I'm not sure how corpuscularians deal with 'every' (i.e. Russell and
Witters) but Pears, in his intro to Russell's philosophy of atomism, is looking
for an example, and chooses indeed an example that features the word
'every'.
For Pears writes -- as cited by McEvoy but also _simpliciter_:
"The two views may be combined without any incoherence. They share the same
conclusion, logical atomism, and they both incorporate the assumption of a
general correspondence between language and reality. They differ only in
their methods of establishing the conclusion. According to one view, it is
established empirically, like the conjecture that every even number is the
sum of two prime numbers, while the other view takes it to be provable, as
it is hoped that the arithmetical conjecture will be proved one day. So
Russell was not wrong when he allowed both views to be represented in his
treatment of logical atomism."
McEvoy seems to find this Oxford-focused, and although we have dealt with
broader issues, we may point now to this very 'every".
McEvoy refers to one of my quotations, to the effect that "the upshot of
the above discussion, albeit based on a single case study, is that
mathematicians ought not to — and in general do not — give weight to
enumerative
induction per se in the justification of mathematical claims."
-- which while relevant, may not be specific enough in dealing with Pears's
simile.
I rather focus on 'every'. In his second William James lecture on logic and
conversation (Grice 1967) Grice cites one one-place operator (negation,
English "not") and a few two-place ones (conjunction, English "and";
disjunction, English "or"; conditional, English "if") and THREE quantifiers:
the
universal quantifier (English "all"), the existential quantifier (English "at
least some") and the definite descriptor (English "the") on which Pears
(reading Russell) spends all the time that 'the' deserves: -- "the king of
France is bald". But McEvoy's commentary now focuses, and rightly so, on the
'every':
In a message dated 9/16/2015 5:27:43 A.M. Eastern Daylight Time,
donalmcevoyuk@xxxxxxxxxxx writes:
JLS writes many things that unwittingly corroborate my conjecture that
there is no proper justification for Pears writing without any reference to
Popper.
These things also corroborate my explanation that Pears is being 'true to
type' and that there are not valid intellectual reasons for such an
attenuated discussion (for example, JLS indicates that Pears would have been
under
the sway of Mill's attenuated view that we can "observe" 'mathematical
facts' in an empiricist way).
"[E]ven if we could "observe" an even number, we cannot "observe" every
even number in an infinite sequence, and Goldbach's conjecture covers every
even number: and moving from a finite sample to a generalization, about
'every' in a larger sample, raises issues of "induction"".
So the keyword I would bring in here is SUBSTITUTIONAL QUANTIFICATION and
having Pears using 'every' in its substitutional 'usage' (I won't say
"sense").
**************** INTERLUDE towards an interpretation of Pears's 'every' as
substitutional.
The model-theoretic interpretation of the language of quantificational
logic relies on the Tarski's definition of satisfaction -- that Popper learned
on a bench in Vienna's most prominent park -- in a model by an assignment
of values to the variables.
But the assignment of an object to a variable is never dependent on the
availability of a term, t, for the OBJECT [or as Russell and Pears might
prefer 'logical atom'] in the language for which one defines truth in a model.
For example, an existentially quantified formula ∃x A may well be true in a
model even if no "atomic" formula of the form A(t/y) is TRUE in the model.
Substitutional quantifiers Πα A and Σα A are interpreted differently --
the implicature being that 'every' has two uses: when used substitutionally
and when not. Guarded utterers (such as Pears was) must be having a
substitutional use in mind.
An interpretation associates with them not a domain of quantification, but
rather a substitution class, C, of linguistic expressions of an appropriate
syntactic category in the initial language.
The truth-conditions for substitutionally quantified sentences of the form
Σα A and Πα A may be given in terms of the truth-conditions of suitable
SUBSTITUTIONA INSTANCES of A(ϵ/α)
-- and let us revise them:
Euler -- he proved Goldbach's conjecture for 103 cases.
Desboves proved Goldbach conjecture for 104 cases.
Pipping proved Goldbach conjecture for 105 cases
Stein & Stein proved Goldbach's conjecture for 108 cases.
Granville proved Goldbach's conjecture for 1010 cases.
Deshouillers proved Goldbach's conjecture for 1014 cases.
Oliveira & Silva proved Goldbach's conjecture for 1018 cases.
We may assume that in the above, each occurrence of the substitutional
variable α has been replaced with a linguistic expression, ϵ, of the
appropriate syntactic category in the substitution class for the quantifier.
Let's start with Grice's "at least some":
A sentence of the form Σα A is true relative to a substitution class C
iff SOME SUBSTITUTION of the variable α for an expression ϵ in the
substitution class C for the quantifier, A(ϵ/α), given the value 1.
Let's now focus on Pears's "every":
A sentence of the form Πα A is true relative to a substitution class C iff
every substitution of the variable α for an expression ϵ in the
substitution class C for the quantifier, A(ϵ/α), is given the value 1.
This characterization of substitutional quantification allows for
substitutional variables of different syntactic categories, whether singular
terms,
predicates or sentences.
Indeed, substitutional quantification is often used to mimic quantification
into predicate and sentence position of the kinds discussed later in this
entry.
Early work on substitutional quantification was developed in Ruth Barcan
Marcus and S. A. Kripke -- whom Pears quotes as the important philosopher
Kripke is -- and constitutes what is perhaps the most influential discussion
of the subject.
A variety of authors have attempted to make use of substitutional
quantification in different areas of philosophy that range from ontology to the
philosophy of language and mathematics.
It is the use of substitutional quantification (as an interpretant for
'every') in the philosophy of mathematics (for cases like Pears's simile of
Goldbach's conjecture) that is most relevant here -- and the cross-reference
with 'induction' (that Russell admired as displayed by Peano).
The consistency of Peano Arithmetic can be proved by induction up to a
transfinite ordinal number.
The basic postulates of arithmetic contain the induction axiom. In
first-order formalizations of arithmetic, this is formulated as a scheme: for
each
first-order arithmetical formula of the language of arithmetic with one
free variable, one instance of the induction principle is included in the
formalization of arithmetic. Elementary cardinality considerations reveal that
there are infinitely many properties of natural numbers that are not
expressed by a first-order formula. But intuitively, it seems that the
induction
principle holds forall properties of natural numbers. So in a first-order
language, the full force of the principle of mathematical induction cannot
be expressed.
Suppose furthermore that A and B regard the collection of predicates for
which mathematical induction is permissible as open-ended, and are both
willing to accept the other's induction scheme as true.
Let us momentarily focus on singular terms.
To keep matters simple, consider an impoverished fragment of the language
of arithmetic with one constant, 0, read: “zero” and one functional symbol,
S, read: “the successor of”.
The domain of the intended interpretation consists of a set of natural
numbers, which are named by singular terms of the form 0, SS0, SSS0, …, etc.
If we now associate the class of all such terms to the substitutional
quantifiers Σα and Πα, a sentence like
Σα α=α
is evaluated as having value 1 ('true') in virtue of the truth of formulas
like 0=0, whereas
Πα α<α
is evaluated as having value 0 ('false') in virtue of falsity of formulas
like 0<0.
In general, a sentence of the form
Σα A
exhibits the same truth conditions as an infinitary disjunction of the
form:
A(0/α)∨A(S0/α)∨…∨A(SS…nS0/α)∨…
whereas a sentence of the form Πα A exhibits the same truth conditions as
an infinitary conjunction:
A(0/α)∧A(S0/α)∧…∧A(SS…nS0/α)∧…
The case of arithmetic is optimal because we have a name for each member
of the intended domain.
This means that in general, a quantified sentence of the form ∃x A will
have value 1 iff its substitutional counterpart Σα A likewise has value 1.
But in this respect, however, the language of arithmetic is the exception
and not the rule.
In real analysis, for example, there are too many objects in the domain to
have a name in a countable language.
In such a situation, we must confront the risk that a sentence of the form
∃x A
may be true even if Σα A remains with value 0 ('false') in virtue of the
lack of a true substitution instance of the form A(t/α).
This may happen, for example, if the objects that satisfy the open formula
A are not denoted by any singular term in the language.
Call the initial language “the object language”.
And call the language in which we explain the truth conditions of sentences
of the initial language “the metalanguage”.
In the metalanguage, we have explained the truth condition for Σα A in
terms of what looks like objectual quantification over linguistic expressions.
To acknowledge this is of course not to claim that the intended
interpretation of substitutional quantification is one on which it is merely
objectual quantification over linguistic expressions.
But just what the intended interpretation of the substitutional quantifier
might be has been the subject of intense controversy.
Indeed, some have each argued that there is no separate intended
interpretation we can understand independently from our grasp of objectual
quantification over linguistic expressions of the relevant sort.
Whatever the philosophical import of substitutional quantification, Kripke,
whom Pears cites and rightly extensively, makes plain that there is no
technical obstacle to introducing substitutional quantifiers into a given
language.
In what is perhaps the canonical treatment of substitutional
quantification, Kripke explains how to extend an interpreted language L into a
substitutional language LΣ equipped with substitutional quantification over a
class
of expressions in L.
One first expands the vocabulary of L with an infinite stock of variables α
1, α2, … and a substitutional quantifier Σ.
Kripke defines Π as the dual of Σ, where Πα A abbreviates: ¬Σα¬ A.
An "atomic" preformula is an expression that results from a sentence of L
when zero or more terms are replaced by a substitutional variable.
A form is an atomic preformula, where the replacement of its substitutional
variables with terms yields back a sentence.
We can now define a formula of LΣ recursively. A is a formula of LΣ iff
(i) A is a form of LΣ
or
(ii) A is ¬B -- where B is a formula of LΣ
or
(iii) A is (B→C)
-- where B and C are each formulas of LΣ or (iv) A is Σα B, where B is a
formula of LΣ. A sentence of LΣ is a formula without free substitutional
variables.
Kripke defines truth for sentences of LΣ recursively in terms of truth in
L.
If A is an "atomic" sentence of LΣ—which may well be a complex sentence of
L, A has value 1 in LΣ iff A has value 1 in L.
If A is a sentence of the form ¬B, then A is true in LΣ if, and only if, B
is not true in LΣ. If A is of the form (B→C), A is true in LΣ iff if, B is
not true or C is true in LΣ.
Finally, and more crucially, if A is of the form Σα B, then A is true in LΣ
if and only if B(ϵ/α) is true in LΣ for some ϵ in the substitution
class C associated with Σ.
Kripke then defines a sentence A of the language of pure substitutional
quantificational logic to be valid iff A comes out true no matter what base
language L and non-empty class of terms C we take as input for the
substitutional expansion and what predicates of L we substitute for the
predicates
of A.
In particular, Kripke notes that when the quantifiers of a valid sentence
of pure quantificational logic are suitably rewritten as substitutional
quantifiers, we obtain a valid sentence in the language of pure substitutional
quantificational logic.
----- END OF INTERLUDE ON 'every' as used substitutionally. (A further
keyword here may be intuitionism, since the idea of 'experience' that McEvoy
takes as basic as defining Empiricism may well be something that is shared
by Mathematical Intuitionism, and therefore, a reading of Goldbach's
Conjecture INTUITIONISTICALLY, alla Dummett, say, another Oxonian, may prove
useful).
McEvoy concludes his post: about "a way that makes clearer how
indefensible is the exclusion of Popper from Pears' narrow Oxbridge-focused
discussion."
Perhaps part of the blame is Russell, who calls himself a 'disciple' of the
most infamous monist (and thus anti-corpuscularianist) that Oxford every
produced: Bradley. If Isaiah Berlin once wrote about the early origins of
linguistic philosophy (in Berlin et al, essays on Austin) as originating in
the Thursday evening Play Group meetings at All Souls) perhaps we can
similarly speak of the Oxonian roots of Russell's philosophy.
So Pears is just being true to these, metaphorical, roots?
Cheers,
Speranza
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