Andreas Flache

Manu Muñoz-Herrera

Introduction to formal logicLecture Week 3 - Application of Theories

Block A 2012/2013

http://manumunozh.wix.com/apptheories

Summary Assignment 1

Assignment 1

You were meant to read Lave & March chapters 2 and 3, and Freakonomics chapter 4.

You were supposed to use the 4-steps in the Lave & March model and apply it to 3 explanations given by Levitt and Dubner for the surprising 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 handed in is that most of them DO NOT contain a discussion of HOW the explanations 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) and omitted (i) and (ii)

The second main limitation was literal transcription from the book to the assignments. Do not copy but use your own words.

Aims of the lecture

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 ‘‘syllogistic logic’’

How to generalize and specify concepts and statements

Part 1: How to formulate valid arguments/explanations.

What is logic?Philosophical discipline established by Aristotle

Aristotle384BC - 322BC

Arguments consist of premises and a conclusion

What is logic?Logic is the analysis and appraisal of arguments

An argument is valid means: if all premises are true it is possible that the conclusion is wrong

Premise. If you are reading this, you aren’t illiterate You are reading thisConclusion. You aren’t illiterate

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)

What is logic?

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 the conclusion is true. If the argument is valid and at least one premise is false, the

conclusion can be false You need empirical research (and further arguments) to test

whether all premises are true.

Note: Arguments are not true or false (statements are!)

Which argument is valid and which is sound?

If economic welfare increases, the rate of unemployment decreases

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 decreases

In the 1990s, in the US, rate of unemployment decreased

In the 1990s, in the US, the rate of violent crimes decreased

1

2

Basic Propositional Logic

The members of group x are integrated The citizens of Leipzig protest People who hold similar opinions tend to form friendships

Basic propositional logicPropositions are statements, for instance:

Propositions are statements which are either true or false (not valid or invalid)

Shut up! (commands) Why did nobody bring cookies? (question) This is a bad song (normative statements)

Hence, statements which are not true or false are not considered propositions. For instance,

e.g., “I am a sociologist”

Propositional language and truth tables

Propositions are translated into so called “wff’s” (pronounce as woof as in wood). Wff stands for ``well formed formula”

s

Propositions are analyzed using truth tables. Truth tables give a logical diagram for a given wff, listing all possible truth-value combinations.

S

1

0

Symbol of the proposition

Truth values: s can be true (1) or false (0)

Truth-functional operatorsPropositions can be combined, forming new propositions. This is done with so called operators

Operators define the truth-value of the combined proposition based on the 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

0 1

If s is true, then the negation is false

If s is false, then the negation is true

Operator 2: Disjunction Symbol: ⋁ (vee) or || or + Read: “or”

p q p⋁q1 1 11 0 10 1 10 0 0

The disjunction of p and q is false if both p and q are false

Operator 3: Conjunction Symbol: ⋅ (dot) or & or ⋀ Read: “and”

p q p ⋅ q1 1 11 0 00 1 00 0 0

The conjunction of p and q is true if both p and q are true

Operator 4: Implication Symbol: ⊃ (horseshoe) or → Read: “if p then q”

p q p⊃q1 1 11 0 00 1 10 0 1

The implication of p and q is false only if p is true and q is false

Example: 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 invalid

Popper is not a sociologist + Popper is a Marxist : wff is valid

Popper 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≡q1 1 11 0 00 1 00 0 1

The equality of p and q is true if either p and q are both true or both false

Example: 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 invalid

Popper is not a sociologist + Popper is a Marxist : wff is invalid

Popper 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 (↓,⊥)

Venn Diagrams

True and False

~p

p

Area inside the circle: possible states where p is true Area outside: possible states where p is false

Disjunction

White area: states where the disjunction of p and q is true Pink area: states where the negation of the disjunction of p and q is true

p q⋁

Conjunction

Pink area: p and not q Blue area: q and not p White area: q and p Purple area: not (q or p)

p qp⋅q

~(p⋁q)

Truth Tables

Working with truth tables

Example: Let us demonstrate for which combination of truth values of 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⊃qp⊃qp⊃q q⊃pq⊃pq⊃p (p⊃q)·(q⊃p)(p⊃q)·(q⊃p)(p⊃q)·(q⊃p)1 1 1 1 1 1 1 1 1 1 11 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

Definition of an equality

This proves that: (p!q)!((p⊃q)·(q⊃p))

Rules of Inference

Rules of inference

When we formulate an argument, we infer the conclusion from the premises.

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 is

correct), we need to demonstrate that the conjunction (·) of all premises implies (⊃) 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 hypothetical syllogism is a valid argument form

Thus, we want to demonstrate that the conjunction (·) of all premises implies (⊃) the conclusion.

Therefore, we need to demonstrate:

if the premises are true, then the conclusion is always true.

This is an implication

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 the truth-values of p, q, and r.

p q r p⊃q q⊃r (p⊃q)·(q⊃r)(p⊃q)·(q⊃r)(p⊃q)·(q⊃r) p⊃r ((p⊃q)·(q⊃r))⊃(p⊃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)

Is ((p⊃q)·(q⊃r))⊃(p⊃r) always valid?

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⊃qp ----------q

pq

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

pq

Venn diagram of an implication

Syllogistic Logic

Syllogistic Logic

Like propositional logic, it is a branch of logic. Propositional logic focuses on propositions which refer to single

objects (i.e., Popper In contrast, syllogistic logic is concerned with domains of objects

Propositional logic: It rains (r) Popper is cool (c)

Syllogistic logic: All swans are white (all S is W) Societies with high anomie suffer

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

Typical wffs from:

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 logic

General 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)

Rules of Inference

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 Mall M is C ----------All S is C

Venn diagram

SMC

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:all S is Mp is S ----------p is M

SM

Venn diagram of an implication

★

Popper

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 Mp is not M ----------p is not S

SM

Venn diagram of an implication

★

Popper

The Star Test

Testing whether a syllogism is valid: The star test

The 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 Bno A is Bx is Ax is not y

Underline the distributed letters

Testing whether a syllogism is valid: The star test

Step 2: Star premises letters which are distributed and conclusion letters which are not distributed

all A* is Bsome C is A-----------------some C* is B*

Testing whether a syllogism is valid: The star test

Step 3: Decide. A syllogism is valid if and only if every capital letter is starred exactly once.&if there is exactly one star on the right hand side

all A* is Bsome C is A-----------------some C* is B*

Each capital letter is starred exactly once

There is exactly one star at the right hand side (see the 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.

Generalizing and Specifying Concepts

Abstract &

Generalize

Specify Classify

Relation between humans

Social relations

Friendships

Friendships between students

Friendships between first-years

dyadsRelation between

humans

Social relations

Friendships

Friendships betweenstudents

Friendships between

first-years

Specify:

Generalize:

Include more characteristics in the definition of the concept

Fewer objects fall under the concept

Abstract more details

More objects fall under the concept

All sociologists (S) are good statisticians (G). (S⊃M)

S=df. Everybody with at least a Doctor’s degree in Sociology

S=df. Everybody with a university degree in Sociology

G=df. Everybody who can interpret a

regression

G=df. Everybody who can explain what a

regression is

1

2

4

3

Generalizing and specifying implications

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 implicationScientists seek to formulate informative statements. Thus, they should inform us about many things and make precise predictions.

Independent variable (if) should be general (true for many cases)

Dependent variable (then) should be very specific (true for few cases)

S=df. Only modern human societies

S=df. All human societiesD=df. Increase in

complexity

D=df. Increase in stratification

⊃

Societies (S) Differentiate (D)

⊃Modern human

societies

Traditional human

societies

All human societies

Anything can happen

Social differentiation

Social differentiation& 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)

Assignment

In the reader you have the second chapter of this book

Read this chapter and do the exercises in the assignment guide in Nestor