Assignment 1 You were meant to read Lave & March chapters 2 and 3, andFreakonomics chapter 4. You were supposed to use the 4-steps in the Lave & March modeland apply it to 3 explanations given by Levitt and Dubner for thesurprising drop in violent crime rates in the US in the 1990s. You were asked to choose a correct, and incorrect and a surprising(and correct) explanation from the text to do the assignment.
Main observations The main (and most frequent) limitation in the assignments handedin is that most of them DO NOT contain a discussion of HOW theexplanations were tested. It is important to state (i) what are the implications of the explanation, (ii) how this explanations were tested to account for the implications, and (iii) what was the ﬁnding of the authors. In most case assignments only contained a reference to (iii) andomitted (i) and (ii) The second main limitation was literal transcription from the bookto the assignments. Do not copy but use your own words.
In this lecture we will learn: How to formulate valid arguments/explanations How to test whether an argument/explanation is valid The core methods of so called ``propositional logic’’ and ‘‘syllogisticlogic’’ How to generalize and specify concepts and statements
Part 1: How to formulate validarguments/explanations.
What is logic?Philosophical discipline established by Aristotle Aristotle 384BC - 322BC
What is logic? Logic is the analysis and appraisal of arguments Arguments consist of premises and a conclusion An argument is valid means: if all premises are true it is possible that the conclusion is wrongPremise. If you are reading this, you aren’t illiterate You are reading thisConclusion. You aren’t illiterate
What is logic? This is wonderful. With the help of logic you can ﬁnd out whether a statement (conclusion) is true if you know whether other statements (premises) are true. Thus, if your assumptions are plausible you can make your hypotheses/predictions plausible too (no matter how counter intuitive they are) No matter how counter intuitive they are???
We distinguish valid from sound arguments. An argument is valid means: If all premises are true it is impossible that the conclusion is wrong. Logic provides techniques to test whether a given argument is valid An argument is sound means: The argument is valid plus all premises are true. Only if an argument is sound, we can be 100% certain that theconclusion is true. If the argument is valid and at least one premise is false, theconclusion can be false You need empirical research (and further arguments) to testwhether all premises are true. Note: Arguments are not true or false (statements are!)
Which argument is valid andwhich is sound? If economic welfare increases, the rate of unemployment decreases1 In the 1990s, in the US, the economic welfare increased In the 1990s, in the US, the rate of unemployment decreased If the rate of unemployment decreases, the rate of violent crimes decreases2 In the 1990s, in the US, rate of unemployment decreased In the 1990s, in the US, the rate of violent crimes decreased
Basic propositional logicPropositions are statements, for instance: The members of group x are integrated The citizens of Leipzig protest People who hold similar opinions tend to form friendships Propositions are statements which are either true or false (not valid or invalid)Hence, statements which are not true or false are not consideredpropositions. For instance, Shut up! (commands) Why did nobody bring cookies? (question) This is a bad song (normative statements)
Propositional language andtruth tablesPropositions are translated into so called “wff’s” (pronounceas woof as in wood). Wff stands for ``well formed formula” e.g., “I am a sociologist” sPropositions are analyzed using truth tables. Truth tablesgive a logical diagram for a given wff, listing all possibletruth-value combinations. S Symbol of the proposition 1 Truth values: s can be 0 true (1) or false (0)
Truth-functional operatorsPropositions can be combined, forming new propositions. This is donewith so called operatorsOperators deﬁne the truth-value of the combined proposition based onthe truth-values of the propositions that it consists of.Operator 1: Negatione.g. Assume, s (“I am a sociologist”) is true (1). Then, the negation of s(~s) is false (“I am not a sociologist”). Symbol: ~ (squiggle) Read: “not” s ~s 1 0 If s is true, then the negation is false 0 1 If s is false, then the negation is true
Operator 2: Disjunction Symbol: ⋁ (vee) or || or + Read: “or” p q p⋁q 1 1 1 1 0 1 The disjunction of p and q is 0 1 1 false if both p and q are false 0 0 0Operator 3: Conjunction Symbol: ⋅ (dot) or & or ⋀ Read: “and” p q p⋅q 1 1 1 The conjunction of p and q is 1 0 0 true if both p and q are true 0 1 0 0 0 0
Operator 4: Implication Symbol: ⊃ (horseshoe) or → Read: “if p then q” p q p⊃q 1 1 1 1 0 0 The implication of p and q is false 0 1 1 only if p is true and q is false 0 0 1Example: If Popper is a sociologist, then he is a Marxist.Popper is a sociologist + Popper is a Marxist : wff is validPopper is a sociologist + Popper is not a Marxist : wff is invalidPopper is not a sociologist + Popper is a Marxist : wff is validPopper is not a sociologist + Popper is not a Marxist : wff is invalid
Operator 5: Equality (biconditional) Symbol: ! (threebar) or = or ↔ Read: “if and only if p then q” p q p!q 1 1 1 The equality of p and q is true if 1 0 0 either p and q are both true or 0 1 0 both false 0 0 1Example: If and only if Popper is a sociologist, then he is a Marxist.Popper is a sociologist + Popper is a Marxist : wff is validPopper is a sociologist + Popper is not a Marxist : wff is invalidPopper is not a sociologist + Popper is a Marxist : wff is invalidPopper is not a sociologist + Popper is not a Marxist : wff is valid
Other operators Exclusive disjunction: true if one but not if both operands are true (XOR, ≠, ⨁) Logical NAND: false if bot operands are true and true if at least one operand is false (↑,|) Logical NOR: true if both operands are false and false if at least one operand is true (↓,⊥)
Working with truth tables Example: Let us demonstrate for which combination of truth valuesof p and q is it is correct to state: “p and q are equivalent (p!q)”. Thus,we want to show that: (if p, then q) and (if q, then p)p q p⊃q q⊃p (p⊃q)·(q⊃p)1 1 1 1 1 1 1 1 1 1 1 Deﬁnition of an equality1 0 1 0 0 0 1 1 0 1 00 1 0 1 1 1 0 0 1 0 00 0 0 0 1 0 0 1 1 1 1 This proves that: (p!q)!((p⊃q)·(q⊃p))
Rules of inference When we formulate an argument, we infer the conclusion from thepremises. An argument is valid means: If all premises are true it is impossible that the conclusion is wrong. Thus, if all premises are true, then the conclusion is true. This is an implication (⊃) In order to show that an argument is valid (that the inference iscorrect), we need to demonstrate that the conjunction (·) of all premisesimplies (⊃) the conclusion. There are three important forms of argument
Rule 1: Hypothetical Syllogism Example: If Popper is a sociologist (p), then he is is a Marxist (q) If Popper is a Marxist (q), then he hates capitalism (r) If Popper is a sociologist (p), then he hates capitalism (r) General form: p⊃q q⊃r ---------- p⊃r
Demonstrations that the hypotheticalsyllogism is a valid argument form Thus, we want to demonstrate that the conjunction (·) of all premisesimplies (⊃) the conclusion. Therefore, we need to demonstrate: This is an implication if the premises are true, then the conclusion is always true. This means: (p⊃q)·(q⊃r) This means: (p⊃r) We need to show that ((p⊃q)·(q⊃r))⊃(p⊃r) is true independent of thetruth-values of p, q, and r.
Is ((p⊃q)·(q⊃r))⊃(p⊃r) always valid?p q r p⊃q q⊃r (p⊃q)·(q⊃r) p⊃r ((p⊃q)·(q⊃r))⊃(p⊃r)1 1 1 1 1 1 1 1 1 1 1 11 1 0 1 0 1 0 0 0 0 0 11 0 1 0 1 0 1 0 1 0 1 11 0 0 0 1 0 1 0 0 0 0 10 1 1 1 1 1 1 1 1 1 1 10 1 0 1 0 1 0 0 1 0 1 10 0 1 1 1 1 1 1 1 1 1 10 0 0 1 1 1 1 1 1 1 1 1 The conjunction of the premises logically implies the conclusion. Thus, the hypothetical syllogism is always valid (independent of the truth-values of the truth of the premises)
Rule 2: Modus Ponens Example: If Popper is a sociologist (p), then he is is a Marxist (q) If Popper is a sociologist (p) Popper is a Marxist (q) General form: p⊃q q p p ---------- q Venn diagram of an implication
Rule 3: Modus Tollens Example: If Popper is a sociologist (p), then he is is a Marxist (q) If Popper is not a Marxist (~q) Popper is not a sociologist (~p) General form: p⊃q q p ~q ---------- ~p Venn diagram of an implication
Syllogistic LogicLike propositional logic, it is a branch of logic. Propositional logic focuses on propositions which refer to singleobjects (i.e., Popper In contrast, syllogistic logic is concerned with domains of objects Typical wffs from: Syllogistic logic: Propositional logic: All swans are white (all S is W) It rains (r) Societies with high anomie suffer Popper is cool (c) from high crime rates (all A is C) With syllogistic logic, we study the implications of general statements (laws). Remember that our theories are general statements
Formulating wffs in syllogistic logic To formulate a correct wff, you need only ﬁve words: all no some is not
Formulating wffs in syllogistic logic There are only eight (8) forms of wffs: all A is B All swans are white no A is B There are no white swans some A is B Some swans are white some A is not B Some swans are not white x is B This swan is white x is not B This swan is not white x is y This is the only white swan x is not y This is not the white swan Any sentence can be translated into a wff of one of these forms
Implications in syllogistic logicGeneral form of an implication: all A is BRead: For all objects in the domain, if an object is A then it is B Use capital letters to refer to domains of objects (all) Use small letters to refer to single objects (me, Popper)Why is “all A are B” an implication? (a1⊃b1)·(a2⊃b2)·(a3⊃b3)...(an⊃bn)
Rule 1: Hypothetical Syllogism Example: All sociologists (S) are Marxists (M) All Marxists (M) are against capitalism (C) All sociologists (S) are against capitalism (C) General form: all S is M all M is C C M S ---------- All S is C Venn diagram
Rule 2: Modus Ponens Example: All sociologists (S) are Marxists (M) Popper (p) is a sociologist (S) [p is S] Popper (p) is a Marxist (M) [p is M] General form: Popper all S is M M S p is S ★ ---------- p is M Venn diagram of an implication
Rule 3: Modus Tollens Example: All sociologists (S) are Marxists (M) Popper (p) is not a Marxist (M) [p is not M] Popper (p) is not a sociologist (S) [p is not S] General form: All S is M Popper M S p is not M ---------- ★ p is not S Venn diagram of an implication
Testing whether a syllogism is valid:The star testThe star test consist of three steps:Step 1: Find the “distributed letters”A letter is distributed if it occurs just after “all” or anywhere after “no”or “not” all A is B Underline the no A is B distributed letters x is A x is not y
Testing whether a syllogism is valid: The star testStep 2: Star premises letters which are distributed and conclusion letters whichare not distributed all A* is B some C is A ----------------- some C* is B*
Testing whether a syllogism is valid: The star testStep 3: Decide. A syllogism is valid if and only if every capital letter isstarred exactly once.&if there is exactly one star on the right hand side Each capital letter is starred exactly once all A* is B some C is A There is exactly one star at the right hand ----------------- side (see the B) some C* is B* Thus, this syllogism is valid.
Second example: no A* is B* no C* is A* ----------------- no C is B Is it a valid syllogism?
Second example: no A* is B* no C* is A* ----------------- no C is B A is starred twice. There are two stars on the right hand side (see A and B) Thus, there are two reasons why this syllogism is not valid.
Abstract &Generalize Relation between humans Social relations Friendships Friendships between students Friendships between ﬁrst-years Specify Classify
Specify: Include more characteristics in the deﬁnition of the concept Friendships Fewer objects fall under the between concept ﬁrst-years Friendships between students Generalize: Friendships Abstract more details More objects fall under the Social relations concept Relation betweendyads humans
Generalizing and specifying implications All sociologists (S) are good statisticians (G). (S⊃M) 2 S=df. Everybody with G=df. Everybody who at least a Doctor’s 1 can interpret a degree in Sociology regression S=df. Everybody with a G=df. Everybody who 4 university degree in Sociology can explain what a regression is 3 Generalize the implication: from 1 to 3, or from 2 to 4 Specify the implication: from 3 to 4, or from 1 to 2
The information content of an implication Scientists seek to formulate informative statements. Thus, they should inform us about many things and make precise predictions. Independent variable Dependent variable (then) (if) should be general should be very speciﬁc (true for many cases) (true for few cases) Societies (S) Differentiate (D) D=df. Increase in S=df. All human societies complexity S=df. Only modern ⊃ D=df. Increase in human societies stratiﬁcation
Anything can happen All human societies Social differentiation TraditionalModern human human societies ⊃ Social differentiationsocieties & conﬂicts Implications are more informative if: You use disjunctions in the if part (if A or B or C) You use conjunctions in the then part (then X and Y and Z)
AssignmentIn the reader you have the second chapter of this bookRead this chapter and do the exercises in the assignment guide in Nestor