Flashcards in Exam 3 (kms 💀) Deck (33)
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1
Affirming the consequent
Whenever an argument is in this form:
If s, then t.
t.
Therefore, s
Example: If Bill Gates owns Fort Knox, then he is rich.
Bill Gates is rich.
Therefore, he owns Fort Knox
2
Denying the consequent
Whenever an argument is in this form:
If s, then t.
Not s.
Therefore, not t
Example: If Bill Gates is Chinese, then he is a human being.
Bill Gates is not Chinese.
Therefore, he is not a human being
3
The undisturbed middle
When someone assumes that two things related to a third thing are related to each other, as in:
All cats are mammals.
All dogs are mammals.
Therefore, all cats are dogs.
If Bill wins the lottery, then he’ll be happy.
If Bill buys a new car, then he’ll be happy.
Therefore, if Bill wins the lottery, then he’ll buy a new car.
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Claims are claims that cannot have the same truth value
Contradictory
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Claims are claims that cannot both be true but can both be false
Contrary
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Occurs in this argument because the word ‘bank’ is ambiguous and used in two different senses:
All banks are alongside rivers, and the place where I keep my money is a bank.Therefore the place where I keep my money is alongside a river.
Equivocation
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Occurs when the structure of a sentence makes the sentence ambiguous:
If you want to take the motor out of the car, I’ll sell it to you cheap
*The pronoun ‘it’ may refer to the car or to the motor. It isn’t clear which. It would be a fallacy to conclude one way or the other, without more information.
Amphiboly
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A fallacy that happens when a speaker or writer assumes that what is true of a group of things taken individually must also be true of those same things taken collectively; or assumes that what is true of the parts must be true of the whole.
Example: “This building is made from rectangular bricks; therefore, it must be rectangular.”
Composition
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A fallacy that happens when a speaker or writer assumes that what is true of a group of things taken collectively must also be true of its individual members; or assumes that what is true of the whole must be true of its parts.
Example: “This building is circular; therefore, it must be made from circular bricks.”
Division
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Bill’s chances of becoming a professional football player are about 1 in 1,000, and Hal’s chances of becoming a professional hockey player are about 1 in 5,000. So the chance of both of them becoming professionals in their respective sports is 1 in 6,000.
* NOPE. The two events, Hal becoming a hockey player and Bill becoming a football player, are independent.
Miscalculating probabilities
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Cannot affect the outcome of another independent event.
To calculate the probability that independent events both occur, we multiply their individual probabilities
The probability of both Hal and Bill becoming pro is 1/1000 times 1/5000 which is 1/5,000,000.
Independent events
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When we don’t realize that independent events really are independent, that past performance of an independent event will not influence a subsequent performance of that kind of event, Then we are at risk
Remember, independent events do not affect one another’s outcome.
Example: No matter how many times a fair coin is flipped, no matter how many times ‘Tails’ has been the outcome of those flips, the probability that the next flip will show ‘Heads’ is still exactly ½. And, for that matter, there is the same probability that it will come up ‘Tails’.
Gambler's Fallacy
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The prior probability of something is its true or actual proportion.
The prior probability of a fair coin coming up ‘Heads’ when it is flipped is one in two, or ½.
The prior probability of an unfair coin coming up ‘Heads’ when it is flipped is a proportion different than ½
This fallacy occurs when failing to take into consideration the likelihood of an event all other things being equal; that is, its likelihood apart from any outside influences.
Example: “Bill is the best football player in our high school, and Hal is the best hockey player in our high school. So it appears that Bill’s chances of becoming a professional football player and Hal’s chances of becoming a professional hockey player are equally good.”
Overlooking prior probabilities
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False alarms
Occurs when probabilities are miscalculated in the following way: When you derive the proportion of Xs (carrot eaters) that are Ys (flunked the midterm) from the proportion of Ys (flunked the midterm) that are Xs (carrot eaters) – while failing to take into consideration the proportion of non-Ys (passed the midterm) that are also Xs (carrot eaters)
Example: 60% of the people who flunked the midterm ate carrots prior to the test. Therefore, 60% of the carrot eaters flunked the midterm; so, avoid carrots before taking the test.
You must also take into account the proportion of carrot-eaters who did not flunk the midterm. Eating carrots is a “false alarm.”
Overlooking false positives
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A term in a language which takes one or more sentences and joins them together to form a new sentence
Examples:(i) ‘but’ is a sentential connective. Given sentences: John is tall and: Fred is short we can join them using ‘but’ to get:John is tall but Fred is short.
(ii) ‘not ’is a sentential connective. Given a sentence: John is tall we can join it to the connective ‘not’ to get: John is not tall
NOTE: In English the joining of sentences with connectives is not always a simple matter of sticking sentences and connectives end-to-end (concatenating sentences and connectives).
Arity: Sentential connectives can be categorized by their arity. The arity of a connective is the number of sentences it joins to form a new sentence. A unary connective connects with a single sentence to make a new sentence. A binary connective joins with two sentences to make a new sentence.
‘not’ is a unary connective.
‘but’ is a binary connective.
Sentential connective
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Negation: unary, 'not'
Conjunction: binary, 'and'
Disjunction: binary, 'or'
Conditional: binary, 'if ... then'
Biconditional: binary, 'if and only if'
Connectives of sentential logic
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The negation symbol joins with a single sentence to form a new sentence called the negated sentence. The negated sentence has the opposite truth value of the original sentence.
Semantic rule: A negated sentence ‘S1’is true if S1 is false; false if S1 is true
Negation
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The conjunction symbol ‘&’joins together two sentences called conjuncts to form a new sentence called a conjunction. The conjunction is true just in case both conjuncts are true.
Semantic rule: A conjunction ‘S1 & S2’is true if S1 and S2 are both true; false if one or both of S1 and S2 are false.
Conjunction
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The disjunction symbol joins together two sentences called disjuncts to form a new sentence called a disjunction. The disjunction is true just in case at least one disjunct is true.
Semantic rule: A disjunction ‘S1 S2’is true if either S1 or S2 (or both) are true; false if both are false.
Disjunction
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The conditional symbol joins together two sentences. The first sentence is called the antecedent; the second sentence is called the consequent. The new sentence is called a conditional. The conditional is false only when the antecedent is true and the consequent is false. The conditional is true if its antecendent is false or if its consequent is true.
Semantic rule:A conditional ‘S1 S2’is true if S1 is false or S2 is true; falsei f S1 is true and S2 is false.
Conditional
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The biconditional symbol joins together two sentences. The first sentence is called the left-hand side; the second sentence is called the right-hand side. The new sentence is called a biconditional. The biconditional is true just in case the two component sentences have the same truth value.
Semantic rule: A biconditional ‘S1 S2’is true if S1 and S2 are both true or if S1 and S2 are both false; false if S1 is true and S2 is false or if S1 is false and S2 is true.
Biconditional
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To define the syntax of LSL, we need to specify what the basic terms of the language are (the lexicon) and how those terms may grammatically be combined (the formation rules)
Syntax
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1) If sis a sentence letter, then sis a sentence of LSL.
(2) If sis a sentence of LSL, then sis a sentence of LSL.
(3) If sand tare sentences of LSL, then (s& t) is a sentence of LSL.
(4) If sand tare sentences of LSL, then (st) isa sentence of LSL.
(5) If sand tare sentences of LSL, then (s→t) is a sentence of LSL.
(6) If sand tare sentences of LSL, then (st) is a sentence of LSL.
(7) Nothing else is a sentence of LSL.
Formation rules
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Examples of sentence formations
(P Q)
((P & Q) (Q →(P R)))
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An interpretation of LSL is an assignment of truth values to each sentence letter of LSL. There are exactly two truth values, true (T) and false (⊥); each sentence letter is assigned one and only one truth value.
An interpretation can thus be thought of as a list of sentence letters and corresponding truth values: A:T B:T C:⊥D:T E:⊥F:⊥G:T etc. (The list will be an infinite one.)
Interpretation
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(1) A sentence letter sis true relative to an interpretation I iff I(s) = T.
(2) A sentence of the form sis true relative to an interpretation I iff sis not true (or sis false) relative to I.
(3) A sentence of the form (s& t) is true relative to an interpretation I iff sis true relative to I and tis true relative to I.
(4) A sentence of the form (st) is true relative to an interpretation I iff either sis true relative to I or tis true relative to I.
(5) A sentence of the form (s→t) is true relative to an interpretation I iff either sis false relative to I or tis true relative to I.
(6) A sentence of the form (st) is true relative to an interpretation I iff either sand tare both true relative to I or sand tare both false relative to I.
Truth Relative to an Interpretation
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Examples of interpretation
Let I be an interpretation such that I(P)= T, I(Q) =⊥, and I(R) = ⊥.
(i) What is the truth value of (P & Q) relative to I? •
By rule (1), P is true relative to I iff I(P) = T. So,P is true relative to I. •
By rule (1), Q is true relative to I iff I(Q) = T. So,Q is false relative to I. •
By rule (2), Q is true relative to I iff Q is false relative to I. So Q is true relative to I. •
By rule (3), (P & Q) is true relative to I iff P is true relative to I and Q is true relative to I. So (P & Q) is true relative to I. •
By rule (2), (P & Q) is true relative to I iff (P & Q) is false relative to I. So (P & Q) is false relative to I. Conclusion: I((P & Q)) = ⊥
(ii) What is the truth value of (P (Q R)) relative to I? •
By rule (1), P is true relative to I iff I(P) = T. So,P is true relative to I. •
By rule (1), Q is true relative to I iff I(Q) = T. So,Q is false relative to I. •By rule (1), R is true relative to I iff I(R) = T. So,R is false relative to I. •
By rule (2), R is true relative to I iff R is false relative to I. So,R is true relative to I. •
By rule (4), (Q R) is true relative to I iff Q is true relative to I or R is true relative to I. So,(Q R) is true relative to I. •
By rule (6), (P (Q R)) is true relative to I iff P and(Q R) are either both true relative to I or P and (Q R) are both false relative to I. So,(P (Q R)) is true relative to I. Conclusion: I(P (Q R)) = T
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If it is necessarily true (or if it is impossible for it to be false)
TAUTOLOGY
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If it is possible for it to be true and it is possible for it to be false.
CONTINGENCY
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