By J. L. Speranza, of the Grice Club.
----
I append more running comments on Bernard Linsky's interesting entry on 'logical construction' in the Stanford Encyclopaedia (online at
http://plato.stanford.edu/entries/logical-construction/
and (c) B. Linsky (2009).
It relates to this item in the Grice Archive as to 'personal identity' as a 'logical construction'. It connects with Carnap's idea of 'matter' as a logical construction out of sense data.
Linksy writes that Russell refers to "several different definitions and philosophical analyses as providing "logical constructions"". These are of certain entities and expressions." The "examples he cites" are:
a. "the Frege/Russell definition of numbers as classes of equinumerous classes".
b. "the theory of definite descriptions".
c. "the construction of matter from sense data,"
-- "and several others", Linsky writes.
"Generally expressions for such entities are called "incomplete symbols" and the entities themselves "logical fictions"."
"The notion originates with Russell's logicist program of reducing mathematics to logic, was widely used by Russell, and led to the later Logical Positivist notion of construction and ultimately the widespread use of set theoretic models in philosophy."
"Russell's most specific formulation of logical construction as a method in Philosophy comes from his essay "Logical Atomism"."
Linksy quotes directly from Russell:
One very important heuristic maxim which
Dr. Whitehead and I found, by experience, to
be applicable in mathematical logic, and have
since applied to various other fields, is a
form of Occam's Razor. When some set of
supposed entities has neat
logical properties, it turns out, in a
great many instances, that the supposed entities
can be replaced by purely logical structures composed
of entities which have not such neat properties. In
that case, in interpreting a body of
propositions hitherto believed to be
about the supposed entities, we can
substitute the logical structures without altering
any of the detail of the body of propositions
in question. This is an economy, because
entities with neat logical properties are
always inferred, and if the propositions
in which they occur can be interpreted without
making this inference, the ground for the
inference fails, and our body of propositions
is secured against the need of a doubtful
step. The principle may be stated in the [following] form.
RUSSELL's formulation of 'logical construction':
Whenever possible, substitute
constructions out of known entities
for inferences to unknown entities
Ruseell 1924:160.
Linksy comments that Russell is speaking of logical constructions in this memorable passage from his "Philosophy of Logical Atomism" lectures." Again Linksy quotes directly from Russell:
The method of `postulating' what we
want has many advantages; they are the
same as the advantages of theft
over honest toil. Let us leave them
to others and proceed with
our honest toil
Russell 1918:71.
Linksy comments:
"The notion of logical construction
appears frequently with the idea that
what is defined is a "logical fiction",
and an "incomplete symbol"."
As Linksy comments:
"The latter term, [incomplete symbol] derives
from the use of contextual definitions, providing an analysis of each sentence in which a defined symbol may occur without, however, giving an explicit definition, an equation or universal statement giving necessary and sufficient conditions for the application of the term in isolation."
Linksy continues:
"The terms "fiction" and "incomplete symbol" apply with differing aptness to various constructions."
Here starts Linksy's brilliant historical analysis. He writes:
"Russell's first use of construction, and
the model for later constructions, is the
Frege/Russell definition of numbers as classes.
This follows the kind of definitions used
in the arithmetization of analysis of
the preceding century, in particular, Dedekind's
earlier construction of real numbers as
bounded classes in the rational numbers. Russell's
logicist program could not rest content with
postulates for the fundamental objects of
mathematics such as the Peano Axioms
for the natural numbers. Instead numbers
were to be defined as classes of
equinumerous classes."
Is this abstraction?
Linsky notes:
"Russell refers to this method as "abstraction",
now known as the abstraction of an
equivalence class. The definition of equinumerosity,
or of the existence of a one to one mapping between two classes, also called "similarity", is solely in terms
of logical notions of quantifiers and identity. With
the numbers defined, for example, two as
the class of all two membered sets, or pairs, the properties of numbers could be derived by logical means alone."
--- Linsky then turns to the second illustration:
"The most influential of Russell's constructions
was the theory of descriptions [in] "On Denoting" in 1905."
"Russell's theory provides an analysis of
sentences of the form `The alpha is beta'
where `The alpha' is called a definite
decription."
"The analysis proposes that
`The alpha is beta' as equivalent to
`There is one and only one alpha and it is beta'".
"With this analysis, the logical properties
of descriptions can now be deduced using just the
logic of quantifiers and identity."
Linksy continues:
"Among the theorems in *14 of PM are those
showing what follows."
"First, if there is just one alpha then `The alpha is alpha' is 1, and if there is not, then `The alpha is beta' is always 0 and, crucially for the logical manipulation of descriptions."
"Second, if the alpha = the beta, and the alpha is gamma, then the beta is gamma. I.e. proper -- uniquely referring -- descriptions behave like singular terms."
"Some of these results are contentious. P. F. Strawson (Grice's tutee at St. John's< Oxford), noted that
`The present king of france is bald'
should be truth valueless since there is no present king of France, rather than "plainly false", as Russell's theory predicts."
But Grice came to the defense of Russell.
Linsky goes on:
"The theory of descriptions introduces
Russell's notion of incomplete symbol.
Definite descriptions like
`The alpha' do not show up in the formal
analysis of sentences in which they occur,
thus
`The alpha is gamma' becomes as per below."
(x) [(y)(Fy y=x) & Hx]
-- "of which no subformula, or continuous segment, can be identified as the analysis of `The alpha'".
Linsky notes:
"Much as talk about "the average family" as in
"The average family has 2.2 children" becomes
"The number of children in families divided by the number of families = 2.2",
there is no portion of that analysis that corresponds with "the average family"."
"Instead, we have a formula for eliminating
such expressions from contexts in which they occur,
hence the notion of "incomplete symbol" and
the related "contextual definition"".
Linsky notes:
"It is standard to see in this the origins of
the distinction between between surface grammatical
form and logical form, and thus the origin of linguistic analysis as a method in philosophy which operates by seeing past superficial linguistic form to underlying philosophical analysis."
----
He adds:
"The theory of descriptions has been criticized by some linguists who see descriptions and other noun phrases as full fledged constituents of sentences, and who see the sharp distinction between grammatical and logical form as a mistake."
Citation needed!
Linsky goes on:
"The theory of descriptions is often described as a model for avoiding ontological commitment to objects such as Meinongian subsistent entities, and so logical constructions in general are often seen as being chiefly aimed at ontological goals."
Cfr. Grice on 'avoiding Meinongian jungles' in "Vacuous Names".
Linsky goes on:
"In fact, that goal is at most peripheral
to most constructions. Rather, the goal is
to allow the proof of propositions that
would otherwise have to be assumed as
axioms or hypotheses."
Linsky adds:
"Nor need the ontological goal be
always elimination of problematic entities. Other
constructions should be seen more as reductions
of one class of entity to another, or replacements of one notion by a more precise, mathematical, substitute."
---
"Russell's 'no-class' theory of classes
from *20 of Whitehead's and Russell's PM
provides a contextual definition
like the theory of descriptions."
How? Well,
"One of Russell's early diagnoses of the
paradoxes was that they showed that
classes could not be objects. Indeed he
seems to have come across his paradox
of the class of all classes that are not members of
themselves by applying Cantor's argument to show that there are more classes of objects than objects."
Linksy goes on:
"Hence, Russell concludes that classes can not be objects."
"Inspired by the theory of descriptions, Russell proposes
that to say something beta of the class of alphas --
BETA{x: ALPHAx}
-- is to say that there is some property beta coextensive with (1 of the same things as) alpha such that gamma is beta."
"Extensionality of sets is thus derivable, rather than postulated: if alpha and gamma are co-extensive then anything true of"
{x: ALPHAx}
"will be true of {x: GAMMAx}."
"Features of sets then follow from the features of the logic of properties, the "ramified theory of types"."
Russell echoing Bentham:
"Because classes would seem to be individuals of some sort, but on analysis are found not to be, Russell speaks of them as "logical fictions", an expression which echoes Jeremy Bentham's notion of a "legal fiction"."
"Because statements attributing a property to particular classes are analyzed by existential sentences saying that there is some propositional function having that property, this construction should not be seen as avoiding ontological commitment entirely, but rather of reducing classes to propositional functions."
"The properties of classes are really properties of propositional functions and for every class said to have a property there really is some propositional function having that property."
Linsky then draws from other illustrations.
"For other constructions such as propositions a contextual definition is not provided."
"In any case, constructions do not appear as the referents of logically proper names, and so by that account are not part of the fundamental "furniture" of the world."
J. WISDOM on constructions.
"Early critical discussions of constructions, such as John Wisdom's, stressed the contrast between logically proper names, which do refer, and constructions, which were thus seen as ontologically innocent."
Linsky notes:
"Beginning with The Problems of Philosophy (1912), Russell turns repeatedly to the problem of matter."
"Part of the problem is to find a refution of Berkeleyan idealism, of showing how the existence and real nature of matter can be proved."
"In Problems Russell argues that matter is a well supported hypothesis that explains our experiences. Matter is known only indirectly, "by description", as the cause, whatever it may be, of our sense data, which we know "by acquaintance". This is the notion of hypothesis which Russell contrasts with construction in the passage above."
"Russell saw an analogy between the case of simply hypothesizing the existence of numbers with certain properties, those described by axioms, and hypothesizing the existence of matter. While we distinguish the certain knowledge we may have of mathematical entities from the contingent knowledge of material objects, Russell says that there are certain "neat" features of matter which are just too tidy to have turned out by accident. Examples include the most general spatiotemporal properties of objects, that no two can occupy the same place at the same time, and so on. Material objects are now to be seen as collections of sense data. Influenced by William James, Russell defended a "neutral monism" by which matter and minds were both to be constructed from sense data, but in different ways. Intuitively, the sense data occuring as they do "in" a mind, are material to construct that mind, the sense data derived from an object from different points of view to constructthat object. Russell saw some support for this in the theory of relativity, and the fundumental importance of frames of reference in the new physics."
"These prominent examples are not the only use of the notion of construction in Russell's thought."
"In Whitehead's and Russell's Principia Mathematica the multiple relation theory of propositions is introduced by saying that propositions are "incomplete symbols"."
"Russell's multiple relation theory, that he held from 1910 to 1919 or so, argued that the constitituents of propositions, say
i. Desdemona loves Cassio
which is 0, are unified in a way that does not make it the case that they constitute a fact by themselves."
"Those constituents occur only in the context of beliefs, say,
ii. Othello judges that Desdemona loves Cassio.
"The real fact consists of a relation of
Belief holding between the constituents
Othello, Desdemona and Cassio."
B(o, d , L, c).
"Because one might also have believed
propositions of other structures, such as --
"B (o, F, a)"
there need to be many such relations B, thus
the "multiple" relation theory."
"Like the construction of numbers, this
construction abstracts out what a
number of occurrences of a belief have in common,
a believer and various objects in a certain order."
"The analysis also makes the proposition an incomplete symbol because there is no constituent in the analysis of
`x believes that p'
that corresponds to `p'"
"Russell also suggests that propositional functions are logical constructions when he says that they are "nothing", but "nonetheless important for that" (1918:96)."
"Propositional functions are abstracted from their values, propositions. The propositional function
x is human.
is abstracted from its values
Socrates is human
Plato is human.
etc.
"Viewing propositional functions as constructions from propositions which are in turn constructions by the multiple relation theory helps to make sense of the theory of types of propositional functions in Whitehead's and Russell's Principia Mathematica".
"The notion of "incomplete symbol" does not make as much sense as "construction" when applied to propositional functions and propositions."
"This usage requires a broadening of the notion."
---- From Russell to Carnap, Grice, and beyond:
"The notion of logical construction
had a great impact on
analytic philosophy."
"One line of influence was via the notion
of a contextual definition, or paraphrase, intended
to minimize ontological commitment and to
be a model of philosophical analysis."
"The distinction between the surface appearance of definite descriptions, as singular terms, and the fully analyzed sentences from which they seem to disappear was seen as a model for making problematic notions disappear upon analysis. The theory of descriptions has been viewed as a paradigm of philosophical analysis."
CARNAP EXPLICITLY CITED BY LINSKY:
"A more technical strand in
analytic philosophy was
influenced by the construction of
matter."
Or, as I prefer, 'physicalism' out of 'phenomenalism'.
Linsky writes:
"Carnap was attempted to carry out the
construction of matter from
sense data."
--- R. B. JONES WILL HAVE LOADS TO SAY ABOUT THAT:
i. It's a pragmatist prescription. Carnap sees phenomenalism as more primitive, but there are prior and prior.
ii. Etc.
----
Linsky goes on:
"Later Nelson Goodman continued the project."
And popularised in Oxford with reviews of "Structure of Appearance" by Dummett. Grice would similarly speak of "The syntax of illusion".
Linsky goes on:
"More generally, however, the use of
set-theoretic constructions became
widespread among philosophers,
and continues in the construction of set-theoretic models,
both in the sense of logic where they model formal theories, and as objects of interest in their own right."
--- And then there's Grice.
All in all, we congratulate Bernard Linsky, recommend interested readers to check his essay from the Stanford site, and report back with any insight as it dwells on our city of eternal truth.
References cited by Linsky include:
Carnap, R. , The Logical Structure of the World & Pseudo Problems in Philosophy, trans. R.George, Berkeley: University of California Press, 1967.
Goodman, N., The Structure of Appearance, Cambridge Mass: Harvard University Press, 1951.
Russell, B., 1905, "On Denoting", in Robert Marsh, Logic and Knowledge: Essays 1901-1950 , London: George Allen and Unwin, 1956, 39-56.
----1918, "The Philosophy of Logical Atomism" in The Philosophy of Logical Atomism , D.F.Pears, ed. Lasalle: Open Court, 1985, 35-155.
----1924, "Logical Atomism", in The Philosophy of Logical Atomism , D.F.Pears, ed., Lasalle: Open Court, 1985, 157-181.
----1912, The Problems of Philosophy , Oxford: Oxford University Press, reprinted 1967.
Whitehead, A.N., and Russell,B.: 1925, Principia Mathematica Vol.I., second ed., Cambridge: Cambridge University Press, 1925.
Wisdom, J., 1931, "Logical Constructions (I.).", Mind , XL , April, 188-216.
Related Entries
Cross-referneces by Stanford: "Russell, Bertrand | Russell's Paradox | definite descriptions | and Carnap, Rudolf"
Subscribe to:
Post Comments (Atom)
No comments:
Post a Comment