LOGIC AND SCIENTIFIC REASONING
LOGIC AND SCIENTIFIC REASONING PHL 313Q
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Date Created: 09/07/15
Summer 03 Talbot INTRODUCTION TO SYMBOLIC LOGIC Prof Rob Koons rkoonsmailutexasedu Text Robert Koons and Daniel Bonevac Tools for Thinking Oxford 2003 Classware Socrates 20 and Aristotle I 0 for Windows Socrates 088 is available for the Macintosh platform Available Aristotle amp Socrates Web pages wwwutexaseducoursesplatoaristotle wwwutexaseducoursessocrates Requirements Homework 20 Logic workshops 10 Exams 50 Term project 20 Why are Logicians so Funny Kierkegaard the comic is the perception of a painless contradiction The logic of Groucho Marx Logic is the study of the universal laws of truth and consequence Valid for everything from physics to poetry Even law Traditionally one of the seven liberal arts The capstone of the seven Distinctive Emphases of the Course 1 Practical realworld application 2 Semantics meanings of the logical symbols not merely rules for manipulation 3 Informal presentation of mathematical proofs 4 Breadth Wide range of forms of reasoning 5 Logic as a multipurpose tool 6 Use of computer for guided self instruction COURSE OBJECTIVES 1 To become adept at using logic in the following ways 1 verifying the correctness of logical inferences 2 exposing hidden assumptions of inferences 3 raising pertinent secondary questions questions that must be answered in the course of answering the principal question 4 exposing inconsistency in information and assumptions 5 recovering from inconsistency by locating questionable information 6 understanding and creating informal proofs in mathematics 6 discovering and explaining anomalies 7 using logic as an adjunct to argumentation 2 To acquire the following skills of argumentation 1 identifying arguments 2 understanding and analyzing arguments 3 evaluating arguments for correctness and completeness 4 repairing incomplete or otherwise defective arguments 5 constructing new arguments 3 To gain uency in the language of symbolic logic through 1 translating English statements in to the language of symbolic logic 2 translating statements in symbolic logic into English 3 using semantic tables to analyze the logical implications of statements in the language of symbolic logic 4 To gain basic competence in the following forms of reasoning I propositional logic 2 predicate logic with identity 3 modal logic the logic of necessity and possibility 4 defeasible logic PART 1 BASIC PROPOSITIONAL LOGIC ARGUMENTS Our first task in logic is that of argument analysis Analysis is applied to a block of text or speech that is argumentative in nature Argumentative text informal presentation of one or more lines of reasoning Argument analysis spelling out explicitly the questions and inferences implicit in an argumentative text The skill of argument analysis is fundamental it is needed for the evaluation criticism repair and construction of arguments Arguments consist of premises conclusions and log1cal inferences Premises a collection of statements known or believed or assumed to be true The starting point of a process of reasoning Conclusion a statement Whose truth is to be derived from the premises by a process of reasoning The goal of the inquiry Statement the use of a sentence to attempt to state a fact or represent information about the world A statement is either true or false and never both Inference the process of making explicit What is implicit in a body of statements Deductive inference a truthpreserving inference if the input is true the output cannot possibly be false Some other important terms Validity the property of a correct deductive inference truthpreserving Implication a set of sentences S implies or entails a sentence A just in case the argument S A is deductively valid Consistency a set of statements S is consistent if it is logically possible for all of them to be true Inconsistency a set of statements S is inconsistent if it is logically impossible for all of them to be true There is a very tight connection between validity and inconsistency An argument A1A2 An B is deductively valid if and only if the set A1A2AnnotB is inconsistent 12 Some examples of logical reasoning G K Chesterton39s quotThe Secret Gardenquot from T he Innocence of Father Brown Cast of characters Valentin chief of police in Paris Lord amp Lady Galloway English ambassador amp Wife Lady Margaret Graham daughter of these Dr Simon French scientist Father Brown Commandant O39Brien of French Foreign Legion Julius K Brayne American millionaire The mystery takes place in Valentin39s home an absolutely secure house with surrounding gardens After dinner the guests disperse throughout gardens Soon a body is discovered the head is completely severed neck and shoulders slashed The face is recognized by Valentin it is that of an American thief Whose twin brother had just been guillotined Julius Brayne has disappeared Dr Simon asks I How did the Victim enter the garden 2 How did Brayne leave the garden 3 Why were neck and shoulders of Victim mutilated Argument Analysis 1 The body of the Victim could not in its entirety have entered or been brought into the garden 2 The only person who was in the garden and who is unaccounted for is Brayne IXI 3 The body is at least in part that of Brayne L1 from 1 2 4 The head is not that of Brayne nor of anyone else who was in the garden IXIS The head and the body do not belong to the same person L1 from 3 4 IXI6 The head was brought into the garden L1 from 4 7 The only person Who could have brought a head into the garden is Valentin IXI 8 Valentin brought the head into the garden L1 from 6 7 9 Whoever brought the head into the garden is the murderer IXI lO Valentin is the murderer LI from 89 A second example Socrates and the slave boy in the Meno 1 The area of a square Whose sides are 2 feet long is 4 sq feet Premise 2 The area of a square Whose sides are 4 feet long is 8 sq feet Premise 3 A square Whose sides are 4 feet long is equal in area to four squares Whose sides are 2 feet long Premise 4 The area of a square Whose sides are 4 feet long is 16 sq feet Logical Inference from 13 5 8 16 L1 from 2 4 6 8 16 New premise Socrates then enables the slave boy to work out a range within which the true answer must lie 1 If there are squares X y and z with areas 4 8 and 16 respectively then the length of the sides of square y will be between the lengths of the sides of squares X and z 1M 2 The area of a square with sides 2 ft long is 4 sq feet 1M 3 The area of a square with sides 4 ft long is 16 sq feet 1M 4 The length of the sides of a square with area 8 sq feet is between 2 and 4 feet L1 from 1 2 3 The Table Method To begin an analysis we construct a table Here are a few of the basic facts about semantic tables Statements on the table are assumed to be true We place all the premises plus the denial or quotnegationquot of the conclusion on the table If these assumptions are compatible then the argument is logically incorrect at least incomplete If these assumptions are incompatible then the argument is logically correct The process of argument analysis consists in making the information implicit in the premises and the conclusion explicit This process involves logical inferences which extend or create paths vertical columns Inference moves preserve truth A path is a vertically arranged sequence of statements on the table Disjunctive information posits more than one alternative For example consider the statement either the butler did it or the maid did There are two ways that this statement could be true Inference moves applied to disjunctive information split the path into two parallel subpaths Each table is a compilation of possibilities Each possibility corresponds to one of the paths of the table A path is closed if it contains logically incompatible bits of information Closed paths are marked with an X In particular if a single path in the table contains both the statement A and the statement 39It is not the case that A39 the negation of A then that path does not represent a real possibility A table is closed if every path in it is closed Consider the following argument analysis 1 Either the butler did it or the maid did it assumption 2 The butler did not do it assumption 3 The maid did not do it assumption 4 The butler did it Themaid didit From 1 S X contradiction 6 X contradiction The mble is closed We have already seen the rule of Disjunction Decomposition illustrated Whenever a disjunction a statement involving the connective 39or39 occurs on a path in the left half of the table we may split this path into two subpaths placing onehalf of the statement the part preceding the 39or39 in the left subpath and the remaining half of the statement in the right subpath 20 The Modus Ponens rule concerns the unpacking of information implicit in an 39if then39 statement called in logic a conditional statement If on one path in the left side of the table we find both a statement of the form 39if A then B39 and another statement asserting simply 39A39 then we may add the statement 39B39 to the same path For example consider the following simple argument 1 If I don t pay my taxes I will go to jail assumption 2 I don t pay my taxes assumption 3 I will not go to jail assumption 4 Iwill g0 tojail Modus Ponens l 2 5 X contradiction The table is closed In this argument there are two premises and a single conclusion From premises 1 and 2 line 4 follows by Modus Ponens wl03 21 A third important inference rule is that of Conditional Decomposition Consider the following argument 1 If everyone is here there is a quorum 2 There is not a quorum 3 Everyone is here 4 Not everyone is here 5 6 X There is a quorum The table is closed assumption assumption assumption CD 1 contradiction contradiction 22 LOGICAL ABBREVIATONS I the negation abbreviating the phrase not gt the conditional abbreviating the phrase if then amp the conjunction abbreviating the phrase and v the disjunction abbreviating the phrase or H the biconditional abbreviating the phrase if and only if Any capital letter A B Z followed by any number of primes C is a well formed sentence If 3Q and 75 are well formed sentences so are u y lamp75 x75 A675 and 675 All of the well formed sentences of sentential logic can be formed by repeated applications of rules 1 and 2 23 24 Thus the following strings of characters are not well formed A A A AVB AampB AampBy A In contrast the following are well formed sentences of sentential logic ltAampBgtvltBAgtgt weAH a BwAn A AvAampAlt gtA 25 The Haitian President does not authorize invasion IThe Haitian President authorizes invasion ID The UN and the OAS have authorized an invasion The UN has authorized an invasion amp The OAS has authorized an invasion AampB The UN has authorized both an embargo and an invasion 26 The UN has authorized an embargo amp The UN has authorized an invasion CampB Logical equivalents to and but although however rnoreover yet Non restrictive relative clauses The UN Which has authorized an embargo has also authorized an invasion Either the UN or the OAS is the appropriate authority 27 The UN is the appropriate authority v The OAS is the appropriate authority EvF Logical equivalents to or unless or else Inclusive vs Exclusive If the invasion is authorized by the appropriate authority then the US should invade The invasion is authorized by the appropriate authority gt The US should invade G gtH 28 The US should invade if the invasion is authorized by the appropriate authority The invasion is authorized by the appropriate authority gt The US should invade G gtH The US should invade only if the invasion is authorized by the appropriate authority The US should invade gt The invasion is authorized by the appropriate authority H gtG Antecedent the first half of a conditional before the gt 29 Consequent the second half of a conditional after the gt The antecedent is asserted to be a condition of the consequent The consequent is asserted to be a condition of the consequent If you take 180 credits then you graduate C gtG You graduate only if you take 180 credits G gtC 30 If you smoke then you have a high risk of cancer You smoke only if you have a high risk of cancer S gt C The US should invade if and only if the invasion is authorized by the appropriate authority The US should invade if the invasion is authorized amp The US should invade only if the invasion is authorized The invasion is authorized gt US should invade amp US should invade gt The invasion is authorized G gtHampH gtG Glt gtH 31 Need parentheses The US should invade if the invasion is authorized the invasion is in our national interest The US should invade if the invasion is authorized and the invasion is in our national interest GmampK GampKyH Similarly A amp B v C is ambiguous AampBvC or AampBvC 32 1 Any capital letter A B Z followed by any number of primes C is a well formed sentence 2 If 3Q and 75 are wellIforrned sentences so are u y lamp75 x75 t gt75 and 675 All of the well formed sentences of propositional logic can be formed by repeated applications of rules 1 and 2 A A A AVB AampB AampB a A AampBVB AD AVAampAHA 33 There should always be exactly as many left parentheses in a sentence as there are right parentheses and there should be exactly one pair of parentheses for each binary connective A A AB AampB T F TT T F T TF F F T F F F F 34 AB AVB AB AaB TT T TT T TF T TF F FT T FT T FF F FF T Material conditional gt is essentially disjunctive If A then B r Either not A or B AFB IAVB IA v B contains the minimal amount of information consistent With the claim that A is in some sense sufficient for B If IA v B is true and A is true then B must be true since A Will be false For mathematical purposes this is all the information we need LLHHH LLHHH mHmHLL ltHHLLLL Er gt ltL air Am r i 3 5 2 Am i r LLFFLL Hume mHmHLL ltHHLLLL Sr is Ami TJ TJ Dgt TJ TJ w TJ TJ Dgt TJ TJ w HDgt 36 IAvB IA amp B F F F F F F T T IAB A amp B F F T T F F F F A F A F T T F 37 Evaluating argument forms 1 List the sentence letters appearing in the argument form 2 Beneath them list all possible interpretations 3 List each premise formula and then the conclusion formula 4 Compute the value of each formula PQPPeQ 39Q TTTT TFTTFF F FTFFTF T FFFFTF F P Q Pamp Q39PHQ TT TFFT T 7T TF TTTF T F FTF FFT F T FF F FTF F F 38 What is Logic Logical Inference Psychologism logic 2 laws of human thought Logic is normative not merely descriptive Logic makes explicit What is implicit What does implicit information mean What does information mean Logically valid inference impossible for the premises to be true While the conclusion is false What does impossible mean here 39 This is gold This is water This is element 59 This is H20 Marie has been decapitated Marie is dead Distinguish logical possibility amp physical possibility Irnaginable possibility There is a lOOO sided figure What about John is a bachelor Mary is an optharn John is unmarried Mary is an MD 40 Ludwig Wittgenstein Tractatus Logica Philosophicus Atomic propositions either true or false Lo gically independent Each assignment of truth values to all atomic sentences is a possible world a point in logical space TJ TJ Dgt TJH TJ W 41 Truth functional proposition a proposition whose truth value in any world is a function of the truth values of certain atomic propositions in that world An argument is logically valid if and only if the conclusion is true in every world in which all the premises are true Straightforward mathematical question The following introduce negative contexts v Denial Antecedent of gt 42 A part of a sentence occurs negatively in that sentence if and only if it occurs Within an odd number of nested negative contexts In the following sentences the sole occurrence of A occurs negatively and the sole occurrence of B occurs positively IA A gtB CvA gtBampD BvA nbAvB BvAPC 43 To weaken a sentence logically do the following within contexts 1 Delete conjuncts 2 Add disjuncts 3 Conditionalize eg replace A with B gtA To weaken a sentence logically do the following within contexts 1 Add conjuncts 2 Delete disjuncts 3 Deconditionalize eg replace B gtA with A 44 To a sentence logically do the following within contexts 1 Add conjuncts 2 Delete disjuncts 3 Deconditionalize To strengthen a sentence logically do the following within contexts 1 Delete conjuncts 2 Add disjuncts 3 Conditionalize eg replace A with B gtA Strengthen the following sentences 1 AampB gtC 2 B gtD 3 AVD gtCVB Now weaken the same sentences 45 Principles of Argument Analysis I Distinguish between what is and what should be Represent by means of different sentences II Distinguish between negation and opposition 111 Find a valid argument whenever possible Even if the argument is invalid you should find a valid argument in the vicinity of what is said IV Eliminate super uous information in the premises and unnecessary weakness in the conclusion Experiment with eliminating or weakening premises if the argument remains valid make the change Try to make the conclusion as strong as possible so long as argument remains valid V Separate separate arguments If you have two sets of premises each of which would separately close the table separate them into two arguments 46 I Distinguishing What is and What should be College football should not have a national playoff because if it does regular season play will be diminished in importance and the regular season should not be diminished Attempted analysis P College football should have a national play off R Regular season play will be diminished in importance Symbolic argument P gtRIR39IP W301 What s wrong a confusion of should and will 47 Q College football have a national play off Improved analysis Q gt R Q r R P P Even better C A national play off would cause a diminution of the importance of regular season play Argument C gt P C IP Something should be done about population control because if nothing is done a famine will occur and famines should not occur Attempted analysis D Something should be done about population control F Famine occurs Argument ID gt F F D 48 Another confusion of should and will Better analysis N Doing nothing about population control will result in famine Argument N N gt D D Also distinguish between quothasquot and quotneedsquot quotThe penal system needs more moneyquot vs quotThe penal system has more moneyquot 11 Distinguish between negation and opposition L you are a lover IL means quotyou are not a loverquot 49 It does not mean You are a beloved You are a hater 111 Find a valid argument Wherever possible Abbreviation G The federal government is running a budget de cit M The money supply is increasing V The total amount available for investments is stable P Interest rates will increase The symbolic argument G M v V GampV gt P P W3 02 What premise needs to be added to make the table close 50 IV Eliminate super uous premises amp unnecessary weakness in conclusion A US Soldiers amp civilians might get hurt B Plan is a disaster C Clinton disarms Cedras D Cedras still running Haiti A B C v A C amp D B W303 Which premise can be weakened Without depriving the argument of validity 51 S The nation remains under a laissez faire system P The nation will prosper H The community works to solve health care 0 The community works as an ef cient organ1zat1on 1 P a o amp H 2 P v O 3 s o amp IH s IP w304 Which premises can be weakened or deleted altogether 52 G animals have a divine right to life A animals have a right to life L animals have a right to life granted them by humans K human laws prevent us from killing animals for food or sport 1 G a A 2 L amp IG 3 A a K 4 K IA w305 What redundancies in the premises can we identify Can we re interpret the premises in such a way to make facts about G and L relevant 53 A better interpretation 1 A gt G v L 2 G amp K 3 L gt K IA W306 V Separate separate arguments 1AaB 2 C 613 3AampC 39B w307 Break this into two separate arguments for B 1 A v D 2 ID gt B 3 A v B 39D W308 54 The quotParadoxesquot of Material Implication The following argument forms are valid A m A Remember B gt A 2 B V A 55 66 99 M The media is biased in favor of the candidate E The candidate Wins English sentences quotThe fact that the media is biased in favor of the candidate is not by itself sufficient to guarantee that the candidate Will Winquot It is not the case that if the media is biased in favor of a candidate that candidate Will Win Mistranslation M gt E This is equivalent to M amp IE Back translation 56 quotThe media is biased in favor of the candidate and the candidate won t Winquot In such cases it may be better to translate the conditional by means of a simple sentence letter like C and represent the denied conditional as C In chapter 5 we Will introduce some new alternatives P There is a national play off Q Regular season play Will be diminished R Every game counts rap 2 P 6 R 39P Q Premise 2 is redundant The conclusion follows from premise 1 alone Solution replace premise l by R Q G Gore Will be re elected in 2000 H Congress passes major health care reform in 98 M Gore receives a majority vote in the Electoral College in 2000 58 quotGore won t be re elected in 2000 since Congress won t pass health care reform in 98 So even if Gore receives a majority vote in the EC he won t be re electedquot l G amp IH M gt G This crazy argument turns out to be logically valid when translated in this way The fundamental problem is this the conclusion is much too weak given these conclusions We can strengthen the conclusion to simply G which eliminates the paradoxical suggestion that even the Electoral College couldn t elect Gore
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