*From*: "" <dmarc-noreply@xxxxxxxxxxxxx> (Redacted sender "Jlsperanza" for DMARC)*To*: lit-ideas@xxxxxxxxxxxxx*Date*: Wed, 16 Sep 2015 08:44:59 -0400

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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