what is not frege’s caesar problem

What is not Frege’s Caesar Problem?

Bruno Bentzen¹

¹ Department of Philosophy, Graduate Student

Federal University of Pernambuco

[email protected]

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

4

• Numbers are value-ranges

• Value-ranges are contextually defined by Basic Law V.


Value-ranges?

• Define the value-range of f as:

• Do we really know what does this name mean?

Value-ranges?

5 What is not Frege’s Caesar Problem? Bruno Bentzen

• Define the value-range of f as:


• No we don’t!

Value-ranges?


• Same objects can be denoted by different names

• The value range of f could be also denoted by, say, q

Value-ranges?


• 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?



Summary

• Value-ranges?




• Caesar Problem




Reconstructive part

Argumentative part


Permutation Argument

• Let Δ be an attribution from objects to “ ” names.

• Let h be a non-trivial permutation of all objects.



• 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.

12

Δ attribution:

Basic Law V: True False



13

Δ’ attribution:




h( ) h( ) h( ) h( )

14

Δ’ attribution:




15

Δ attribution:






• The reference of “ ” names is left undetermined by Basic Law V

• How can this be solved?


Summary

• Value-ranges?




• Caesar Problem




Reconstructive part

Argumentative part


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


Extra Stipulation

• Stipulate truth-values to be singletons (then value ranges)

• Then Every object denotable by the names of the theory are value ranges


Summary

• Value-ranges?




• Caesar Problem




Reconstructive part

Argumentative part


Dummett’s View

• What about The Moon, the Sugar Loaf – they are not also objects?


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).


Dummett’s View

• Can we quantify without “be afraid” of Caesar? (Heck, 1999)

• Take this sentence. Is it true or false?


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!


Summary

• Value-ranges?




• Caesar Problem




Reconstructive part

Argumentative part


Caesar Problem

• Numbers are logical objects

• Logical objects oppose ordinary objects


Caesar Problem

• Definitions have to establish:

• Numbers Ordinary object

• Define the number of f as:


Caesar Problem


• Define the number of f as:


• No we don’t!

Caesar Problem


• Same objects can be denoted by different names

• The number of f could be also denoted by, say, q

Caesar Problem


• 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



Summary

• Value-ranges?




• Caesar Problem




Reconstructive part

Argumentative part

• Dummett’s View:

Splitting the problems


Value-ranges problem

Caesar Problem

• Dummett’s View: The objective of this paper



Value-ranges problem

Caesar Problem

• Concerning value-ranges, the problem is technical

• Caesar Problem is philosophical



• Concerning value-ranges, the problem is technical – semantic

• Caesar Problem is philosophical – metaphysical



Proposition:



• Concerning value-ranges, the problem is solvable

• Caesar Problem keeps unsolvable


Summary

• Value-ranges?




• Caesar Problem




Reconstructive part

Argumentative part

• 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


• “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)



• 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!




Summary

• Value-ranges?




• Caesar Problem




Reconstructive part

Argumentative part

• We can’t ask anymore whether Caesar is a value range or not in the theory!

• Is Caesar Problem then solved?

Caesar Persists!


• We can’t ask anymore whether Caesar is a value range or not in the theory!

• Is Caesar Problem then solved?

• No!

Caesar Persists!


• We still have a metalinguistic problem (Dummett, 1981):

• is “the referent of ‘ ’ is Julius Caesar” True or False?

Caesar Persists!


• 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!


• 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!


• 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


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Science

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