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What is not Frege’s Caesar Problem?
Bruno Bentzen¹
¹ Department of Philosophy, Graduate Student
Federal University of Pernambuco
What is not Frege’s Caesar Problem? Bruno Bentzen
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What is not Frege’s Caesar Problem? Bruno Bentzen
Summary
• Value-ranges?
• Permutation Argument
• Extra Stipulation
• Dummett’s View
• Caesar Problem
• Splitting the problems
• Solving value-ranges problem
• Caesar Persists!
Reconstructive part
Argumentative part
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What is not Frege’s Caesar Problem? Bruno Bentzen
Summary
• Value-ranges?
• Permutation Argument
• Extra Stipulation
• Dummett’s View
• Caesar Problem
• Splitting the problems
• Solving value-ranges problem
• Caesar Persists!
Reconstructive part
Argumentative part
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• Numbers are value-ranges
• Value-ranges are contextually defined by Basic Law V.
What is not Frege’s Caesar Problem? Bruno Bentzen
Value-ranges?
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• Define the value-range of f as:
• Do we really know what does this name mean?
Value-ranges?
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• Define the value-range of f as:
• Do we really know what does this name mean?
• No we don’t!
Value-ranges?
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• Same objects can be denoted by different names
• The value range of f could be also denoted by, say, q
Value-ranges?
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• Now ask if eventually is to be the case.
• We have no means to find q’s corresponding function to apply our criteria of identity!
Value-ranges?
8 What is not Frege’s Caesar Problem? Bruno Bentzen
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What is not Frege’s Caesar Problem? Bruno Bentzen
Summary
• Value-ranges?
• Permutation Argument
• Extra Stipulation
• Dummett’s View
• Caesar Problem
• Splitting the problems
• Solving value-ranges problem
• Caesar Persists!
Reconstructive part
Argumentative part
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10 What is not Frege’s Caesar Problem? Bruno Bentzen
Permutation Argument
• Let Δ be an attribution from objects to “ ” names.
• Let h be a non-trivial permutation of all objects.
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11 What is not Frege’s Caesar Problem? Bruno Bentzen
Permutation Argument
• Let Δ be an attribution from objects to “ ” names.
• Let h be a non-trivial permutation of all objects.
• Consider a very similar attribution Δ’ except that:
• For every object a assigned by Δ to “ ”, Δ’ will assign h(a) instead.
• Thus: Δ’ is a different than Δ, but it is also consistent with BLV if so is Δ.
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Δ attribution:
Basic Law V: True False
What is not Frege’s Caesar Problem? Bruno Bentzen
Permutation Argument
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Δ’ attribution:
Basic Law V: True False
What is not Frege’s Caesar Problem? Bruno Bentzen
Permutation Argument
h( ) h( ) h( ) h( )
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Δ’ attribution:
Basic Law V: True False
What is not Frege’s Caesar Problem? Bruno Bentzen
Permutation Argument
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Δ attribution:
Basic Law V: True False
What is not Frege’s Caesar Problem? Bruno Bentzen
Permutation Argument
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Permutation Argument
• The reference of “ ” names is left undetermined by Basic Law V
• How can this be solved?
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What is not Frege’s Caesar Problem? Bruno Bentzen
Summary
• Value-ranges?
• Permutation Argument
• Extra Stipulation
• Dummett’s View
• Caesar Problem
• Splitting the problems
• Solving value-ranges problem
• Caesar Persists!
Reconstructive part
Argumentative part
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Extra Stipulation
• Frege’s theory contains either value ranges names or truth value names
• Indeterminacies could only occur in identity statements between these two names
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Extra Stipulation
• Stipulate truth-values to be singletons (then value ranges)
• Then Every object denotable by the names of the theory are value ranges
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What is not Frege’s Caesar Problem? Bruno Bentzen
Summary
• Value-ranges?
• Permutation Argument
• Extra Stipulation
• Dummett’s View
• Caesar Problem
• Splitting the problems
• Solving value-ranges problem
• Caesar Persists!
Reconstructive part
Argumentative part
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Dummett’s View
• What about The Moon, the Sugar Loaf – they are not also objects?
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Dummett’s View
• What about The Moon, the Sugar Loaf – they are not also objects?
• Stipulations still leave open whether “Caesar” is a value-range or not (1981).
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23 What is not Frege’s Caesar Problem? Bruno Bentzen
Dummett’s View
• Can we quantify without “be afraid” of Caesar? (Heck, 1999)
• Take this sentence. Is it true or false?
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24 What is not Frege’s Caesar Problem? Bruno Bentzen
Dummett’s View
• Can we quantify without “be afraid” of Caesar? (Heck, 1999)
• Take this sentence. Is it true or false?
• Problem is left unsolved!
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What is not Frege’s Caesar Problem? Bruno Bentzen
Summary
• Value-ranges?
• Permutation Argument
• Extra Stipulation
• Dummett’s View
• Caesar Problem
• Splitting the problems
• Solving value-ranges problem
• Caesar Persists!
Reconstructive part
Argumentative part
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26 What is not Frege’s Caesar Problem? Bruno Bentzen
Caesar Problem
• Numbers are logical objects
• Logical objects oppose ordinary objects
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27 What is not Frege’s Caesar Problem? Bruno Bentzen
Caesar Problem
• Definitions have to establish:
• Numbers Ordinary object
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• Define the number of f as:
• Do we really know what does this name mean?
Caesar Problem
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• Define the number of f as:
• Do we really know what does this name mean?
• No we don’t!
Caesar Problem
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• Same objects can be denoted by different names
• The number of f could be also denoted by, say, q
Caesar Problem
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• Now ask if eventually is to be the case.
• We have no means to find q’s corresponding function to apply our criteria of identity!
Caesar Problem
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What is not Frege’s Caesar Problem? Bruno Bentzen
Summary
• Value-ranges?
• Permutation Argument
• Extra Stipulation
• Dummett’s View
• Caesar Problem
• Splitting the problems
• Solving value-ranges problem
• Caesar Persists!
Reconstructive part
Argumentative part
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• Dummett’s View:
Splitting the problems
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Value-ranges problem
Caesar Problem
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• Dummett’s View: The objective of this paper
Splitting the problems
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Value-ranges problem
Caesar Problem
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• Concerning value-ranges, the problem is technical
• Caesar Problem is philosophical
Splitting the problems
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• Concerning value-ranges, the problem is technical – semantic
• Caesar Problem is philosophical – metaphysical
Splitting the problems
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Proposition:
Splitting the problems
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• Concerning value-ranges, the problem is solvable
• Caesar Problem keeps unsolvable
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What is not Frege’s Caesar Problem? Bruno Bentzen
Summary
• Value-ranges?
• Permutation Argument
• Extra Stipulation
• Dummett’s View
• Caesar Problem
• Splitting the problems
• Solving value-ranges problem
• Caesar Persists!
Reconstructive part
Argumentative part
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• Against Dummett there’s no (semantical) problem regarding value-ranges.
• Caesar could only appear as a problem in quantified sentences
• Frege’s criteria of referentiality of quantified sentences is substitutional
Solving value-ranges problem
39 What is not Frege’s Caesar Problem? Bruno Bentzen
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• “To investigate whether the name “ ” of a second-level function denotes something, we ask whether it follows universally from the fact that the function-name “ ” denotes something, that “ ” succeeds in denoting. Now “ ” has a denotation if, for every denoting proper name “a”,
“ F(x) ” denotes something. If this is the case, then this denotation either always is the True (whatever “a” denotes), or not always. In the first case ” “ denotes the True, in the second the False.”
Basic Laws of Arithmetic, §31)
Solving value-ranges problem
40 What is not Frege’s Caesar Problem? Bruno Bentzen
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• Whether ordinary objects are or not in the domain is irrelevant to fix the meaning of
• What is at stake is what names there are!
Solving value-ranges problem
41 What is not Frege’s Caesar Problem? Bruno Bentzen
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What is not Frege’s Caesar Problem? Bruno Bentzen
Summary
• Value-ranges?
• Permutation Argument
• Extra Stipulation
• Dummett’s View
• Caesar Problem
• Splitting the problems
• Solving value-ranges problem
• Caesar Persists!
Reconstructive part
Argumentative part
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• We can’t ask anymore whether Caesar is a value range or not in the theory!
• Is Caesar Problem then solved?
Caesar Persists!
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• We can’t ask anymore whether Caesar is a value range or not in the theory!
• Is Caesar Problem then solved?
• No!
Caesar Persists!
44 What is not Frege’s Caesar Problem? Bruno Bentzen
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• We still have a metalinguistic problem (Dummett, 1981):
• is “the referent of ‘ ’ is Julius Caesar” True or False?
Caesar Persists!
45 What is not Frege’s Caesar Problem? Bruno Bentzen
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• To establish that extensions are uncontroversially logical objects then they could never be ordinary objects in advance would be to beg the question!
Caesar Persists!
46 What is not Frege’s Caesar Problem? Bruno Bentzen
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• To establish that extensions are uncontroversially logical objects then they could never be ordinary objects in advance would be to beg the question!
Thank you for your attention!
Caesar Persists!
47 What is not Frege’s Caesar Problem? Bruno Bentzen
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• Frege, G. (1987). Die Grundlagen der Arithmetik: Eine logisch mathematische Untersuchung über den Begriff der Zahl. Stuttgard: Philipp Reclam
• ______. (1998). Die Grundgesetze der Arithmetik I/II. Georg Olms: Hildesheim
• Dummett, M. (1981). Frege: Philosophy of Language. U.K: Duckworth
• Heck, R. (1999). Grundgesetze der Arithmetic I §10. Philosophia Mathematica, v. 7, n. 3, p. 258-292, out
• Ruffino, M. (2002). Logical Objects in Frege’s Grundgesetze, Section 10. In: Reck (Ed.). From Frege to Wittgenstein, Oxford: Oxford University Press. p. 125-48
References
48 What is not Frege’s Caesar Problem? Bruno Bentzen