Logic, reasoning and fallacies - and some puzzling · A categorical syllogism is a deductive argument with two categorical propositions as premisses and a categorical conclusion

Logic, reasoning and fallaciesand some puzzling

Karst Koymans

Informatics InstituteUniversity of Amsterdam

(version 16.3, 2016/11/21 12:58:26)

Wednesday, November 23, 2016

Karst Koymans (UvA) Logic, reasoning and fallacies Wednesday, November 23, 2016 1 / 51

1 Before we start

2 Introductory Examples

3 Arguments and Reasoning

4 Paradoxes and Fallacies


Before we start

Outline

1 Before we start





Before we start

Decide how to make a random choice

Random choice of X, Y, Z, nX, Y, Z ∈ {A, E, I, O}, not necessarily all different

n ∈ {1, 2, 3, 4}


Introductory Examples

Outline

1 Before we start






Example 0: valid reasoning

Valid reasoningNiels is younger than Jaap. (premiss)

Jaap is younger than Karst. (premiss)

Hence Niels is younger than Karst. (conclusion)

The validity of this reasoning depends on Logic, but also

on the property of “is younger than” being transitive.



Example 1: invalid reasoning

Invalid reasoningA bad lecture is better than nothing. (premiss)

Nothing is better than a good lecture. (premiss)

Thus a bad lecture is better than a good lecture. (conclusion)

Although “is better than” is also supposed to be transitive,

this is an invalid argument. Why?

Language and Logic are more than just form (syntax).

Different contexts may give different meaning (semantics).Karst Koymans (UvA) Logic, reasoning and fallacies Wednesday, November 23, 2016 7 / 51


Example 2: iff

Meaning of iff“A iff B” abbreviates “A if and only if B”

“A if B” means “If B then A” (Notation: B → A)

“A only if B” means “If A then B” (Notation: A → B)

“A iff B” means “A if B and A only if B”

(Notation: B → A ∧ A → B ≡ B ↔ A)



Example 3: desda (Dutch iff)

Meaning of desda“A desda B” abbreviates “A dan en slechts dan als B”“A (dan) als B” means “Als B dan A” (Notation: B → A)(Note: It is “A dan als B” and not “A dan B”, which would mean“Als A dan B”)“A slechts (dan) als B” means “Als A dan B” (Notation: A → B)“A desda B” means “A dan als B en A slechts dan als B”(Notation: B → A ∧ A → B ≡ B ↔ A)

What about the meaning of “necessary and sufficient”

(in Dutch: “noodzakelijk en voldoende”)?


Example 4: Retrograde Analysis

http://www.mathpuzzle.com/retrograde.html

http://www.mathpuzzle.com/retrograde.html

Example 5: Dead Reckoning puzzle

http://anselan.com/tutorial.html

http://anselan.com/tutorial.html


Example 5: Dead Reckoning solution

Chess Law 5.2bThe game is drawn when a position has arisen in which neither player cancheckmate the opponent’s king with any series of legal moves. The gameis said to end in a ‘dead position’. This immediately ends the game,provided that the move producing the position was legal.

Karst Koymans (UvA) Logic, reasoning and fallaciesWednesday, November 23, 2016 12 /

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Example 6: Slitherlink puzzle

What are the rules of this game?

Example 6: Slitherlink solution

http://www.brainbashers.com/slitherlink.asp

http://www.brainbashers.com/slitherlink.asp

Example 7: Nurikabe puzzle


Example 7: Nurikabe solution

http://www.brainbashers.com/nurikabe.asp

http://www.brainbashers.com/nurikabe.asp

Example 8: Masyu puzzle


Example 8: Masyu solution

http://www.chiark.greenend.org.uk/~sgtatham/puzzles/js/pearl.html

http://www.chiark.greenend.org.uk/~sgtatham/puzzles/js/pearl.html

Example 9: Link-A-Pix tutorial


Example 9: Link-A-Pix example

http://www.conceptispuzzles.com/index.aspx?uri=puzzle/euid/

01000000d31337c45c1ce9e3d8dd757af93cc72b3182e00e89d2330ff4f2973b4381c154889f23e5263f107e430bc15d92f5ce4c/play

http://www.conceptispuzzles.com/index.aspx?uri=puzzle/euid/01000000d31337c45c1ce9e3d8dd757af93cc72b3182e00e89d2330ff4f2973b4381c154889f23e5263f107e430bc15d92f5ce4c/play

http://www.conceptispuzzles.com/index.aspx?uri=puzzle/euid/01000000d31337c45c1ce9e3d8dd757af93cc72b3182e00e89d2330ff4f2973b4381c154889f23e5263f107e430bc15d92f5ce4c/play

Arguments and Reasoning

Outline

1 Before we start





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Definition of Argument

Definition (Proposition)A proposition is a sentence with a truth value, being either true or false.

Definition (Simple Argument)A simple argument is a series of propositions.

One proposition (the conclusion) should be

evidenced by the other propositions (premisses).

Definition (Complex Argument)A complex argument is a series of arguments.

The conclusion of one argument may be used

as premiss in another argument.Karst Koymans (UvA) Logic, reasoning and fallacies

Wednesday, November 23, 2016 22 /51


Example of an argument

Women for OS3[We do not have many female students at OS3]1.

Hence [we should intensify our PR]2,

because [it is good for society that women graduate at our master]3.

1 is a premiss

2 is the conclusion

3 is a premiss


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1 is a premiss

2 is the conclusion

3 is a premiss


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1 is a premiss

2 is the conclusion

3 is a premiss


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1 is a premiss

2 is the conclusion

3 is a premiss


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Deductive and Inductive Arguments

Definition (Deductive argument)A deductive argument is an argument where

the conclusion follows 100% certain from the premisses.

Definition (Inductive argument)An inductive argument is an argument where the conclusion

follows with (preferably a high) probability from the premisses.

Can you give examples of both kind of arguments?


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Reasoning Example 0

Valid reasoning with true propositionsAll Dutch people are humans

All humans are mortal

∴ All Dutch people are mortal


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Reasoning Example 1

Valid reasoning with false propositionsAll Dutch people are smart

All smart people live forever

∴ All Dutch people live forever


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Reasoning Example 2

Invalid reasoning with true propositionsIf I am God, I will live forever

I am not God

∴ I will not live forever

This is an example of a fallacy: “Denying the Antecedent”


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Reasoning Example 3

Valid reasoning with false premisses and a true conclusionAll part-time OS3 students are smart

All smart people are male

∴ All part-time OS3 students are male


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Valid arguments

Theorem (Soundness)A valid argument with true premisses always leads to a true conclusion


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Categorical propositions

Definition (Aristoteles)Given a subject term S and a predicate P we have

the four classic categorical propositions:

(A) All S are P

(E) No S is P (≡ All S are not P)

(I) Some S is P

(O) Some S is not P

AffIrmo (universal/particular positive)

nEgO (universal/particular negative)Karst Koymans (UvA) Logic, reasoning and fallacies



Syllogisms

Definition (Categorical Syllogism)A categorical syllogism is a deductive argument with two categorical

propositions as premisses and a categorical conclusion.

Example of form AAAAll M are P

All S are M

∴ All S are P

M is called the “middle term”Karst Koymans (UvA) Logic, reasoning and fallacies



Position and role of the middle term

First roleM—P

S—M

S—P

Third roleM—P

M—S

S—P

Second roleP—M

S—M

S—P

Fourth roleP—M

M—S

S—P


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Validity of Categorical Syllogisms

256 Possible schemesThe possible schemes are XYZ-n where

X, Y, Z ∈ {A, E, I, O}

n ∈ {1, 2, 3, 4}

Of these schemes only 15 are valid.


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Venn diagrams

Paradoxes and Fallacies

Outline

1 Before we start





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Famous paradoxes

Liar paradoxThis sentence is false

Drinker paradoxThere is a person such thatif that person drinks then everybody drinks

Grelling-Nelson paradoxAutological ≡ describes itself correctlyHeterological ≡ does not describe itself correctlyIs heterological heterological?

Berry paradoxThe smallest positive integer not definable in under eleven wordsDoes such a number exist or not?


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Mathematical paradox

Russell paradoxV = {x | x /∈ x}

Does V ∈ V hold?

Zwicker’s HypergameA finite game is a game that always endsafter a finite number of alternating moves from both playersHypergame is played by first choosing a finite gameby the first player (A) and than playing that gamestarting with the second player (B)Is Hypergame a finite game or not?


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Fallacies

Fallacies of Relevance

Fallacies of Presumption

Logical/Formal Fallacies


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Fallacies of Relevance

Argument from Ignorance

Argument Ad Hominem

Appeal to Inappropriate Authority

Appeal to Authority or Force

Appeal to Emotion

Drawing an Irrelevant Conclusion


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Argumentum ad IgnorantiamThe fallacy that a proposition is true simply on the basis that

it has not been proved false or that it is false simply because

it has not been proved true.


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Argumentum ad IgnorantiamThe fallacy that a proposition is true simply on the basis that

it has not been proved false or that it is false simply because

it has not been proved true.

Example Argument from IgnoranceScientists cannot prove that a meteor will cause a catastrophe.

Therefore it won’t occur.


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Argument Ad Hominem

Argument Ad HominemThe fallacy of attacking the character or circumstances of an individual

who is advancing a statement or an argument instead of trying to

disprove the truth of the statement or the soundness of the argument.


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Argument Ad Hominem

Argument Ad HominemThe fallacy of attacking the character or circumstances of an individual

who is advancing a statement or an argument instead of trying to

disprove the truth of the statement or the soundness of the argument.

Example Argument Ad HominemThis cannot be true, since you belong to that radical party.


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Argumentum ad VerecundiamThe fallacy of appealing to the testimony

of an authority outside his special field.


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Argumentum ad VerecundiamThe fallacy of appealing to the testimony

of an authority outside his special field.

Example Appeal to Inappropriate AuthorityWe from ”WC Eend” advise ”WC Eend”.


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Argumentum ad BaculumThe fallacy committed when one appeals to (the threat of) force

to bring about the acceptance of a conclusion.


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Argumentum ad BaculumThe fallacy committed when one appeals to (the threat of) force

to bring about the acceptance of a conclusion.

Example Appeal to Authority or ForceYou will lower your price, because otherwise

I might reconsider your bonus.


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Appeal to Emotion

Argumentum ad MisericordiamThe fallacy committed when pity or a related emotion

such as sympathy or compassion is appealed to

for the sake of getting a conclusion accepted.


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Appeal to Emotion

Argumentum ad MisericordiamThe fallacy committed when pity or a related emotion

such as sympathy or compassion is appealed to

for the sake of getting a conclusion accepted.

Example Appeal to EmotionEverybody else thinks cloud computing is a good idea.

You should too.


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Ignoratio ElenchiThe fallacy of proving a conclusion not pertinent

and quite different from that which was intended or required.


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Ignoratio ElenchiThe fallacy of proving a conclusion not pertinent

and quite different from that which was intended or required.

Example Drawing an Irrelevant ConclusionA prisoner pleaded guilty as a thief.

The judge allowed him to change this plea to not guilty.

The jury on the trial accused the prisoner of lying.

Judge reasons: “You said you were a thief;

the jury calls you a liar; hence you are discharged.”


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Fallacies of Presumption

Fallacy of False Cause

Begging the Question


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Fallacy of False CauseThe fallacy committed when an argument mistakenly attempts

to establish a causal connection.


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Fallacy of False CauseThe fallacy committed when an argument mistakenly attempts

to establish a causal connection.

Example Fallacy of False CauseThe introduction of the death penalty has caused the highest

criminal rate and number of prisoners of the civilized world.


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Petitio PrincipiiThe fallacy of assuming as a premiss a statement

which has the same meaning as the conclusion.


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Petitio PrincipiiThe fallacy of assuming as a premiss a statement

which has the same meaning as the conclusion.

Example Begging the QuestionThe Bible says God exists.

The Bible is God’s word, and God never lies.

So what the Bible says is true.

Therefore God exists.


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Logical Fallacies

Affirming the Consequent

Denying the Antecedent


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Affirming the ConsequentThe invalid reasoning: P → Q and Q, therefore P.


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Affirming the ConsequentThe invalid reasoning: P → Q and Q, therefore P.

Example Affirming the ConsequentIf it’s raining then the streets are wet.

The streets are wet.

Therefore, it’s raining.


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Denying the AntecedentThe invalid reasoning: P → Q and ¬P, therefore ¬Q.


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Denying the AntecedentThe invalid reasoning: P → Q and ¬P, therefore ¬Q.

Example Denying the AntecedentIf it’s raining, then the streets are wet.

It isn’t raining.

Therefore, the streets aren’t wet.


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Logic, reasoning and fallacies - and some puzzling · A categorical syllogism is a deductive argument with two categorical propositions as premisses and a categorical conclusion