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Logic Workshop

Logic Workshop

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Logic Workshop

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  1. Logic Workshop

  2. Expectations? • What do you expect to get out of this workshop? • What do you currently know about logic?

  3. Schedule • Day One • Introduction to logic (what is it? why is it important?) • Historical connections within the context of philosophy • Major philosophers and their contributions to logic • The building blocks of logic • Day Two • Applications of logic • Symbolic logic • Paradoxes and other interesting philosophical questions

  4. Logic 101 λόγος

  5. Greek • λόγος – logos • From Classical Greek • Essentially means “thought, idea, argument, account, reason, or principle” • A very important term in philosophy (as well as in analytic psychology, rhetoric, & religion) • Derives from • Logic is about just this (logos) – it’s about developing an understanding of the methods and principles used to distinguish between correct and incorrect reasoning • Traditionally considered a branch of philosophy • Today it has also spread to other fields such as mathematics (e.g. mathematical logic) and computer science (e.g. programming, writing code)

  6. Everyday Use • We use “logic” everyday • Arguments, justifications, reasons, most all rational functions • Listen to just how many logical arguments (with premises and conclusions) you hear on TV shows, the radio, etc. • Therefore it’s important to develop our logical knowledge…

  7. …because if we don’t • then we might be the victim of Bad Reasoning • But, as the Stanford Encyclopedia of Philosophy notes, logic “does not, however, cover good reasoning as a whole. That is the job of the theory of rationality. Rather it deals with inferences whose validity can be traced back to the formal features of the representations that are involved in that inference, be they linguistic, mental, or other representations”

  8. Brief Introduction to Logic • Logic: the organized body of knowledge, or science, that evaluates arguments • Our aim then is to develop system of methods to use as criteria for evaluating arguments of others and for constructing our own; to determine good arguments from bad arguments

  9. Logic Evaluates Arguments • Argument: a group of statements, one or more of which (premises) are claimed to provide support for, or reasons to believe, one of the others (conclusions) • Good argument: premises support the conclusion • Bad argument: premises do not support conclusion (even if they claim to) • Syllogism: a kind of logical argument in which one proposition (the conclusion) is inferred from at least two others (the premises) of a certain form

  10. Arguments are Made up of Statements • Statement: a sentence that is either true (T) or false (F) • Melatonin helps relieve jet lag. (T) • No wives ever cheat on their husbands. (F) • and is made up of… • 1. Premises • Statements that set forth the reasons or evidence • 2. Conclusions • Statements that the evidence is claimed to support or imply (claimed to follow from the premises)

  11. Propositions • Proposition: the meaning or information content of a statement (the ‘material of our reasoning’) • “Proposition=Statement” • Can be simple (making only one assertion) or compound (containing two or more “simple” propositions • Inference: a process of linking propositions by affirming one proposition on the basis of one or more other propositions • “Inference=Argument” • Many sentences, unlike statements, cannot be said to be T or F • Questions (Where is Tom?); Proposals (Let’s go to a movie.); Suggestions (I suggest you get contact lenses.); Commands (Turn off the TV.); Exclamations (Wow!)

  12. Good vs. Bad Arguments • Good Argument: 1. All film stars are celebrities. (Premise 1) 2. Halle Berry is a film star. (Premise 2) C. Therefore, Halle Berry is a celebrity. (Conclusion) • Bad Argument: 1. Some film stars are men. 2. Cameron Diaz is a film star. C. Therefore, Cameron Diaz is a man.

  13. Example The space program deserves increased expenditures in the years ahead. Not only does the national defense depend upon it, but the program will more than pay for itself in terms of technological spinoffs. Furthermore, at current funding levels the program cannot fulfill its anticipated potential.

  14. Argument Reconstruction • Break up compound statements • List premises first, then conclusions:P1: The national defense is dependent upon the space program.P2: The space program will more than pay for itself in terms of technological spinoffs.P3: At current funding levels the space program cannot fulfill its anticipated potential.C: The space program deserves increased expenditures in the years ahead.

  15. Classes of Arguments • Inductive Argument: • Claim that the premises support the conclusion with a certain degree of probability • If the premises are true, then it’s likely the conclusion is true • Ex. This cat is black. That cat is black. A third cat is black. Therefore all cats are black. • Deductive Argument: • Claim that the premises support the conclusion conclusively – there is no probability involved, but logical certainty • If the premises are true, then it’s impossible the conclusion is false • Ex. All men are mortal. Joe is a man. Therefore Joe is mortal.

  16. Deductive Arguments • Look for… • Validity: if all premises are true, then the conclusion must be true (follows from the structure, not the content - no matter how ridiculous it may seem - of an argument) • Soundness: if the argument is valid, and the premises are true, then the argument is sound • All sound arguments are valid, but not all valid arguments are sound • Cogency: if the argument is valid and sound, is it cogent (convincing) to a community of listeners • If all sound arguments are valid, must all cogent arguments be sound?

  17. Belvedere by M.C. Escher • A structure in which the relations of the base to the middle and upper portions are not rational • A deductive argument rests upon premises that serve as its foundation • To succeed, its parts must be held firmly in place by the reasoning that connections those premises to all that is built upon them – this way the argument can stand

  18. Waterfall by M.C. Escher • Doesn’t make sense: the water flows away, and going away comes closer • Flows downward, and going down it comes up, returning to the point from which it began • As perception may be tricked by a clever picture, our thinking may be tricked by a clever argument • In the picture we confront disorder in seeing, and then with scrutiny detect its cause • In logic we confront many bad arguments, and then with scrutiny learn what makes them bad

  19. History • Logic as a discipline started with the transition from the unreflective use of logical methods and argument patterns to the reflection on and inquiry into these and their elements, including the syntax and semantics of sentences • Mesopotamia (11th century BC): • Medical diagnostic handbook containing numerous axioms and assumptions (i.e. through examination and inspection of the symptoms of a patient, it is possible to determine the patient's disease, its aetiology and future development, and the chances of the patient's recovery) • Babylon (8th – 7th centuries BC): • Babylonian astronomers began employing an internal logic within their predictive planetary systems, which was an important contribution to logic and the philosophy of science. • Babylonian thought had a considerable influence on early Greek thought

  20. Greece (4th century BC) • Logic as a fully systematic discipline begins with Aristotle (384-322 BC) • He systematized much of the logical inquiries of his predecessors • Main achievements: • “Term Logic” • Fundamental elements of his syllogistic logic are “terms,” which are not true or false in themselves, such as ‘human being,’ ‘animal,’ ‘white’ (i.e. H, A, and W) • For Aristotle, a term is simply a “thing,” a part of a proposition; and a proposition is just a particular kind of sentence, in which the subject and predicate are combined so as to assert something true or false • The essential feature of term logic is that, of the four terms in the two premises, one must occur twice. Thus • All Greeks are men • All men are mortal. • This system became known as the “categorical syllogistic”

  21. Another System Emerges • Chrysippus (279-206 BC) • Was a stoic • {Stoicism: teaches the development of self-control and fortitude as a means of overcoming destructive emotions; holds that becoming a clear and unbiased thinker allows one to understand the universal reason, logos} • Main achievements: • “Propositional Logic” • Fundamental elements of logic are “whole propositions” - something over and above the terms are true • Studies ways of joining and/or modifying entire propositions, statements, or sentences to form more complicated ones • Also looks at the logical properties derived from these methods of combining or altering statements (e.g. logical connectives in symbolic logic). • Does not study those logical properties that depend upon parts of statements that are not themselves statements on their own, such as the subject and predicate of a statement • This system became known as the “hypothetical syllogistic”

  22. 1300-year Intermission • 200 BC – 1100 AD • Lack of logical innovations • Mostly just commentators either preserving Aristotelian logic, such as Boethius (480-524 AD), or criticizing it, such as Islamic philosopher Avicenna (980-1037 AD)

  23. Peter Abelard (1079-1142 AD) • Theory of universals in the mind rather than natures outside the mind (as Aristotle argued) • Distinguishes between arguments valid in form and arguments valid in content • Abelard observes that the same propositional content can be expressed with different force in different contexts: • the content that Socrates is in the house is expressed in an assertion in “Socrates is in the house”; in a question in “Is Socrates in the house?”; in a wish in “If only Socrates were in the house!” and so on. • Hence Abelard can distinguish the assertive force of a sentence from its propositional content, a distinction that allows him to point out that the component sentences in a conditional statement are not asserted, though they have the same content they do when asserted • Ex. "If Socrates is in the kitchen, then Socrates is in the house" does not assert that Socrates is in the kitchen or that he is in the house • Likewise, the distinction allows Abelard to define negation, and other propositional connectives, purely truth-functionally in terms of content, so that negation, for instance, is treated: not-p is false/true if and only ifp is true/false

  24. Gottfried Leibniz (1646 – 1716 AD) • Leibniz is the most important logician between Aristotle and 1847, when George Boole and Augustus De Morgan each published books that began modern formal logic • Leibniz enunciated the principal properties of what we now call conjunction, disjunction, negation, identity, set inclusion, and the empty set, essentially building the foundation for symbolic logic as we know it today • Two main principles: • All our ideas are compounded from a very small number of simple ideas, which form the alphabet of human thought. • Complex ideas proceed from these simple ideas by a uniform and symmetrical combination, much like arithmetical multiplication. • This world is the best of all possible worlds. - Leibniz

  25. Symbolic Logic AP • Is a technique for analysis of deductive arguments • English (or any) language can make any argument appear vague, ambiguous; especially with use of things like metaphors, idioms, emotional appeals, etc. • We want to avoid these difficulties to move into logical heart of argument: use symbolic language • Now can formulate an argument with precision • Symbols facilitate our thinking about an argument • These are called “logical connectives”

  26. The Logical Connectives • We can translate arguments from sentences and simple or compound propositions/statements into symbolic logical form using: • Conjunction: “(conjunct 1) and (conjunct 2)” • p • q p & q • Disjunction: “(disjunct 1) or (disjunct 2)” • p v q • Negation: “It is not the case that…” • ~ p • Conditional: “If (antecedent), then (consequent)” • p q p  q ∩

  27. Basic Abbreviation & Translation Rules • “Charlie’s neat and Charlie’s sweet.” • N • S • Dictionary: N=“Charlie’s neat” S=“Charlie’s sweet” • Can choose any letter to symbolize each statement (in this case, two conjuncts), but it is best to choose one relating to the content of that conjunct to make it easier to remember

  28. Punctuation • As in mathematics, it is important to correctly punctuate logical parts of an argument • Ex. (2x3)+6 = 12 whereas 2x(3+6)= 18 • Ex. p • q v r (this is ambiguous) • To avoid ambiguity and make meaning clear • Make sure to order sets of parentheses according to how the argument reads: • { A • [(B v C) • (C v D)] } • ~E would be a different read than [(A • B) v (C • C)] v [D • ~E]

  29. Example “There were three people involved in the accident, and no one was injured.” (Steps: Translate into logic by making a dictionary then arrange statements correctly in logical form) (Hint: Note: When symbolizing statements, always make the statement a positive one. If you have a negative statement in the sentence, put its positive in the dictionary – then when you translate, simply negate that sentence)

  30. Solution There were three people involved in the accident, and no one was injured. • T • ~OT=Three people were involved in the accident.O=Someone was injured.

  31. One more… “Either you are male or female but not both.”

  32. Solution “Either you are male or female but not both.” (M v F) • ~(M • F) M=You are male. F=You are female.

  33. Inference Rules • There are many different systems of formal logic, each one with its own set of well-formed formulas, rules of inference, and even semantics (e.g. temporal logic, modal logic, intuitionistic logic, quantum logic) • These rules are like shortcuts for us when doing logical proofs • Take a look at the handout you have on Inference Rules

  34. Nine Basic Inference Rules • These nine rules of inference correspond to elementary argument forms whose validity is easily established by truth tables. With their use, formal proofs of validity can be constructed for a wide range of more complicated arguments.

  35. Let’s put this all together now:First, translate the following argument from English into symbolic logic: • If Anderson was nominated, the she went to Boston. • If she went to Boston, then she campaigned there. • If she campaigned there, she met Douglas. • Anderson did not meet Douglas. • Either Anderson was nominated or someone more eligible was selected. C. Therefore: someone more eligible was selected.

  36. Solution (Part 1) • A B • B C • C D • ~D • A v E • Therefore E ∩ ∩ ∩

  37. Now let’s try to prove the conclusion from only the given premises: • This is now where our inference rules may (or may not) be of help to us

  38. Once translated… • Write the premises and the statements that we deduce from them in a single column – to the right of this column, for each statement, its “justification” is written (e.g. the reason why we include that statement in the proof) • Here’s what it looks like:

  39. Formal Proof ∩ • A B Premise • B C Premise • C D Premise • ~D Premise • A v E Premise • A C 1,2 H.S. • A D 6,3 H.S. • ~A 7,4 M.T. • E 5,8 D.S. ∩ ∩ The justification for each statement (the right most column) consists of the numbers of the preceding statements from which that line is inferred, together with the abbreviation for the rule of inference used to get it ∩ ∩

  40. Proofs • There are many other examples of simpler (and more difficult) proofs, but we don’t have time to get into the rules associated with proving even the most basic of them, so we will skip over that for now • The main point here is to show the application of symbolic logic in the context of symbolic translations and in constructing formal proofs

  41. Paradoxes and Questions • Paradox – seemingly sound piece of reasoning based on seemingly true assumptions that leads to a contradiction or another obviously false conclusion • Zeno's paradoxes are a set of problems devised by Zeno of Elea to support Parmenides' doctrine that "all is one" and that, contrary to the evidence of our senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion

  42. Zeno’s Paradoxes • Story of a race between Achilles and the tortoise • Achilles can run faster so tortoise given big head start • When race starts, Achilles’s first goal is to get to the point where the tortoise started • By the time he gets there, the tortoise has moved again (but only a little) – so Achilles must now get to that spot • No matter how many times Achilles reaches the tortoise’s prior location, even if he does it an infinite number of times, he’ll never catch up with the tortoise, although he’ll get awfully close • Tortoise just has to not stop then in order to win “In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.” -Aristotle

  43. Racetrack paradox: • In order to get to the end of a racetrack, a runner must first complete an infinite number of journeys • He must run to the midpoint • Then he must run to the midpoint of the remaining distance • Then to the midpoint of the still remaining distance, etc etc • Therefore he can never get to the end of the track

  44. Arrow paradox • If everything, when it occupies an equal space, is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless.- Aristotle • Imagine an arrow in flight • Divide up time into a series of indivisible moments • At any given moment, if we look at the arrow it has an exact location, so it is not moving • Yet movement has to happen in the present (it can't be that there's no movement in the present yet movement in the past or future) • So throughout all time, the arrow is at rest • Thus motion cannot happen

  45. Zen Koans • A kōan is a story, dialogue, question, or statement in the history and lore of Zen Buddhism, generally containing aspects that are inaccessible to rational understanding (yet may be accessible to intuition)

  46. The Gateless Gate • Joshu’s Dog A monk asked Joshu, a Chinese Zen master:“Has a dog Buddha-nature or not?” Joshu answered: “Mu.” • Mumon’s comment: Enlightenment always comes after the road of thinking is blocked. ‘Mu’ is not nothingness, just concentrate your whole energy into this Mu, and do not allow any discontinuation.

  47. The Gateless Gate • Wakuan complained when he saw a picture of bearded Bodhidharma: “Why hasn’t that fellow a beard?” • Daibai asked Baso: “What is Buddha?” Baso said: “This mind is Buddha.” A monk asked Baso: “What is Buddha?” Baso said: “This mind is not Buddha.” • Two hands clapping make a sound. What is the sound of one hand clapping?

  48. Thank you! • Were your expectations met? If not, why? • What do you now know about logic?