philosophy discussion 3
Valerielee
ExplainingtheSoritesParadox.pdf
2/8/2018
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Phil 2: Puzzles and Paradoxes
Prof. Sven Bernecker
University of California, Irvine
Explaining the
Sorites Paradox
History of the Sorites Paradox
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• The sorites paradox is attributed to Eubulides of Miletus
(4th century BC) who is also came up with the liar paradox.
• Miletus was part of the Ancient Greek empire and is part of
modern day Turkey.
• “Sorites” derives from the Greek word soros (meaning
“heap”). This is why this paradox is also called the
“paradox of the heap.”
Miletus
Theater of Miletus
• The paradox of the heap: Would you describe a single grain of
sand as a heap? No. Would you describe two grains of sand as a
heap? No. … You must admit the presence of a heap sooner or
later, so where do you draw the line?
https://www.youtube.com/watch?v=9DXVDPiiUN8
• The paradox of the heap is the name given to a class of
paradoxical arguments, also known as “little-by-little arguments,”
which arise as a result of the indeterminacy surrounding limits of
application of the predicates involved.
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Sorites Paradox Example:
1) A man with 1 hair on his head is bald.
2) If a man with 1 hair on his head is bald, then a man with 2 hairs on his head is bald.
3) If a man with 2 hairs on his head is bald, then a man with 3 hairs on his head is bald.
….
C) Therefore, a man with 100,000 hairs on his head is bald. Everyone is bald.
•The “..... “ stands for a long list of premises which can be summed up as follows: For any number of hairs h, if someone's number of hairs is h and he is bald, then someone whose number of hairs is h + 1 is also bald.
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• The conclusion (C) is obviously false, but the reasoning seems valid and the premises seem plausible.
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• The reason that the paradox of the heap occurs in natural language is
because of the existence of vague predicates. Words such as bald
and poor are only partially defined.
• But, classical logic says that ~(b & ~b) ≡ b v ~b. This is not the case
with DeVito! So, predicates such as bald don’t conform to the usual
logical rules.
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• Because bald is only partially defined, it would be
wrong to say that Danny DeVito is either bald or non-
bald, so ~(b v ~b). People would also agree that it
would be correct to say that DeVito is not both bald
and non-bald, so ~(b & ~b).
More Examples of Sorites Paradoxes
1) Someone who is 7 feet in height is tall.
2) If someone who is 7 feet in height is tall, then someone 6'11.9” in height
is tall.
3) If someone who is 6'11.9” in height is tall, then someone 6'11.8” in height
is tall.
......
C) Therefore, someone who is 3' in height is tall.
The “..... “ stands for a long list of premises which can be summed up as follows:
For any height h, if someone's height is h and she is tall, then someone whose
height is h – 0.1” is also tall.
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1) 10,000 grains of sand is a heap of sand.
2) If 10,000 grains of sand is a heap of sand, then 9999 grains of
sand is a heap of sand.
3) If 9999 grains of sand is a heap of sand, then 9998 grains of
sand is a heap of sand.
......
C) Therefore, 1 grain of sand is a heap of sand.
Sorites premise: For any number n, if n grains of sand is a heap, then n
-1 grains of sand is a heap.
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Mathematical Induction
• A conclusion reached through ordinary inductive reasoning is not certain; it is at best probable.
• Mathematical induction is different. Mathematical induction is a method of proof typically used to establish a given statement for all natural numbers. Example:
1) 0 is a natural number, and it is divisible by 1
2) If natural number n is divisible by 1, so is n+1
C) Every natural number is divisible by 1
• The paradox of the heap is an example of the failure of mathematical induction in natural language.
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Vagueness, Ambiguity, Equivocation
• A statement is vague if it uses concepts that have indefinite
application to particular cases.
• A statement is ambiguous if it can have two or more distinct
meanings.
• A statement is both vague and ambiguous if it has multiple
meanings, some (or all) of which have indefinite applications.
• Ambiguity creates equivocation.
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• Lexical ambiguity: when a word in a sentence has multiple meanings: “His
house is next to the bank.”
1: the land alongside a river...
2: a financial establishment...
“Time flies like an arrow, fruit flies like a banana.” (Groucho Marx)
• Syntactic ambiguity: when sentence-structure (syntax) allows for different
interpretations: “The killing of tyrants is justified.”
1: The killing DONE BY tyrants...
2: The killing OF tyrants...
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• Contextual ambiguity: when
conversational context creates the
possibility of multiple interpretations.
“John wanted him to leave the band.”
Who does “him” refer to?
Vagueness is (almost) ubiquitous
• Gradable adjectives: tall, red, fun
• Non-gradable adjectives: circular, level
• Nouns: striped apple, stone lion
• Verbs: running, singing, sleeping
• An important factor affecting the location of the threshold is the
set of objects with respect to which the property in question is
being judged – the comparison class
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