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In mathematics, the '''Iwahori–Hecke algebra''', or '''Hecke algebra''',  named for [[Erich Hecke]] and [[Nagayoshi Iwahori]], is a one-parameter deformation of the [[group algebra]] of a [[Coxeter group]].
{{DISPLAYTITLE:''De dicto'' and ''de re''}}
'''''De dicto''''' and '''''de re''''' are two phrases used to mark important distinctions in [[intensional statement]]s, associated with the intensional operators in many such statements.  The distinctions are most recognized in [[philosophy of language]] and [[metaphysics]].[http://semanticsarchive.net/Archive/DU3YTgyN/Attributive,%20referential,%20de%20dicto%20and%20de%20re.pdf]


Hecke algebras are quotients of the group rings of [[Artin braid group]]s. This connection found a spectacular application in [[Vaughan Jones]]' construction of [[Jones polynomial|new invariants of knots]]. Representations of Hecke algebras led to discovery of [[quantum group]]s by [[Michio Jimbo]]. [[Michael Freedman]] proposed Hecke algebras as a foundation for [[topological quantum computer|topological quantum computation]].
The literal translation of the phrase "''de dicto''" is "of (the) word", whereas ''de re'' translates to "of (the) thing". The original meaning of the Latin locutions is useful for understanding the living meaning of the phrases, in the distinctions they mark. The distinction is best understood by examples of intensional contexts of which we will consider three: a context of thought, a context of desire, and a context of [[modal logic|modality]].


==Hecke algebras of Coxeter groups==
==Context of thought==
Start with the following data:
There are two possible interpretations of the sentence “Peter believes someone is out to get him”.  On one interpretation, ‘someone’ is unspecific and Peter suffers a general paranoia; he believes that it is true that a person is out to get him, but does not necessarily have any beliefs about who this person may be.  What Peter believes is that the predicate ‘is out to get Peter’ is satisfied.  This is the ''de dicto'' interpretation.


* ''(W,S)'' is a [[Coxeter system]] with the Coxeter matrix ''M = (m<sub>st</sub>)'',  
On the ''de re'' interpretation, ‘someone’ is specific, picking out some particular individual. There is some person Peter has in mind, and Peter believes that person is out to get him.
* ''R'' is a commutative ring with identity.
* {''q<sub>s</sub>'' | ''s'' &isin; ''S''} is a family of units of ''R'' such that ''q<sub>s</sub>'' = ''q<sub>t</sub>'' whenever ''s'' and ''t'' are conjugate in ''W''
* ''A'' is the ring of [[Laurent polynomial]]s over '''Z''' with indeterminates ''q<sub>s</sub>'' (and the above restriction that ''q<sub>s</sub>'' = ''q<sub>t</sub>'' whenever ''s'' and ''t'' are conjugated), that is ''A'' = '''Z''' [''q''{{su|p=±1|b=s}}]


===Multiparameter Hecke Algebras===
In the context of thought, the distinction helps us explain how people can hold seemingly self-contradicting beliefs. Say Lois Lane believes Clark Kent is weaker than Superman. Since Clark Kent is Superman, taken ''de re'', Lois’s belief is untenable; the names ‘Clark Kent’ and ‘Superman’ pick out an individual in the world, and a person (or super-person) cannot be stronger than himself.  Understood ''de dicto'', however, this may be a perfectly reasonable belief, since Lois is not aware that Clark and Superman are one and the same.
The ''multiparameter Hecke algebra'' ''H<sub>R</sub>(W,S,q)'' is a unital, associative ''R''-algebra with generators ''T<sub>s</sub>'' for all ''s'' &isin; ''S'' and relations:
* '''Braid Relations:''' ''T<sub>s</sub> T<sub>t</sub> T<sub>s</sub>'' ... = ''T<sub>t</sub> T<sub>s</sub> T<sub>t</sub>'' ..., where each side has ''m<sub>st</sub>'' < &infin; factors and ''s,t'' belong to ''S''.
* '''Quadratic Relation:''' For all ''s'' in ''S'' we have: (''T<sub>s</sub>'' - ''q<sub>s</sub>'')(''T<sub>s</sub>'' + 1) = 0.


'''Warning''': in recent books and papers, Lusztig has been using a modified form of the quadratic relation that reads <math>(T_s-q_s^{1/2})(T_s+q_s^{-1/2})=0.</math> After extending the scalars to include the half integer powers ''q''{{su|p=±½|b=s}} the resulting Hecke algebra is isomorphic to the previously defined one (but the ''T<sub>s</sub>'' here corresponds to ''q''{{su|p=½|b=s}} ''T''<sub>s</sub> in our notation). While this does not change the general theory, many formulae look different.
==Context of desire==


===Generic Multiparameter Hecke Algebras===
Consider the sentence "Jana wants to marry the tallest man in Fulsom County". It could be read either ''de dicto'' or ''de re''; the meanings would be different. One interpretation is that Jana wants to marry the tallest man in Fulsom County, whomever he might be.  On this interpretation, what the statement tells us is that Jana has a certain unspecific desire; what she desires is that a certain situation should obtain, namely, ''Jana's marrying the tallest man in Fulsom County''.  The desire is directed at that situation, regardless of how it is to be achieved.  The other interpretation is that Jana wants to marry a certain man, who in fact happens to be the tallest man in Fulsom County.  Her desire is for ''that man'', and she desires herself to marry ''him''. Again, the first interpretation, "Jana desires that she marry the tallest man in Fulsom County", is the ''de dicto'' interpretation. The second interpretation, "Of the tallest man in Fulsom County, Jana desires that she marry him", is the ''de re'' interpretation.
''H<sub>A</sub>(W,S,q)'' is the ''generic'' multiparameter Hecke algebra. This algebra is universal in the sense that every other multiparameter Hecke algebra can be obtained from it via the (unique) ring homomorphism ''A'' → ''R'' which maps the indeterminate ''q<sub>s</sub>'' &isin; ''A'' to the unit ''q<sub>s</sub>'' &isin; ''R''. This homomorphism turns ''R'' into a ''A''-algebra and the scalar extension ''H<sub>A</sub>(W,S)'' &otimes;<sub>''A''</sub> ''R'' is canonically isomorphic to the Hecke algebra ''H<sub>R</sub>(W,S,q)'' as constructed above. One calls this process ''specialization'' of the generic algebra.


=== One-parameter Hecke Algebras ===
Another way to understand the distinction is to ask what Jana would want if the man who was the tallest man in Fulsom County at the time the original statement was made were to lose his accolade to a 9 foot tall immigrant, such that he was no longer the tallest man in Fulsom County.  If she continued to want to marry that man &ndash; and, importantly, perceived this as representing no change in her desires &ndash; then she could be taken to have meant the original statement in a ''de re'' sense.  If she no longer wanted to marry that man but instead wanted to marry the ''new'' tallest man in Fulsom County, and saw this as a continuation of her earlier desire, then she meant the original statement in a ''de dicto'' sense.
If one specializes every indeterminant ''q<sub>s</sub>'' to a single indeterminant ''q'' over the integers (or ''q''{{su|p=½|b=s}} to ''q''<sup>½</sup> respectively), then one obtains the so-called generic one-parameter Hecke algebra of ''(W,S)''.


Since in Coxeter groups with single laced Dynkin diagrams (for example groups of type A and D) every pair of Coxeter generators is conjugated, the above mentioned restriction of ''q<sub>s</sub>'' being equal ''q<sub>t</sub>'' whenever ''s'' and ''t'' are conjugated in ''W'' forces the multiparameter and the one-parameter Hecke algebras to be equal. Therefore it is also very common to only look at one-parameter Hecke algebras.
==Context of modality==
The number of discovered chemical elements is 117.
Take the sentence "The number of [[chemical elements]] is necessarily greater than 100". Again, there are two interpretations as per the ''de dicto / de re'' distinction. The first interpretation is that things could not have gone differently, with the number of elements fewer than 100. If the inner workings of the atom could differ, there could be fewer than 100 elements. The second interpretation is that things could not have gone differently with the number 117 turning out to be fewer than 100. Intuitively, this claim is true. Of all the ways the world could have turned out, presumably there are no possibilities wherein 117 is fewer than 100. That 117 is greater than 100 is a necessary fact. The first interpretation, which seems to yield a false statement, is the ''de dicto'' interpretation. The second interpretation, which seems to yield a true statement, is the ''de re'' interpretation.


=== Coxeter groups with weights ===
Another example: "The [[President of the USA]] in 2001 could not have been [[Al Gore]]". This claim seems false on a ''de dicto'' reading. Presumably, things could have gone differently, with the [[SCOTUS|Supreme Court]] not [[Bush v. Gore|claiming]] that Bush had won the [[United States presidential election, 2000|election]]. But it looks more plausible on a ''de re'' reading. After all, we might skeptically wonder of [[George W. Bush]] whether ''he'' could have been Al Gore. Indeed, assuming that ''being George Bush'' is an essential feature of George Bush and that this feature is incompatible with being Al Gore, a ''de re'' reading of the statement is true.
If an integral weight function is defined on ''W'' (i.e. a map ''L:W'' → '''Z''' with ''L(vw)=L(v)+L(w)'' for all ''v,w'' &isin; ''W'' with ''l(vw)=l(v)+l(w)''), then a common specialization to look at is the one induced by the homomorphism ''q<sub>s</sub>'' ↦ ''q<sup>L(s)</sup>'', where ''q'' is a single indeterminant over '''Z'''.


If one uses the convention with half-integer powers, then weight function ''L:W'' → ½'''Z''' may be permitted as well. For technical reasons it is also often convenient only to consider positive weight functions.
==Representing ''de dicto'' and ''de re'' in modal logic==


== Properties ==
In [[modal logic]] the distinction between ''de dicto'' and ''de re'' is one of scope. In ''de dicto'' claims, any [[quantification|quantifiers]] are within the scope of the modal operator, whereas in ''de re'' claims the modal operator falls within the scope of the quantifier. For example:
1. The Hecke algebra has a basis <math>(T_w)_{w\in W}</math> over ''A'' indexed by the elements of the Coxeter group ''W''. In particular, ''H'' is a free ''A''-module. If <math>w=s_1 s_2 \ldots s_n</math> is a [[reduced decomposition]] of ''w'' &isin; ''W'', then <math>T_w=T_{s_1}T_{s_2}\ldots T_{s_n}</math>. This basis of Hecke algebra is sometimes called the '''natural basis'''. The [[neutral element]] of ''W'' corresponds to the identity of ''H'': ''T<sub>e</sub>'' = 1.


2. The elements of the natural basis are ''multiplicative'', namely, ''T''<sub>yw</sub>=''T''<sub>y</sub> ''T''<sub>w</sub> whenever ''l(yw)=l(y)+l(w)'', where ''l'' denotes the [[length function]] on the Coxeter group ''W''.
{| border="0"
| ''De dicto'':
| <math>\Box \exists{x} Ax</math>
| Necessarily, some ''x'' is such that it is ''A''
|-
| ''De re'':
| <math>\exists{x} \Box Ax</math>
| Some ''x'' is such that it is necessarily ''A''
|}


3. Elements of the natural basis are invertible. For example, from the quadratic relation we conclude that ''T''{{su|p=-1|b=s}} = ''q''{{su|p=-1|b=s}} ''T<sub>s</sub>'' + (''q''{{su|p=-1|b=s}}-1).
=== Willard van Orman Quine ===


4. Suppose that ''W'' is a finite group and the ground ring is the field '''C''' of complex numbers. [[Jacques Tits]] has proved that if the indeterminate ''q'' is specialized to any complex number outside of an explicitly given list (consisting of roots of unity), then the resulting one parameter Hecke algebra is [[semisimple algebra|semisimple]] and isomorphic to the complex group algebra '''C'''[''W''] (which also corresponds to the specialization ''q'' ↦ 1.
[[W.V.O. Quine|Willard Van Orman Quine]] refers to [[David Kaplan (philosopher)|D. Kaplan]], who in turn credits [[Montgomery Furth]] for the term [[vivid designator]] in his paper ''Reference Modality''. He examines the separation between ''de re'' and ''de dicto'' statements and does away with ''de re'' statements, because ''de re'' statements can only work for names that are used [[Reference|referentially]].<ref>''On Quine, Transparency and Specificity in Intentional Contexts'', Andrea Bonomi, p.183</ref> In fact, both [[rigid designators]] and vivid designators are similarly dependent on context and empty otherwise.  The same is true of the whole [[Quantification|quantified]] [[modal logic]] of [[necessity]]; for it collapses if [[essence]] is withdrawn.<ref>Quine, W.V.O., ''Quintessence, Reference and Modality'', pp.356-357</ref>


5. More generally, if ''W'' is a finite group and the ground ring ''R'' is a field of [[characteristic zero]], then the one parameter Hecke algebra is a [[semisimple algebra|semisimple associative algebra]] over ''R''[''q''<sup>±1</sup>]. Moreover, extending earlier results of Benson and Curtis, George Lusztig provided an explicit isomorphism between the Hecke algebra and the group algebra after the extension of scalars to the quotient field of ''R''[''q''<sup>±½</sup>]
==See also==
<!--
* [[Barcan formula]]
that if ''A'' is extended to the field <math>K=R(q^{\frac12}) then the ''K''-algebra <math>H_K=H\otimes_A K</math> obtained from ''H'' by the change of scalars is isomorphic over ''K'' to the group algebra ''K[W]'' of the Coxeter group ''W''.
* [[Latitudinarianism (philosophy)]]
* ''[[De se]]''


Lusztig, George. On a theorem of Benson and Curtis. J. Algebra 71 (1981), no. 2, 490–498.
==References==
However, it seems excessive to give this reference in an article in an encyclopedia!
{{Reflist}}
-->


== Canonical basis ==
==Bibliography==
{{main|Kazhdan–Lusztig polynomial}}
*Burge, Tyler. 1977. Belief de re. ''Journal of Philosophy'' 74, 338-362.
A great discovery of Kazhdan and Lusztig was that a Hecke algebra admits a ''different'' basis, which in a way controls representation theory of a variety of related objects.
*Donnellan, Keith S. 1966. Reference and definite descriptions. ''Philosophical Review'' 75, 281-304.
*Frege, Gottlob. 1892. Über Sinn und Bedeutung. Zeitschrift für Philosophie und philosophische Kritik 100, 25-50. Translated as On sense and reference by Peter Geach & Max Black, 1970, in Translations from the philosophical writings of Gottlob Frege. Oxford, Blackwell, 56-78.
*Kaplan, David. 1978. Dthat. In Peter Cole, ed., ''Syntax and Semantics'', vol. 9: Pragmatics. New York: Academic Press, 221-243
*Kripke, Saul. 1977. Speaker’s reference and semantic reference. In Peter A. French, Theodore E. Uehling, Jr., and Howard K. Wettstein, eds., Midwest Studies in Philosophy vol. II: Studies in the philosophy of language. Morris, MN: University of Minnesota, 255-276.
*Larson, Richard & Gabriel Segal. 1995. Definite descriptions. In Knowledge of meaning: An introduction to semantic theory. Cambridge, MA: MIT Press, 319-359.
*Ludlow, Peter & Stephen Neale. 1991. Indefinite descriptions: In defense of Russell. ''Linguistics and Philosophy'' 14, 171-202.
*Ostertag, Gary. 1998. Introduction. In Gary Ostertag, ed., Definite descriptions: a reader. Cambridge, MA: MIT Press, 1-34.
*Russell, Bertrand. 1905. On denoting. ''Mind'' 14, 479-493.
*Wettstein, Howard. 1981. Demonstrative reference and definite descriptions. ''Philosophical Studies'' 40, 241-257.
*Wilson, George M. 1991. Reference and pronominal descriptions. ''Journal of Philosophy'' 88, 359-387.


The generic multiparameter Hecke algebra, ''H<sub>A</sub>(W,S,q)'', has an involution ''bar'' that maps ''q''<sup>½</sup> to ''q''<sup>-½</sup> and acts as identity on '''Z'''. Then ''H'' admits a unique ring automorphism ''i'' that is [[semilinear transformation|semilinear]] with respect to the bar involution of ''A'' and maps ''T<sub>s</sub>'' to ''T{{su|p=-1|b=s}}''. It can further be proved that this automorphism is involutive (has order two) and takes any ''T<sub>w</sub>'' to <math>T^{-1}_{w^{-1}}.</math>
==External links==
* [http://plato.stanford.edu/entries/prop-attitude-reports/dere.html The ''De Re/De Dicto'' Distinction], Stanford Encyclopedia of Philosophy


<blockquote> '''Kazhdan - Lusztig Theorem:''' For each ''w'' ∈ ''W'' there exists a unique element <math>C^{\prime}_w</math> which is invariant under the involution ''i'' and if one writes its expansion in terms of the natural basis:
[[Category:Latin logical phrases]]
::<math> C'_w= \left (q^{-1/2} \right )^{l(w)}\sum_{y\leq w}P_{y,w}T_y, </math>
[[Category:Philosophical concepts]]
one has the following:
[[Category:Philosophy of language]]
* ''P''<sub>w,w</sub>=1,
[[Category:Modal logic]]
* ''P''<sub>y,w</sub> in '''Z'''[''q''] has degree less than or equal to ½''(l(w)-l(y)-1)'' if ''y<w'' in the [[Bruhat order]],
[[Category:Dichotomies]]
* ''P''<sub>y,w</sub>=0 if <math>y\nleq w.</math></blockquote>


The elements <math>C^{\prime}_w</math> where ''w'' varies over ''W'' form a basis of the algebra ''H'', which is called the ''dual canonical basis'' of the Hecke algebra ''H''. The ''canonical basis'' {''C''<sub>w</sub> | ''w'' &isin; ''W''} is obtained in a similar way. The polynomials ''P''<sub>y,w</sub>(''q'') making appearance in this theorem are the [[Kazhdan–Lusztig polynomials]].
[[de:De re und de dicto]]
 
[[es:De dicto y de re]]
The Kazhdan–Lusztig notions of left, right and two-sided ''cells'' in Coxeter groups are defined through the behavior of the canonical basis under the action of ''H''.
[[fr:De dicto et de re]]
 
[[fi:De dicto ja de re]]
== Hecke algebra of a locally compact group ==
Iwahori–Hecke algebras first appeared as an important special case of a very general construction in group theory. Let ''(G,K)'' be a pair consisting of a [[unimodular group|unimodular]] [[locally compact topological group]] ''G'' and a closed subgroup ''K'' of ''G''. Then the space of ''K''-biinvariant [[continuous function]]s of [[compact support]], ''C<sub>c</sub>(K\G/K)'', can be endowed with a structure of an associative algebra under the operation of [[convolution]]. This algebra is denoted by ''H(G//K)'' and called the '''Hecke ring''' of the pair ''(G,K)''.
 
'''Example:''' If ''G'' = SL(''n'','''Q'''<sub>''p''</sub>) and ''K'' = SL(''n'','''Z'''<sub>''p''</sub>) then the Hecke ring is commutative and its representations were studied by [[Ian G. Macdonald]]. More generally if ''(G,K)'' is a [[Gelfand pair]] then the resulting algebra turns out to be commutative.
 
'''Example:''' If ''G'' = SL(2,'''Q''') and ''K'' = SL(2,'''Z''') we get the abstract ring behind [[Hecke operators]] in the theory of [[modular forms]], which gave the name to Hecke algebras in general.
 
The case leading to the Hecke algebra of a finite Weyl group is when ''G'' is the finite [[Chevalley group]] over a [[finite field]] with ''p''<sup>k</sup> elements, and ''B'' is its [[Borel subgroup]]. Iwahori showed that the Hecke ring ''H(G//B)'' is obtained from the generic Hecke algebra ''H''<sub>q</sub> of the [[Weyl group]] ''W'' of ''G'' by specializing the indeterminate ''q'' of the latter algebra to ''p''<sup>k</sup>, the cardinality of the finite field. George Lusztig remarked in 1984 (''Characters of reductive groups over a finite field'', xi, footnote):
 
:''I think it would be most appropriate to call it the Iwahori algebra, but the name Hecke ring (or algebra) given by Iwahori himself has been in use for almost 20 years and it is probably too late to change it now.''
 
Iwahori and Matsumoto (1965) considered the case when ''G'' is a group of points of a [[reductive algebraic group]] over a non-archimedean [[local field]] ''K'', such as '''Q'''<sub>''p''</sub>, and ''K'' is what is now called an [[Iwahori subgroup]] of ''G''. The resulting Hecke ring is isomorphic to the Hecke algebra of the [[affine Weyl group]] of ''G'', or the [[affine Hecke algebra]], where the indeterminate ''q'' has been specialized to the cardinality of the [[residue field]] of ''K''.
 
Work of Roger Howe in the 1970s and his papers with Allen Moy on representations of ''p''-adic GL(''n'') opened a possibility of classifying irreducible admissible representations of reductive groups over local fields in terms of appropriately constructed Hecke algebras. (Important contributions were also made by Joseph Bernstein and [[Andrey Zelevinsky]].) These ideas were taken much further in [[Colin Bushnell]] and [[Philip Kutzko]]'s ''[[theory of types (mathematics)|theory of types]]'', allowing them to complete the classification in the general linear case. Many of the techniques can be extended to other reductive groups, which remains an area of active research. It has been conjectured that all Hecke algebras that are ever needed are mild generalizations of affine Hecke algebras.
 
== Representations of Hecke algebras ==
It follows from Iwahori's work that complex representations of Hecke algebras of finite type are intimately related with the structure of the spherical [[principal series representation]]s of finite Chevalley groups.
 
George Lusztig pushed this connection much further and was able to describe most of the characters of finite groups of Lie type in terms of representation theory of Hecke algebras. This work used a mixture of geometric techniques and various reductions, led to introduction of various objects generalizing Hecke algebras and detailed understanding of their representations (for ''q'' not a root of unity). [[Modular representation]]s of Hecke algebras and representations at roots of unity turned out to be related with the theory of canonical bases in [[affine quantum group]]s and very interesting combinatorics.
 
Representation theory of affine Hecke algebras was developed by Lusztig with a view towards applying it to description of representations of ''p''-adic groups. It is in many ways quite different in flavor from the finite case. A generalization of affine Hecke algebras, called ''double affine Hecke algebra'', was used by [[Ivan Cherednik]] in his proof of the [[Macdonald conjectures]].
 
== References ==
*David Goldschmidt [http://www.ams.org/online_bks/ulect4/ Group Characters, Symmetric Functions, and the Hecke Algebra] {{MR|1225799}},ISBN 0-8218-3220-4
*Iwahori, Nagayoshi; Matsumoto, Hideya [http://www.numdam.org/item?id=PMIHES_1965__25__5_0 ''On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups.''] Publications Mathématiques de l'IHÉS, 25 (1965), pp.&nbsp;5–48. {{MR|0185016}}
* Alexander Kleshchev, ''Linear and projective representations of symmetric groups'', Cambridge tracts in mathematics, vol. 163. Cambridge University Press, 2005. {{MR|2165457}}, ISBN 0-521-83703-0
* George Lusztig, [http://www.ams.org/bookstore-getitem/item=CRMM-18 Hecke algebras with unequal parameters], CRM monograph series, vol.18, American Mathematical Society, 2003. {{MR|1658581}}, ISBN 0-8218-3356-1
* Andrew Mathas, [http://www.ams.org/bookstore-getitem/item=ULECT-15 Iwahori-Hecke algebras and Schur algebras of the symmetric group], University Lecture Series, vol.15, American Mathematical Society, 1999. {{MR|1711316}}, ISBN 0-8218-1926-7
* Lusztig, George, ''On a theorem of Benson and Curtis'', J. Algebra 71 (1981), no. 2, 490–498. {{MR|0630610}}, {{DOI|10.1016/0021-8693(81)90188-5}}
* Colin Bushnell and Philip Kutzko, ''The admissible dual of GL(n) via compact open subgroups'', Annals of Mathematics Studies, vol. 129, Princeton University Press, 1993. {{MR|1204652}}, ISBN 0-691-02114-7
 
{{DEFAULTSORT:Iwahori-Hecke algebra}}
[[Category:Algebras]]
[[Category:Representation theory]]

Revision as of 06:21, 12 August 2014

De dicto and de re are two phrases used to mark important distinctions in intensional statements, associated with the intensional operators in many such statements. The distinctions are most recognized in philosophy of language and metaphysics.[1]

The literal translation of the phrase "de dicto" is "of (the) word", whereas de re translates to "of (the) thing". The original meaning of the Latin locutions is useful for understanding the living meaning of the phrases, in the distinctions they mark. The distinction is best understood by examples of intensional contexts of which we will consider three: a context of thought, a context of desire, and a context of modality.

Context of thought

There are two possible interpretations of the sentence “Peter believes someone is out to get him”. On one interpretation, ‘someone’ is unspecific and Peter suffers a general paranoia; he believes that it is true that a person is out to get him, but does not necessarily have any beliefs about who this person may be. What Peter believes is that the predicate ‘is out to get Peter’ is satisfied. This is the de dicto interpretation.

On the de re interpretation, ‘someone’ is specific, picking out some particular individual. There is some person Peter has in mind, and Peter believes that person is out to get him.

In the context of thought, the distinction helps us explain how people can hold seemingly self-contradicting beliefs. Say Lois Lane believes Clark Kent is weaker than Superman. Since Clark Kent is Superman, taken de re, Lois’s belief is untenable; the names ‘Clark Kent’ and ‘Superman’ pick out an individual in the world, and a person (or super-person) cannot be stronger than himself. Understood de dicto, however, this may be a perfectly reasonable belief, since Lois is not aware that Clark and Superman are one and the same.

Context of desire

Consider the sentence "Jana wants to marry the tallest man in Fulsom County". It could be read either de dicto or de re; the meanings would be different. One interpretation is that Jana wants to marry the tallest man in Fulsom County, whomever he might be. On this interpretation, what the statement tells us is that Jana has a certain unspecific desire; what she desires is that a certain situation should obtain, namely, Jana's marrying the tallest man in Fulsom County. The desire is directed at that situation, regardless of how it is to be achieved. The other interpretation is that Jana wants to marry a certain man, who in fact happens to be the tallest man in Fulsom County. Her desire is for that man, and she desires herself to marry him. Again, the first interpretation, "Jana desires that she marry the tallest man in Fulsom County", is the de dicto interpretation. The second interpretation, "Of the tallest man in Fulsom County, Jana desires that she marry him", is the de re interpretation.

Another way to understand the distinction is to ask what Jana would want if the man who was the tallest man in Fulsom County at the time the original statement was made were to lose his accolade to a 9 foot tall immigrant, such that he was no longer the tallest man in Fulsom County. If she continued to want to marry that man – and, importantly, perceived this as representing no change in her desires – then she could be taken to have meant the original statement in a de re sense. If she no longer wanted to marry that man but instead wanted to marry the new tallest man in Fulsom County, and saw this as a continuation of her earlier desire, then she meant the original statement in a de dicto sense.

Context of modality

The number of discovered chemical elements is 117. Take the sentence "The number of chemical elements is necessarily greater than 100". Again, there are two interpretations as per the de dicto / de re distinction. The first interpretation is that things could not have gone differently, with the number of elements fewer than 100. If the inner workings of the atom could differ, there could be fewer than 100 elements. The second interpretation is that things could not have gone differently with the number 117 turning out to be fewer than 100. Intuitively, this claim is true. Of all the ways the world could have turned out, presumably there are no possibilities wherein 117 is fewer than 100. That 117 is greater than 100 is a necessary fact. The first interpretation, which seems to yield a false statement, is the de dicto interpretation. The second interpretation, which seems to yield a true statement, is the de re interpretation.

Another example: "The President of the USA in 2001 could not have been Al Gore". This claim seems false on a de dicto reading. Presumably, things could have gone differently, with the Supreme Court not claiming that Bush had won the election. But it looks more plausible on a de re reading. After all, we might skeptically wonder of George W. Bush whether he could have been Al Gore. Indeed, assuming that being George Bush is an essential feature of George Bush and that this feature is incompatible with being Al Gore, a de re reading of the statement is true.

Representing de dicto and de re in modal logic

In modal logic the distinction between de dicto and de re is one of scope. In de dicto claims, any quantifiers are within the scope of the modal operator, whereas in de re claims the modal operator falls within the scope of the quantifier. For example:

De dicto: ◻∃xAx Necessarily, some x is such that it is A
De re: ∃x◻Ax Some x is such that it is necessarily A

Willard van Orman Quine

Willard Van Orman Quine refers to D. Kaplan, who in turn credits Montgomery Furth for the term vivid designator in his paper Reference Modality. He examines the separation between de re and de dicto statements and does away with de re statements, because de re statements can only work for names that are used referentially.[1] In fact, both rigid designators and vivid designators are similarly dependent on context and empty otherwise. The same is true of the whole quantified modal logic of necessity; for it collapses if essence is withdrawn.[2]

See also

References

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Bibliography

  • Burge, Tyler. 1977. Belief de re. Journal of Philosophy 74, 338-362.
  • Donnellan, Keith S. 1966. Reference and definite descriptions. Philosophical Review 75, 281-304.
  • Frege, Gottlob. 1892. Über Sinn und Bedeutung. Zeitschrift für Philosophie und philosophische Kritik 100, 25-50. Translated as On sense and reference by Peter Geach & Max Black, 1970, in Translations from the philosophical writings of Gottlob Frege. Oxford, Blackwell, 56-78.
  • Kaplan, David. 1978. Dthat. In Peter Cole, ed., Syntax and Semantics, vol. 9: Pragmatics. New York: Academic Press, 221-243
  • Kripke, Saul. 1977. Speaker’s reference and semantic reference. In Peter A. French, Theodore E. Uehling, Jr., and Howard K. Wettstein, eds., Midwest Studies in Philosophy vol. II: Studies in the philosophy of language. Morris, MN: University of Minnesota, 255-276.
  • Larson, Richard & Gabriel Segal. 1995. Definite descriptions. In Knowledge of meaning: An introduction to semantic theory. Cambridge, MA: MIT Press, 319-359.
  • Ludlow, Peter & Stephen Neale. 1991. Indefinite descriptions: In defense of Russell. Linguistics and Philosophy 14, 171-202.
  • Ostertag, Gary. 1998. Introduction. In Gary Ostertag, ed., Definite descriptions: a reader. Cambridge, MA: MIT Press, 1-34.
  • Russell, Bertrand. 1905. On denoting. Mind 14, 479-493.
  • Wettstein, Howard. 1981. Demonstrative reference and definite descriptions. Philosophical Studies 40, 241-257.
  • Wilson, George M. 1991. Reference and pronominal descriptions. Journal of Philosophy 88, 359-387.

External links

de:De re und de dicto es:De dicto y de re fr:De dicto et de re fi:De dicto ja de re

  1. ↑ On Quine, Transparency and Specificity in Intentional Contexts, Andrea Bonomi, p.183
  2. ↑ Quine, W.V.O., Quintessence, Reference and Modality, pp.356-357
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