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Logic

Lesson 1: Basic Concepts

Flavio Zelazek

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October 8, 2014

Flavio Zelazek ([email protected]) Logic (Lesson 1: Basic Concepts) October 8, 2014 1 / 37

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Introduction

Lesson 1

1 Introduction

Course overview

Uses and definitions of Logic

2 Basic concepts

Arguments

ValidityNormal form

Enthymemes

Soriteses

Fallacies

What’s next



Introduction Course overview

Course overview

Lesson 1: Introduction to Logic – Arguments – Validity – Normal form

Lesson 2: Introduction to artificial languages – Translation into

propositional and predicative languages – Logical connectives and

truth tables

Lesson 3: More on truth tables: tautology and validity valuation –

Introduction to derivations

Lesson 4: Fitch-style derivations for propositional logic

Lesson 5: Fitch-style derivations for predicative logic




The textbook

P. Teller. A Modern Formal Logic Primer. 2014 PDF edition.

Downloadable from: http://tiny.cc/logicprimer2014.


http://tiny.cc/logicprimer2014



Exercise 0 (1/5)

Summers L. (2014, October 6). Why public investment really is a free

lunch. Financial Times. Retrieved from http://tiny.cc/ft-freelunch.

It has been joked that the letters IMF stand for “it’s mostly fiscal”. The

International Monetary Fund has long been a stalwart advocate of

austerity as the route out of financial crisis, and every year it chastises

dozens of countries for their fiscal indiscipline. Fiscal consolidation – a

euphemism for cuts to government spending – is a staple of the fun’srescue programmes. A year ago the IMF was suggesting that the US

had a fiscal gap of as much as 10 per cent of gross domestic product.


http://tiny.cc/ft-freelunch



Exercise 0 (2/5)

All of this makes the IMF’s recently published World Economic Outlooka remarkable and important document. In its flagship publication, the

IMF advocates substantially increased public infrastructure investment,

and not just in the US but much of the world. It asserts that when

unemployment is high, as it is in much of the industrialised world, the

stimulative impact will be greater if investment is paid for by borrowing,

rather than cutting other spending or raising taxes. Most notably, the

IMF asserts that properly designed infrastructure investment will

reduce rather than increase government debt burdens. Public

infrastructure investments can pay for themselves.




Exercise 0 (3/5)

Why does the IMF reach these conclusions? Consider a hypothetical

investment in a new highway financed entirely with debt. Assume –

counterfactually and conservatively – that the process of building thehighway provides no stimulative benefit. Further assume that the

investment earns only a 6 per cent real return, also a very conservative

assumption given widely accepted estimates of the benefits of public

investment. Then, annual tax collections adjusted for inflation would

increase by 1.5 per cent of the amount invested, since the government

claims about 25 cents out of every additional dollar of income. Real

interest costs, that is interest costs less inflation, are below 1 per cent

in the US and much of the industrialised world over horizons of up to

30 years. So infrastructure investment actually makes it possible to

reduce burdens on future generations.




Exercise 0 (4/5)

In fact, this calculation understates the positive budgetary impact of

well-designed infrastructure investment, as the IMF recognised. It

neglects the tax revenue that comes from the stimulative benefit of

putting people to work constructing infrastructure, as well as the

possible long-run benefits that come from combating recession. It

neglects the reality that deferring infrastructure renewal places aburden on future generations just as surely as does government

borrowing.

It ignores the fact that by increasing the economy’s capacity,

infrastructure investment increases the ability to handle any given level

of debt. Critically, it takes no account of the fact that in many casesgovernment can catalyse a dollar of infrastructure investment at a cost

of much less than a dollar by providing a tranche of equity financing, a

tax subsidy or a loan guarantee.




Exercise 0 (5/5)

When it takes these factors into account, the IMF finds that a dollar of

investment increases output by nearly $3. The budgetary arithmetic

associated with infrastructure investment is especially attractive at a

time when there are enough unused resources that greater

infrastructure investment need not come at the expense of other

spending. If we are entering a period of secular stagnation,

unemployed resources could be available in much of the industrial

world for quite some time.

While the case for investment applies almost everywhere – possibly

excepting China, where infrastructure investment has been used a

stimulus tool for some time – the appropriate strategy for doing more

differs around the world.

[. . . ]



Introduction Uses and definitions of Logic

Logic

“Dialectica est ars artium, ad omnium methodorum principia viam

habens.” (Petrus Hispanus, 1250 c.)

“Logic [is] the analysis of methods of reasoning. In studying thesemethods, logic is interested in the form rather than the content of

the argument.” (Mendelson, 1964)

“Ordinarily, the premises of an argument are supposed to support,

or give us reasons, for believing the conclusion. [. . . ] Logic is the

study of this reason-giving connection. Logic sets out the

important properties of arguments, especially the ways in which

arguments can be good or bad.” (Teller, 1989)

“Logic deals with what is common to every branch of our cognitive

activity, and thus to every scientific discipline.”

(Abrusci, 2009, transl.)




Logic and clarity: Example 1.1 (1/2)

Apollo 13 Technical Air-to-Ground Voice Transcription. Manned

Spacecraft Center. Houston, TX 1970.

[. . . ]

CMP (Command module pilot): Houston, Apollo 13.

CC (Capsule communicator): Apollo 13, Houston. Go ahead.

[. . . ]

CC: Okay. On checklist G/8-2 under step 5, there’s a statement,

“disable all Jets on two adjacent quads,” and that’s what we were

referring to when we called up saying “disable quads A and B.” So

that’s all fine; however, going down now to step 7 where it says “enable

all Jets,” we hoped that you didn’t think we meant leave A and B

disabled there. In that case, it’s as written. All jets should be enabled.Over.




Logic and clarity: Example 1.1 (2/2)

CMP: Okay. Our checklist has “all jets” scratched out, and it says

“enable couples on all axes.” So what we’ll do is – we’ll reestablish this

thing and come down to that step 7, and we’ll enable all jets. That

means enable quads A and B. Is that right?

CC: That’s affirm. You’ll have A, B, C, and D enabled for step 7.

CMP: Okay, real fine. I’ll start on it now.

CC: Okay. Understand you’re going to reestablish it.

[. . . ]




Understanding humour: Exercise 1

Figure: xkcd n. 552



Basic concepts

Lesson 1

1 Introduction

Course overview

Uses and definitions of Logic

2 Basic concepts

Arguments

ValidityNormal form

Enthymemes

Soriteses

Fallacies

What’s next



Basic concepts Arguments

Argument

An Argument is a collection of declarative sentences one of which iscalled the conclusion and the rest of which are called the premises.

[Teller 2014, p. 1 (PDF p. 7)]

More precisely, the sentence that an argument attempts to prove is the

Conclusion of the argument, while the sentences which are used in the

attempt to prove the conclusion, or to support it, are called Premises of

the argument.




Example 1.2

(1) Mary will not pass the exam.




Example 1.2


(2) In order to pass the exam one has to study a lot.




Example 1.2



(3) Mary hasn’t been studying at all.




Deduction vs. Induction

Valid Deductive Argument: An argument in which, without fail, if the

premises are true, the conclusion will also be true.

Good Inductive Argument: An argument in which the premises provide

good reasons for believing the conclusion. In an inductive argument,

the premises make the conclusion likely [or plausible], but the

conclusion might be false even if the premises are true.

We tend to call an argument “Deductive” when we claim, or suggest, or

just hope that it is deductively valid. And we tend to call an argument

“Inductive” when we want to acknowledge that it is not deductively valid

but want its premises to aspire to making the conclusion likely.

[Teller 2014, p. 3 (PDF p. 8)]




Exercise 2

Identify the implicit conclusion:

No logic textbook is boring.Some logic textbook is difficult.




Exercise 2

Identify the implicit conclusion:

No logic textbook is boring.Some logic textbook is difficult.

Therefore, some difficult textbook is not boring.



Basic concepts Validity

Possible worlds

The World in which we live contains, or is constituted by, Facts.

Each of these facts may be described by one or more Sentences, or

Propositions, and a proposition that describes a fact is True.

On the other hand, a proposition which doesn”t describe any fact is

False.

The world went on in a certain way, but it might have gone on in many

other ways. Let’s call each of those ways a Possible World; while the

way in which actually things went on is the Real World, or the Actual

World.



Basic concepts Validity

Validity

A (deductive) argument is Valid when it is impossible that its premises

are true and its conclusion false, i.e. when, if its premises are true, the

conclusion must be true.

This means that no possible world exists in which the premises are

true and the conclusion false.

A proposition is Logically True if it is true in every possible world.

A proposition is Logically False if it is true in no possible world.

A proposition is Logically Undetermined, or Contingent, if it is not

logically true logically nor logically false.

A set S of propositions Entails a proposition A if A is true in every

possible world in which all the propositions of S are true.



Basic concepts Normal form

The normal form of arguments

An argument is in Normal Form when the following conditions are

satisfied:

a) Every premise and the conclusion of the argument are explicitly

stated and associated to a progressive number or letter.

b) The premises precede the conclusion.

c) The conclusion is preceded by a conclusion marker (“therefore”,

“so”, etc.).



Basic concepts Normal form

Example 1.2 put in normal form


(3) Mary hasn’t been studying at all.Therefore, (1) Mary will not pass the exam.



Basic concepts Enthymemes

Enthymemes

In ordinary language, some of the premises of an argument need not

be mentioned: such an incomplete argument is called an Enthymeme.

Moreover, an argument may also be incomplete inasmuch as it lacks

an explicit conclusion.




Principle of charity

Principle of Charity (or of Interpretational Benevolence): if an

argument, in the way it has been expressed by a speaker X , is not

valid (or not well-supported, or at any rate incomplete), and if theaddition of further premises, of whose truth (or likeliness, or

plausibility) we can presume that X is convinced, would make it valid

(or well-supported), then, most of the times, it is legitimate to add

these premises in the construction of the normal form of the argument.




Exercise 3

Find the hidden assumption:

Some journalists are homeopaths.

Therefore, some superficial people are homeopaths.




Exercise 3

Find the hidden assumption:

Every journalist is superficial.

Some journalists are homeopaths.

Therefore, some superficial people are homeopaths.




Exercise 4

In attempting to reduce the high injury toll from road offences such as drinkdriving,successive governments have tended to rely on the deterrent effect of increasinglysevere penalties. However, severe penalties can act as a deterrent only if potentialoffenders believe that the chances of being detected are moderate or high. Recentsurveys have shown that people perceive the chance of detection for the offence ofdrink-driving as being very small.

• Which one of the following can be concluded from the passage above?

A Severe penalties for the offence of drink-driving do not constitute an effectivedeterrent.

B Successive governments have not tackled the problem of drink-driving.

C Drink-driving offenders have a misguided perception of their chances ofdetection.

D Penalties for road offences such as drink-driving should be even more severe.

E There must be better ways to tackle drink-driving, for example education.




Exercise 4

(1) In attempting to reduce the high injury toll from road of-

fences such as drinkdriving, successive governments have

tended to rely on the deterrent effect of increasingly severe

penalties. (2) However, severe penalties can act as a de-

terrent only if potential offenders believe that the chances

of being detected are moderate or high. (3) Recent sur-

veys have shown that people perceive the chance of de-

tection for the offence of drink-driving as being very small.


A Severe penalties for the offence of drink-driving do not constitute an effective deterrent.


C Drink-driving offenders have a misguided perception of their chances of detection.






Exercise 4

(1) In attempting to reduce the high injury toll from road of-

fences such as drinkdriving, successive governments have

tended to rely on the deterrent effect of increasingly severe

penalties. (2) However, severe penalties can act as a de-

terrent only if potential offenders believe that the chances

of being detected are moderate or high. (3) Recent sur-

veys have shown that people perceive the chance of de-

tection for the offence of drink-driving as being very small.

1 2 3

A


A Severe penalties for the offence of drink-driving do not constitute an effective deterrent.


C Drink-driving offenders have a misguided perception of their chances of detection.





Basic concepts Soriteses

Soriteses

An argument may be composed of smaller arguments, such that their

conclusions are used as premises in the main argument: this kind of

argument can be called a Sorites (from the ancient Greek word for“heap”).




Exercise 5

In Britain, wetland areas have traditionally been used for cattle farming,

which requires drainage of the land to allow cattle to graze in summer.

However, maintaining the marshy conditions in wetlands is vital to the

survival of wildlife and plants which are exclusive to such areas. Be-

cause the interests of agriculture and of conservation are thus in conflict

in wetland areas, water levels should be regulated by an independent

body to enable a balance to be achieved.




Exercise 5

(1) In Britain, wetland areas have traditionally been used for cattle farm-

ing, which requires drainage of the land to allow cattle to graze in sum-

mer. (2) However, maintaining the marshy conditions in wetlands is

vital to the survival of wildlife and plants which are exclusive to such

areas. (3) Because the interests of agriculture and of conservation are

thus in conflict in wetland areas, (4) water levels should be regulated

by an independent body to enable a balance to be achieved.

1 2

3

4




Exercise 6

The so-called “baby-boomer” generation (people born in Britain in the late 1940’s/early1950’s) will enjoy a good chance of reaching a ripe old-age. They benefited from a dietthat was austere but healthy when they were children and this has influenced their adulttastes. In addition, people in this age group have given up unhealthy activities such assmoking cigarettes. In contrast, the generation represented by those currently in their20’s and early 30’s are in danger of experiencing serious health problems which woulddecrease their life expectancy. This is the “Big Mac” generation whose childhood dietwas high in fat and sugar, which they continue to crave. They also show worrying ten-dencies to indulge in “binge drinking” and other health-threatening activities. Therefore,the ‘Big Mac’ generation would be wise to change their lifestyle.




Exercise 6(1) The so-called “baby-boomer” generation (people born in Britain in the late 1940’s/early 1950’s)

will enjoy a good chance of reaching a ripe old-age. (2) They benefited from a diet that was

austere but healthy when they were children and this has influenced their adult tastes. (3) In

addition, people in this age group have given up unhealthy activities such as smoking cigarettes.

(4) In contrast, the generation represented by those currently in their 20’s and early 30’s are

in danger of experiencing serious health problems which would decrease their life expectancy.

(5) This is the “Big Mac” generation whose childhood diet was high in fat and sugar, which they

continue to crave. (6) They also show worrying tendencies to indulge in “binge drinking” and other

health-threatening activities. (7) Therefore, the “Big Mac” generation would be wise to change

their lifestyle.

2 3 5 6

1 4

7



Basic concepts Fallacies

Fallacies

A Formal Fallacy is an argument that is not valid (and not even

well-supported).

An Informal Fallacy is an argument that, while not being a formal

fallacy, is however flawed, faulty, or anyhow unreliable (regardless of its

logical structure, which may even make the argument formally valid).




“Ignoratio elenchi”

Figure: xkcd n. 1081




Exercise 7

Spot the (formal) fallacy:

Some politicians are corrupt.

Some stakeholders are politicians.

Therefore, some stakeholders are corrupt.




Exercise 7

Spot the (formal) fallacy:

Every politician is corrupt.

Some politicians are corrupt.

Some stakeholders are politicians.

Therefore, some stakeholders are corrupt.



Basic concepts What’s next

Exercise 8: a taste of things to come. . .

Kinsley, M. (2007, July 5) The Lying Game. The New York Times.

Retrieved from http://tiny.cc/nyt-perjurytrap.

[. . . ]

So when Mr. Libby was questioned by federal investigators pursuing

the leaks, he too was caught in a perjury trap. He could either tell the

truth, thereby implicating colleagues and very possibly himself, inleaking classified security information (the identity of Mr. Wilson’s

wife), or he could lie. In either case he would be breaking the law or

admitting to having done so, and in either case he could have gone to

prison. Mr. Libby, like Mr. Clinton, made the wrong choice.

[. . . ]


http://tiny.cc/nyt-perjurytrap


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