosophy 109, Modern Logic, Queens College Russell Marcus, Instructor email: [email protected] website: http://philosophy.thatmarcusfamily.org Office phone: (718) 997-5287 Rules of Replacement II, §7.4 I. The Last Five Rules of Replacement See the appendix at the end of the lesson for truth tables proving equivalence for each. Transposition (Trans) P e Q :: -Q e -P You may switch the antecedent and consequent of a conditional statement, as long as you negate (or un-negate) both. Often used with (HS). Also, traditionally, called the ‘contrapositive’. Sample Derivation: 1. A e B 2. D e -B / A e -D 3. --B e -D 2, Trans 4. A e -D 1, 3, DN, HS QED Transposition can be tricky when only one term is negated: A e -B becomes, by Trans: --B e -A which becomes, by DN B e -A Equivalently, but doing the double negation first: A e -B becomes, by DN: --A e -B becomes, by Trans: B e -A Either way, you can include the DN on the line with Trans. Material Implication (Impl) P e Q :: -P w Q Implication allows you to change a statement from a disjunction to a conditional, or vice versa. It’s often easier to work with disjunctions. You can use (DM) to get conjunctions. You may be able to use distribution, which doesn’t apply to conditionals. On the other hand, sometimes, you just want to work with conditionals. You can use (HS) and (MP). Proofs are overdetermined by our system - there are many ways to do them. -
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. -
Tables of Implications and Tautologies from Symbolic Logic
Tables of Implications and Tautologies from Symbolic Logic Dr. Robert B. Heckendorn Computer Science Department, University of Idaho March 17, 2021 Here are some tables of logical equivalents and implications that I have found useful over the years. Where there are classical names for things I have included them. \Isolation by Parts" is my own invention. By tautology I mean equivalent left and right hand side and by implication I mean the left hand expression implies the right hand. I use the tilde and overbar interchangeably to represent negation e.g. ∼x is the same as x. Enjoy! Table 1: Properties of All Two-bit Operators. The Comm. is short for commutative and Assoc. is short for associative. Iff is short for \if and only if". Truth Name Comm./ Binary And/Or/Not Nands Only Table Assoc. Op 0000 False CA 0 0 (a " (a " a)) " (a " (a " a)) 0001 And CA a ^ b a ^ b (a " b) " (a " b) 0010 Minus b − a a ^ b (b " (a " a)) " (a " (a " a)) 0011 B A b b b 0100 Minus a − b a ^ b (a " (a " a)) " (a " (a " b)) 0101 A A a a a 0110 Xor/NotEqual CA a ⊕ b (a ^ b) _ (a ^ b)(b " (a " a)) " (a " (a " b)) (a _ b) ^ (a _ b) 0111 Or CA a _ b a _ b (a " a) " (b " b) 1000 Nor C a # b a ^ b ((a " a) " (b " b)) " ((a " a) " a) 1001 Iff/Equal CA a $ b (a _ b) ^ (a _ b) ((a " a) " (b " b)) " (a " b) (a ^ b) _ (a ^ b) 1010 Not A a a a " a 1011 Imply a ! b a _ b (a " (a " b)) 1100 Not B b b b " b 1101 Imply b ! a a _ b (b " (a " a)) 1110 Nand C a " b a _ b a " b 1111 True CA 1 1 (a " a) " a 1 Table 2: Tautologies (Logical Identities) Commutative Property: p ^ q $ q -
A New Logic for Uncertainty
A New Logic for Uncertainty ∗ LUO Maokang and HE Wei† Institute of Mathematics Institute of Mathematics Sichuan University Nanjing Normal University Chengdu, 610064 Nanjing, 210046 P.R.China P.R.China Abstract Fuzziness and randomicity widespread exist in natural science, engineering, technology and social science. The purpose of this paper is to present a new logic - uncertain propositional logic which can deal with both fuzziness by taking truth value semantics and randomicity by taking probabilistic semantics or possibility semantics. As the first step for purpose of establishing a logic system which completely reflect the uncertainty of the objective world, this logic will lead to a set of logical foundations for uncertainty theory as what classical logic done in certain or definite situations or circumstances. Keywords: Fuzziness; randomicity; UL-algebra; uncertain propositional logic. Mathematics Subject Classifications(2000): 03B60. 1 Introduction As one of the most important and one of the most widely used concepts in the whole area of modern academic or technologic research, uncertainty has been involved into study and applications more than twenty years. Now along with the quickly expanding requirements of developments of science and technology, people are having to face and deal with more and more problems tangled with uncertainty in the fields of natural science, engineering or technology or even in social science. To these uncertain prob- arXiv:1506.03123v1 [math.LO] 9 Jun 2015 lems, many traditional theories and methods based on certain conditions or certain circumstances are not so effective and powerful as them in the past. So the importance of research on uncertainty is emerging more and more obviously and imminently. -
Russell's Theory of Descriptions
Russell’s theory of descriptions PHIL 83104 September 5, 2011 1. Denoting phrases and names ...........................................................................................1 2. Russell’s theory of denoting phrases ................................................................................3 2.1. Propositions and propositional functions 2.2. Indefinite descriptions 2.3. Definite descriptions 3. The three puzzles of ‘On denoting’ ..................................................................................7 3.1. The substitution of identicals 3.2. The law of the excluded middle 3.3. The problem of negative existentials 4. Objections to Russell’s theory .......................................................................................11 4.1. Incomplete definite descriptions 4.2. Referential uses of definite descriptions 4.3. Other uses of ‘the’: generics 4.4. The contrast between descriptions and names [The main reading I gave you was Russell’s 1919 paper, “Descriptions,” which is in some ways clearer than his classic exposition of the theory of descriptions, which was in his 1905 paper “On Denoting.” The latter is one of the optional readings on the web site, and I reference it below sometimes as well.] 1. DENOTING PHRASES AND NAMES Russell defines the class of denoting phrases as follows: “By ‘denoting phrase’ I mean a phrase such as any one of the following: a man, some man, any man, every man, all men, the present king of England, the centre of mass of the Solar System at the first instant of the twentieth century, the revolution of the earth around the sun, the revolution of the sun around the earth. Thus a phrase is denoting solely in virtue of its form.” (‘On Denoting’, 479) Russell’s aim in this article is to explain how expressions like this work — what they contribute to the meanings of sentences containing them. -
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. -
Argument Forms and Fallacies
6.6 Common Argument Forms and Fallacies 1. Common Valid Argument Forms: In the previous section (6.4), we learned how to determine whether or not an argument is valid using truth tables. There are certain forms of valid and invalid argument that are extremely common. If we memorize some of these common argument forms, it will save us time because we will be able to immediately recognize whether or not certain arguments are valid or invalid without having to draw out a truth table. Let’s begin: 1. Disjunctive Syllogism: The following argument is valid: “The coin is either in my right hand or my left hand. It’s not in my right hand. So, it must be in my left hand.” Let “R”=”The coin is in my right hand” and let “L”=”The coin is in my left hand”. The argument in symbolic form is this: R ˅ L ~R __________________________________________________ L Any argument with the form just stated is valid. This form of argument is called a disjunctive syllogism. Basically, the argument gives you two options and says that, since one option is FALSE, the other option must be TRUE. 2. Pure Hypothetical Syllogism: The following argument is valid: “If you hit the ball in on this turn, you’ll get a hole in one; and if you get a hole in one you’ll win the game. So, if you hit the ball in on this turn, you’ll win the game.” Let “B”=”You hit the ball in on this turn”, “H”=”You get a hole in one”, and “W”=”you win the game”. -
Philosophy 109, Modern Logic Russell Marcus
Philosophy 240: Symbolic Logic Hamilton College Fall 2014 Russell Marcus Reference Sheeet for What Follows Names of Languages PL: Propositional Logic M: Monadic (First-Order) Predicate Logic F: Full (First-Order) Predicate Logic FF: Full (First-Order) Predicate Logic with functors S: Second-Order Predicate Logic Basic Truth Tables - á á @ â á w â á e â á / â 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 1 0 1 0 0 1 0 0 0 0 1 0 1 1 0 1 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 Rules of Inference Modus Ponens (MP) Conjunction (Conj) á e â á á / â â / á A â Modus Tollens (MT) Addition (Add) á e â á / á w â -â / -á Simplification (Simp) Disjunctive Syllogism (DS) á A â / á á w â -á / â Constructive Dilemma (CD) (á e â) Hypothetical Syllogism (HS) (ã e ä) á e â á w ã / â w ä â e ã / á e ã Philosophy 240: Symbolic Logic, Prof. Marcus; Reference Sheet for What Follows, page 2 Rules of Equivalence DeMorgan’s Laws (DM) Contraposition (Cont) -(á A â) W -á w -â á e â W -â e -á -(á w â) W -á A -â Material Implication (Impl) Association (Assoc) á e â W -á w â á w (â w ã) W (á w â) w ã á A (â A ã) W (á A â) A ã Material Equivalence (Equiv) á / â W (á e â) A (â e á) Distribution (Dist) á / â W (á A â) w (-á A -â) á A (â w ã) W (á A â) w (á A ã) á w (â A ã) W (á w â) A (á w ã) Exportation (Exp) á e (â e ã) W (á A â) e ã Commutativity (Com) á w â W â w á Tautology (Taut) á A â W â A á á W á A á á W á w á Double Negation (DN) á W --á Six Derived Rules for the Biconditional Rules of Inference Rules of Equivalence Biconditional Modus Ponens (BMP) Biconditional DeMorgan’s Law (BDM) á / â -(á / â) W -á / â á / â Biconditional Modus Tollens (BMT) Biconditional Commutativity (BCom) á / â á / â W â / á -á / -â Biconditional Hypothetical Syllogism (BHS) Biconditional Contraposition (BCont) á / â á / â W -á / -â â / ã / á / ã Philosophy 240: Symbolic Logic, Prof. -
Florensky and Frank
View metadata, citation and similar papers at core.ac.uk brought to you by CORE Po zn ań skie Studia Teologiczne Tom 22, 2008 ADAM DROZDEK Duquesne University, Pittsburgh, Pennsylvania Defying Rationality: Florensky and Frank Przezwyciężyć racjonalność: Fiorenski i Frank What should be the basis of knowledge in theological matters? The question has been pondered upon by theologians for centuries and not a few proposals were made. One of these was made by the Orthodox priest and scientist Pavel Florensky in The pillar and ground of the truth (1914) considered to be the most original and influen tial work of the Russian religious renaissance1, marking “the beginning of a new era in Russian theology”2 and even “one of the most significant accomplishments of ec clesiastical thinking in the twentieth century”3. Proposals made in a work deemed to be so important certainly arrest attention. I. THE LAW OF IDENTITY A centerpiece of rationalism, according to Florensky, is the law of identity, A = A, which is the source of powerlessness of rational reasoning. “The law A = A be comes a completely empty schema of self-affirmation” so that “I = I turns out to be ... a cry of naked egotism: ‘I!’ For where there is no difference, there can be no con- 1 R. Slesinski, Pavel Florensky: a metaphysics o f love, Crestwood: St. Vladimir’s Seminary Press 1984, 22. 2 N. Zernov, The Russian religious renaissance of the twentieth century, London: Darton, Longman & Todd 1963, 101. 3 M. Silberer, Die Trinitatsidee im Werk von Pavel A. Florenskij: Versuch einer systematischen Darstellung in Begegnung mit Thomas von Aquin, Wtirzburg: Augustinus-Verlag 1984, 254. -
Overturning the Paradigm of Identity with Gilles Deleuze's Differential
A Thesis entitled Difference Over Identity: Overturning the Paradigm of Identity With Gilles Deleuze’s Differential Ontology by Matthew G. Eckel Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Master of Arts Degree in Philosophy Dr. Ammon Allred, Committee Chair Dr. Benjamin Grazzini, Committee Member Dr. Benjamin Pryor, Committee Member Dr. Patricia R. Komuniecki, Dean College of Graduate Studies The University of Toledo May 2014 An Abstract of Difference Over Identity: Overturning the Paradigm of Identity With Gilles Deleuze’s Differential Ontology by Matthew G. Eckel Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Master of Arts Degree in Philosophy The University of Toledo May 2014 Taking Gilles Deleuze to be a philosopher who is most concerned with articulating a ‘philosophy of difference’, Deleuze’s thought represents a fundamental shift in the history of philosophy, a shift which asserts ontological difference as independent of any prior ontological identity, even going as far as suggesting that identity is only possible when grounded by difference. Deleuze reconstructs a ‘minor’ history of philosophy, mobilizing thinkers from Spinoza and Nietzsche to Duns Scotus and Bergson, in his attempt to assert that philosophy has always been, underneath its canonical manifestations, a project concerned with ontology, and that ontological difference deserves the kind of philosophical attention, and privilege, which ontological identity has been given since Aristotle. -
The Development of Laws of Formal Logic of Aristotle
This paper is dedicated to my unforgettable friend Boris Isaevich Lamdon. The Development of Laws of Formal Logic of Aristotle The essence of formal logic The aim of every science is to discover the laws that could explain one or another phenomenon. Once these laws are discovered, then science proceed to study the other phenomena, which in the nature are of an infinite set. It is interesting to note that in the process of discovering a law, for example in physics, people make thousands of experiments, build proves, among them some experience, or evidence - useful for understanding a certain phenomenon, but other experiments or evidence proved fruitless. But it is found only with hindsight, when the law is already discovered. Therefore, with the discovering of the law it is enough to show 2 - 3 experiments or prove to verify its correctness. All other experiments were the ways of study and there is no need to repeat them, to understand how the law works. In the exact sciences, it is understood, and therefore the students studies only the information that is necessary to understand specific phenomena. By no means this is the case with the study of formal logic. Formal logic, as opposed to other sciences: physics, chemistry, mathematics, biology and so on, studies not an infinite number of phenomena in nature, but only one how a man thinks, how he learns the world surrounding us, and how people understand each other. In other words, what laws govern the logic of our thinking, i.e., our reasoning and judgments in any science or in everyday life. -
The Physics and Metaphysics of Identity and Individuality Steven French and De´Cio Krause: Identity in Physics: a Historical, Philosophical, and Formal Analysis
Metascience (2011) 20:225–251 DOI 10.1007/s11016-010-9463-7 BOOK SYMPOSIUM The physics and metaphysics of identity and individuality Steven French and De´cio Krause: Identity in physics: A historical, philosophical, and formal analysis. Oxford: Clarendon Press, 2006, 440 pp, £68.00 HB Don Howard • Bas C. van Fraassen • Ota´vio Bueno • Elena Castellani • Laura Crosilla • Steven French • De´cio Krause Published online: 3 November 2010 Ó Springer Science+Business Media B.V. 2010 Don Howard Steven French and De´cio Krause have written what bids fair to be, for years to come, the definitive philosophical treatment of the problem of the individuality of elementary particles in quantum mechanics (QM) and quantum-field theory (QFT). The book begins with a long and dense argument for the view that elementary particles are most helpfully regarded as non-individuals, and it concludes with an earnest attempt to develop a formal apparatus for describing such non-individual entities better suited to the task than our customary set theory. D. Howard (&) Department of Philosophy and Graduate Program in History and Philosophy of Science, University of Notre Dame, Notre Dame, IN 46556, USA e-mail: [email protected] B. C. van Fraassen Philosophy Department, San Francisco State University, 1600 Holloway Avenue, San Francisco, CA 94132, USA e-mail: [email protected] O. Bueno Department of Philosophy, University of Miami, Coral Gables, FL 33124, USA e-mail: [email protected] E. Castellani Department of Philosophy, University of Florence, Via Bolognese 52, 50139 Florence, Italy e-mail: elena.castellani@unifi.it L. Crosilla Department of Pure Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT, UK e-mail: [email protected] 123 226 Metascience (2011) 20:225–251 Are elementary particles individuals? I do not know.