4-3. Physics ~ Metaphysics. Meta-physics

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1 1 4-3 A Brief History of Formal Logic The idea of reducing reasoning to computation in some kind of formal calculus is an old dream. Some trace the idea back to Raymond Lull. Certainly Hobbes made explicit the analogy in the slogan Reason [...] is nothing but Reckoning. This parallel was developed by Leibniz, who envisaged a characteristica universalis (universal language) and a calculus ratiocinator (calculus of reasoning). His idea was that disputes of all kinds, not merely mathematical ones, could be settled if the parties translated their dispute into the characteristica and then simply calculated. Leibniz even made some steps towards realizing this lofty goal, but his work was largely forgotten. Meanwhile, mathematics continued to acquire new symbolisms. It was to be expected that the use of symbolism would eventually extend beyond the subject matter of mathematics, to the reasoning used in mathematics. Boole developed the first really successful formal system for logical and set-theoretic reasoning. What s more, he was one of the first to emphasize the possibility of applying formal calculi to several different situations, and doing calculations according to formal rules without regard to the underlying interpretation. In this way he anticipated important parts of the modern axiomatic method. However Boole s logic was limited to propositional reasoning, and it was not until the much later development of quantifiers that formal logic was ready to be applied to general mathematics. The first systems adequate for this purpose were developed by Frege and Peano. Both were mathematicians, and to some extent they had a common motive of making mathematical reasoning more precise. But there were many contrasts between them. Frege devised his Begriffsschrift ( concept-script or ideography ) for formal logic and mathematics, with the particular aim of carrying through what later became known as the logicist programme. He wanted to prove that not only the reasoning used in mathematics, but the underlying assumptions too, and therefore the whole of mathematics, are just pure logic. This would show that mathematics is analytic, refuting Kant s dictum that it is synthetic a priori. Frege made all his deductions using a precisely defined formal deductive system. For this reason above all others he is nowadays commonly regarded as the founding father of modern logic. Moreover, he independently invented quantifiers, and drew attention to numerous important distinctions. Peano s teaching experience led him to be interested in stating mathematical results and arguments precisely. He developed a formal notation for expressing mathematical propositions, which was rather closer to conventional symbolism than was Frege s concept-script. He was not so interested in a foundational reduction, but concentrated on rewriting mathematics in a formal framework. Together with his colleagues and assistants he published a substantial amount of formalized mathematics: his journal Rivista di Matematica was published from 1891 until However, although he stated mathematical propositions in symbolic form, and seemed to be striving towards a way of transforming assertions by analogy with algebraic equations, he never developed a formal deductive system. Russell had started a similar program to Frege, at first independently, before becoming aware, via Peano, of how much Frege had already done. On studying his works, Russell found a logical inconsistency at the core of Frege s system (the Russell paradox ). The story of how he informed Frege of this just as the second volume of Frege s book was in the press is well-known. Russell developed his own logic, which was distinguished from others by its introduction of the notion of type. His logic combined, to some extent, the notational convenience of Peano s system with the precision and foundational aspirations of Frege s work; moreover it wasn't obviously inconsistent. In their monumental work Principia Mathematica, they began a formal development of mathematics from this basis. Without the flexibility which typelessness allowed, it was no longer possible to derive the existence of an infinite set from logic alone, and a separate Axiom of Infinity needed to be posited (as well as an Axiom of Reducibility, but that was a consequence of the unnecessary complication of ramified types). For this reason, Principia failed to establish the logicist thesis that mathematics is completely reducible to logic. Metaphysics Physics ~ Meta-physics

2 2 20 n n 1 n 1 n n < n A B ( ) (1) x + y = z 2z = x + y + z (2) (1) x + y = z 2z = x + y + z (1) (2) (1) (2) if then a x (paradigmatic scientific question) x x

3 Plato was a prolific writer -- scripting some 650,000 words. His texts, furthermore, represent the only writings from antiquity which have been preserved whole and intact by a single author. In other words, there is no reference to a writing attributed to Plato which we today do not possess. This is rather fortunate. Plato s legacy is a collection of thirty-five Socratic Dialogues and thirteen public letters. Plato s writing style possesses infinite variety and a great command of Greek prose -- perhaps the greatest command of the language from antiquity. His work is often tinged with poetic devices, packed with metaphors (especially from music), while also being manneristic, fluid, interlaced, and full of assonance. Plato appears to give each dialogue a separate treatment, adapting its topic to a collection of specific interlocutors. And yet, all of the dialogues have common threads -- e.g., Sokrates who appears in every work save one. Nevertheless, it must be stressed constantly that Plato, now standing at the head of Western Philosophy, left no clearly articulated, well-rounded philosophical system. He apparently lived by example. Plato indeed shunned a political career and lead a life dedicated to the pursuit of truth while, at the same time, heading a cultic body whose purpose was to produce philosophical statesmen. b theory of causality: formal, material, efficient, and final causes Logic: 1.Categories (10 classifications of terms) 2.On Interpretation (propositions, truth, modality) 3.Prior Analytics (syllogistic logic) 4.Posterior Analytics (scientific method and syllogism) 5.Topics (rules for effective arguments and debate) 6.On Sophistical Refutations (informal fallacies) Physical works: 1.Physics (explains change, motion, void, time) 2.On the Heavens (structure of heaven, earth, elements) 3.On Generation (through combining material constituents) 4.Meteorologics (origin of comets, weather, disasters) Psychological works: 1.On the Soul (explains faculties, senses, mind, imagination) 2.On Memory, Reminiscence, Dreams, and Prophesying Works on natural history: 1.History of Animals (physical/mental qualities, habits) 2.On the parts of Animals 3.On the Movement of Animals 4.On the Progression of Animals 5.On the Generation of Animals 6.Minor treatises 7.Problems Philosophical works:1.metaphysics (substance, cause, form, potentiality) 2.Nicomachean Ethics (soul, happiness, virtue, friendship) 3.Eudemain Ethics 4.Magna Moralia 5.Politics (best states, utopias, constitutions, revolutions) 6.Rhetoric (elements of forensic and political debate) 7.Poetics (tragedy, epic poetry) c MP PM MP PM SM SM MS MS SP SP SP SP 3 AAA AEE AI I AEE AI I AOO IAI IAI EAE EAE OAO EIO EIO EIO EIO

4 rhetoric 17 the ideal of certainty the ideal of rational belief with uncertainty, exemplified by the laws of probability and the laws of nature (Newton s law of gravitation) intellectual interest from a theory of demonstration toward theories of rational belief and rational inference, the mathematics of rationality, the quantitative theories of ideas Galilei ( ), Bacon ( ), Descartes ( ), Hobbes ( ), Pascal ( ), Leibniz ( ), Newton ( ), Locke ( ) Science, mathematics, and philosophy were simply different aspects of a common enterprise of knowledge. 1. The theory of deductive reasoning and demonstration is part of psychology. It should provide part of the law of thought, just as physics provides the laws of motion. 2. The laws of thought have an algebraic structure, just as do the laws of arithmetic or the laws of motion. ( 1 turned out to he false, but 2 to be correct later.) combination (human = rational + animal) Any kind or property that can be the object of scientific knowledge can be analyzed into a combination of simple properties that cannot be further analyzed. This gave rise to the mathematical subject of combinatorics, the study of the numbers of possible combinations satisfying given conditions. Ramon Lull (d. 1315), a Franciscan monk Reasoning can be done by a mechanical process. Reasoning does not proceed by syllogism but by combinatorics. Reasoning is the decomposition and recombination of representations. Hobbes: Reasoning is a psychological process, so that a theory of logical inference should be a theory of the operations of the mind. The theory of reasoning is a theory of appropriate combinations. Descartes method What is clearly and distinctly conceived to be true cannot be false. The separation of thoughts of properties is a perfect indicator of the possible separation of the properties: properties that cannot be conceived of separately are necessarily coextensive, and properties that can be conceived of separately are not necessarily coextensive.

5 5 A genuine recollection of a sequence of clear and distinct ideas cannot be false. 1. Descartes argues that some thoughts, some clear and distinct ideas, are indubitable. 2. He claims that we can know with complete certainty that a benevolent God exists. The binomial theorem Pascal: a systematic connection between the theory of combinations and ordinary algebra (x + y) n Binomial theorem For a positive integral n, the binomial coefficient of x (n - r) y r is exactly the number of ways of choosing r things from n things. In other words, the binomial coefficient is n!/(r!(n - r)!) Use the Binomial Theorem to expand the given expressions: ( x + y 2 ) 5 ( 1 + 1/x ) 6 ( x + 2/x ) 4 Pascal s Triangle: 1 n=0 1 1 n= n= n= n= n= n= Leibniz and the mathematics of reason Gottfried Wilhelm Leibniz (b. 1646, d. 1716) was a German philosopher, mathematician, and logician who is probably most well known for having invented the differential and integral calculus (independently of Sir Isaac Newton). In his correspondence with the leading intellectual and political figures of his era, he discussed mathematics, logic, science, history, law, and theology. Leibniz is known among philosophers for his wide range of thought about fundamental philosophical ideas and principles, including truth, necessary and contingent truths, possible worlds, the principle of sufficient reason (i.e., that nothing occurs without a reason), the principle of pre-established harmony (i.e., that God constructed the universe in such a way that corresponding mental and physical events occur simultaneously), and the principle of noncontradiction (i.e., that any proposition from which a contradiction can be derived is false). Leibniz had a lifelong interest in and pursuit of the idea that the principles of reasoning could be reduced to a formal symbolic system, an algebra or calculus of thought, in which controversy would be settled by calculations. De Arte Combinatoria (1666) He formulated the notion of a decision procedure for logic: a mechanical or algorithmic procedure that will determine whether or not an inference is valid. He even attempted to give such a procedure for the theory of the syllogism. He made clear the notion of an incomplete axiomatic theory. An axiomatic theory is incomplete provided there is some

6 6 sentence in its language that can be neither proved nor disproved from its axioms. He introduced the idea that pieces of language can be coded by abstract symbols, including numbers, and that logical relations among the symbols or numbers. He introduced and furthered the idea that logical relations among propositions have an algebraic structure. He developed the thought that universal subject-predicate propositions do not presuppose the existence of things satisfying their predicate or subject terms. The Laws of Thought Boole s conception of logic Logic consists of a set of laws, Like the laws of physics or the laws of geometry. The laws have an algebraic form. The laws have to do with the correct operation of the mind. The Universe of Discourse: (def.) A field of sets over a nonempty set U is any collection of subsets of U that contain U, contains the complement relative to U of any set it contains, contains the intersection of any two sets it contains, and contains the union of any two sets it contains. The Laws of Boolean Algebra x + y = y + x xy = yx x (y + z) = (xy) + (xz) x + (yz) = (x + y) (x + z) x + 0 = x x1 = x x(1 - x) = 0 and x + (1 - x) = 1 x + (y + z) = (x + y) + z x(yz) = (xy)z not 0 = 1 Every field of sets is a Boolean algebra. Two element Boolean algebra No A are B. AB = 0 All C are B. C(1 - B) = 0 No A are C. AC = 0 George Boole Boole was mainly self-educated. Despite not having an academic degree, he was appointed to the chair of mathematics at Queens College, Cork in He taught there for the rest of his life, gaining a reputation as an outstanding and dedicated teacher. In 1854 he published An investigation into the Laws of Thought, on Which are founded the Mathematical Theories of Logic and Probabilities. Boole approached logic in a new way reducing it to a simple algebra, incorporating logic into mathematics. He pointed out the analogy between algebraic symbols and those that represent logical forms. It began the algebra of logic called Boolean algebra which now finds application in computer construction, switching circuits etc. He also worked on differential equations, the calculus of finite differences and general methods in probability publishing around 50 papers. Boolean algebra has wide applications in the design of modern computers and Boole s work has to be seen as a fundamental step in today's computer revolution. Boole was not primarily concerned with the automation of calculating activities. His motivation for developing the system was to assist in expressing and evaluating the soundness of logical propositions. For example, suppose we have a number of assertions P, Q, R, S, etc. Boole considered the question of what could be said of assertions made by combining these in various ways, e.g. P AND Q AND R, ( P OR S ) AND ( Q OR R ). By considering assertions to by true or false, Boole developed an algebraic calculus to interpret whether composite assertions were true or false in terms of how they composition was formed (i.e. the combination of AND, OR, etc) and the truth or falsity of the atomic assertions. Boole published his work in the mid 19th Century An Investigation of the Laws of Thought, Dover Publications, Inc., 1854 Its importance with respect to computer design was realised in the 20th century when approaches to constructing digital computer were being investigated. Boole s calculus was instrumental in breaking the tradition of reducing computation to addition: the route that had, in effect, dominated the design of automatic calculating machines from Schickard through to Babbage, since by exploiting Boolean algebra and electrical (or electro-mechanical) switching components it became possible to build reasonably reliable systems capable of carrying out complex computing tasks. In 1847 in The Mathematical Analysis of Logic [4] Boole writes that

7 7... the validity of the processes of analysis does not depend upon the interpretation of the symbols which are employed, but solely upon the laws of their combination. Every system of interpretation which does not affect the truth of the relations supposed is equally admissible.

Lecture Notes for MATH Mathematical Logic 1

Lecture Notes for MATH Mathematical Logic 1 Lecture Notes for MATH2040 - Mathematical Logic 1 Michael Rathjen School of Mathematics University of Leeds Autumn 2009 Chapter 0. Introduction Maybe not all areas of human endeavour, but certainly the