logic, reasoning and fallacies - and some puzzling · a categorical syllogism is a deductive...
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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
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Before we start
Outline
1 Before we start
2 Introductory Examples
3 Arguments and Reasoning
4 Paradoxes and Fallacies
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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}
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Introductory Examples
Outline
1 Before we start
2 Introductory Examples
3 Arguments and Reasoning
4 Paradoxes and Fallacies
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Introductory Examples
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.
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Introductory Examples
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
Introductory Examples
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)
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Introductory Examples
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”)?
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Example 4: Retrograde Analysis
http://www.mathpuzzle.com/retrograde.html
Introductory Examples
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.
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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
Example 7: Nurikabe puzzle
What are the rules of this game?
Example 7: Nurikabe solution
http://www.brainbashers.com/nurikabe.asp
Example 8: Masyu puzzle
What are the rules of this game?
Example 8: Masyu solution
http://www.chiark.greenend.org.uk/~sgtatham/puzzles/js/pearl.html
Example 9: Link-A-Pix tutorial
What are the rules of this game?
Example 9: Link-A-Pix example
http://www.conceptispuzzles.com/index.aspx?uri=puzzle/euid/
01000000d31337c45c1ce9e3d8dd757af93cc72b3182e00e89d2330ff4f2973b4381c154889f23e5263f107e430bc15d92f5ce4c/play
Arguments and Reasoning
Outline
1 Before we start
2 Introductory Examples
3 Arguments and Reasoning
4 Paradoxes and Fallacies
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Arguments and Reasoning
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
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Arguments and Reasoning
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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Arguments and Reasoning
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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Arguments and Reasoning
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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Arguments and Reasoning
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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Arguments and Reasoning
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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Arguments and Reasoning
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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Arguments and Reasoning
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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Arguments and Reasoning
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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Arguments and Reasoning
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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Arguments and Reasoning
Valid arguments
Theorem (Soundness)A valid argument with true premisses always leads to a true conclusion
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Arguments and Reasoning
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
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Arguments and Reasoning
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
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Arguments and Reasoning
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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Arguments and Reasoning
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
2 Introductory Examples
3 Arguments and Reasoning
4 Paradoxes and Fallacies
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Paradoxes and Fallacies
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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Paradoxes and Fallacies
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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Paradoxes and Fallacies
Fallacies
Fallacies of Relevance
Fallacies of Presumption
Logical/Formal Fallacies
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Paradoxes and Fallacies
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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Paradoxes and Fallacies
Argument from Ignorance
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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Paradoxes and Fallacies
Argument from Ignorance
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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Paradoxes and Fallacies
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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Paradoxes and Fallacies
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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Paradoxes and Fallacies
Appeal to Inappropriate Authority
Argumentum ad VerecundiamThe fallacy of appealing to the testimony
of an authority outside his special field.
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Paradoxes and Fallacies
Appeal to Inappropriate Authority
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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Paradoxes and Fallacies
Appeal to Authority or Force
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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Paradoxes and Fallacies
Appeal to Authority or Force
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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Paradoxes and Fallacies
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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Paradoxes and Fallacies
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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Paradoxes and Fallacies
Drawing an Irrelevant Conclusion
Ignoratio ElenchiThe fallacy of proving a conclusion not pertinent
and quite different from that which was intended or required.
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Paradoxes and Fallacies
Drawing an Irrelevant Conclusion
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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Paradoxes and Fallacies
Fallacies of Presumption
Fallacy of False Cause
Begging the Question
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Paradoxes and Fallacies
Fallacy of False Cause
Fallacy of False CauseThe fallacy committed when an argument mistakenly attempts
to establish a causal connection.
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Paradoxes and Fallacies
Fallacy of False Cause
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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Paradoxes and Fallacies
Begging the Question
Petitio PrincipiiThe fallacy of assuming as a premiss a statement
which has the same meaning as the conclusion.
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Paradoxes and Fallacies
Begging the Question
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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Paradoxes and Fallacies
Logical Fallacies
Affirming the Consequent
Denying the Antecedent
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Paradoxes and Fallacies
Affirming the Consequent
Affirming the ConsequentThe invalid reasoning: P → Q and Q, therefore P.
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Paradoxes and Fallacies
Affirming the Consequent
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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Paradoxes and Fallacies
Denying the Antecedent
Denying the AntecedentThe invalid reasoning: P → Q and ¬P, therefore ¬Q.
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Paradoxes and Fallacies
Denying the Antecedent
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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