Statistical and Inductive Probabilities
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Beginning with a survey of the essentials of sentence theory and of set theory, author Hugues Leblanc examines statistical probabilities (which are allotted to sets by von Mises' followers), showing that statistical probabilities may be passed on to sentences, and thereby qualify as truth-values. Leblanc concludes with an exploration of inductive probabilities (which Keynes' followers allot to sentences), demonstrating their reinterpretation as estimates of truth-values.
Each chapter is preceded by a summary of its contents. Illustrations accompany most definitions and theorems, and footnotes elucidate technicalities and bibliographical references.
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Statistical and Inductive Probabilities - Hugues Leblanc
THE LANGUAGES L
In this first chapter I construct a family of languages, the languages L, which is to consist of a master language, called L∞, and various fragments or sublanguages of L∞, respectively called L¹, L², L³, and so on ad infinitum. The sets allotted probabilities in Chapter 2, for one thing, can all be mentioned in the languages L; the sentences allotted probabilities in Chapters 3 and 4, for another, will all be sentences of these languages. First, I study the vocabulary and the grammar of the languages L (Section 1–2); then, I study the interpretation to be placed upon the languages L (Section 3–4); finally, I graft onto the languages L a set theory of sorts (Section 5). The material, made up in equal parts of definitions and marginal comments thereon, should be familiar to most; it may, however, be novel to some, who will perhaps welcome a briefing on the logic of sentences and the logic of sets.¹
1.THE VOCABULARY OF THE LANGUAGES L
My first group of definitions deals with the vocabulary of the languages L. The so-called primitive signs of L∞ are listed in D1.1; the primitive signs of the sublanguages of L∞ are listed in D1.2; the order in which some of these signs are listed or presumed to be listed in Dl.1–2 is, for technical convenience, given a name in D1.3. Further signs, to be known as the defined signs of L∞ and its sublanguages, will be supplied in Sections 2 and 5.
D1.1. The primitive signs of L∞ consist of the following:
(a) The two connectives ‘~’ and ‘⊃’
(b) The quantifier letter ‘∀*;
(c) The identity sign ‘=’
(d) The comma ‘,’
(e) The two parentheses ‘(’ and ‘)’;
(f) A finite set of predicates (each identified as a one-place predicate or a two-place predicate or a three-place predicate, and so on);
(g) A denumerably infinite set of individual constants;²
(h) A denumerably infinite set of individual variables.:
D1.2. For each N from 1 on, the primitive signs of the sublanguage LN of L∞ consist of the following:
(a) -(f) The various signs listed in Dl.1(a)-(D1.1(h)f);
(g) The first N individual constants listed in Dl.1(g);
(h) The individual variables listed in Dl.1(h).
D1.3. (a) The order in which the individual constants of L∞ and its sublanguages are presumed to be listed in Dl.1(g) and Dl.2(g) is the alphabetical order of those constants.
(b) The order in which the individual variables of L∞ and its sublanguages are listed in Dl.1(h) and Dl.2(h) is the alphabetical order of those variables³
A word of explanation on each one of clauses Dl.l (a)-(h) may be in order. The connective ~ in Dl.l (a) may be read ‘It is not the case thať or, when the occasion warrants, ‘Not.’ The connective ‘⊃’ in Dl.l (a) may be read ‘If … , then.’ The quantifier letter ‘∀’ in D1.1(b) may be read ‘For any.’ The identity sign ‘=’ in Dl.l (c) may be read ‘is identical with,’ ‘is the same as,’ or, when the occasion warrants, ‘is.’ The comma ‘,’ and the two parentheses ‘(’ and ‘)’ in Dl.l (d)-(e) are punctuation signs of a sort; the role played by ‘,’ should be clear from D2.2(a) below, the one played by ‘(’ and ‘)’ clear from D2.2(c)-(e) and the explanations appended thereto. The predicates presumed to be listed in D1.1(f) are expressions like ‘won the 1948 presidential election,’ ‘is one of…’s satellites,’ or ‘extends from … to,’ which, once supplied at appropriate places with nouns like ‘Truman,’ ‘Enceladus,’ ‘Saturn,’ ‘Canada,’ ‘the Atlantic Ocean,’ or ‘the Pacific Ocean,’ yield what are called closed sentences:⁴ ‘Truman won the 1948 presidential election,’ ‘Enceladus is one of Saturn’s satellites,’ ‘Canada extends from the Atlantic Ocean to the Pacific Ocean.’ Among the expressions in question, those which must be supplied with one noun (and one only) to yield a closed sentence are called one-place predicates; those which must be supplied with two nouns (and two only) to yield such a sentence are called two-place predicates; and so on. The individual constants presumed to be listed in D1.1(g) are nouns like the above ‘Truman,’ ‘Enceladus,’ and ‘Saturn’; they are meant to go with the predicates and the identity sign ‘=’ of the languages L; they are also meant to designate among themselves all the individuals making up what are called the universes of discourse of the languages L. Finally, the individual variables listed in D1.1(h) are to fill various roles: they will go (as individual constants do) with the predicates and the identity sign ‘=’ of the languages L and yield what are called open sentences such as ‘Peter knows w,’ ‘w is a man) ⊃ (w is mortal),’ ‘w = w,’ and so on; they will serve with the aid of the quantifier letter ‘∀’ to turn such open sentences into closed ones like ‘(∀w) (Peter knows w),’ ‘(∀w)((w is a man) ⊃ (w is mortal)),’ and (∀w)(w = w)’;⁵ and they will indiscriminately refer to or, as the technical phrase goes, range over the individuals designated in the languages L by the individual constants of those languages.
As the reader will have noticed, I do not produce the finitely many predicates of L∞; I simply assume that L∞ is fitted with such predicates. Neither do I produce the infinitely many individual constants of L∞; I simply assume that L∞ is fitted with such constants.⁶ The procedure, standard in studies of this kind, makes for greater generality. It also gives one greater leeway when it comes to illustrating sundry points about L∞ and its sublanguages.
So much for D1.1. According to D1.2 the sublanguages L¹, L², L³, and so on, of L∞ are like L∞ except for boasting only the first one, the first two, the first three, and so on, of the infinitely many individual constants of L∞. The sublanguages in question will serve various technical purposes; they will also be of theoretical interest in themselves. The universe of discourse of L¹ will, for example, be of size 1, that of L² of size 2, that of L³ of size 3, and so on, as opposed to the universe of discouse of L∞, which will be denumerably infinite in size or, as the matter is often put, of size 0·⁷
2.THE GRAMMAR OF THE LANGUAGES L
My next group of definitions deals with the grammar of (the languages) L.⁸ First, I single out from among all the sequences of primitive signs of L those which are to be known as the expressions of L (D2.1). Then, I single out from among all the expressions of L those which are to be known as the sentences of L (D2.2). Next, I sort the sentences of L into two groups: the closed sentences of L (D2.4) and the open sentences of L (D2.5).⁹ Finally, I define what I understand by an instance in L of a sentence of L (D2.6). The sorting of the sentences of L into closed sentences and open ones calls for an auxiliary notion, that of a free individual variable of L. I define the notion in D2.3.
The capitals with which D2.2, D2.6, and so on, fairly bristle are so-called metalinguistic variables. Four of those variables, ‘P,’ ‘Q,’ ‘R,’ and ‘S,’ range over the sentences of L; four more, ‘W,’ ‘X,’ ‘Y,’ and ‘Z,’ range over the individual signs (that is, the individual constants and the individual variables) of L; and a ninth one, ‘G,’ ranges over the predicates of L. Finally, sequences of metalinguistic variables and signs of L range over the results of substituting for the variables in the sequences the various expressions of L over which the variables range. In D2.2(a), for example, ‘G(W1, W2, … , Wn)’ ranges over the results of substituting a predicate of L for ‘G’ and individual signs of L for ‘W1,’ ‘W2,’ … , and ‘Wn’ in ‘G(W1, W2, … , Wn)’; in D2.2(d), ‘(P) ⊃ (Q)’ ranges over the results of substituting sentences of L for ‘P’ and ‘Q’ in ‘(P) ⊃ (Q)’; and so on.
D2.1. An expression of L is a finite sequence of primitive signs of L.
D2.2. (a) G(W1, W2, … , Wn), where G is an n-place (n ≥ 1) predicate of L and W1 W2, … , and Wn are n individual signs of L, is a sentence of L;
(b) W = X, where W and X are two individual signs of L, is a sentence of L;
(c) If P is a sentence of L, then so is ~ (P);
(d) If P and Q are two sentences of L, then so is (P) ⊃ (Q);
(e) If P is a sentence of L, then so is (∀W) (P), where W is an individual variable of L;
(f) No expression of L is a sentence of L unless its being so follows from (α)-(e).¹⁰
D2.3. (a) An occurrence of an individual variable W of L in a sentence P of L is bound in P if it is in a subsequence (∀W)(Q) of P;¹¹
(b) An occurrence of an individual variable W of L in a sentence P of L is free in P if it is not bound in P;
(c) Any occurrence of an individual constant W of L in a sentence P of L is free in P;
(d) An individual sign W of L is bound in a sentence P of L if at least one occurrence of W in P is bound in P;
(e) An individual sign W of L is free in a sentence P of L if at least one occurrence of W in P is free in P.
D2.4. A closed sentence of L is a sentence of L in which no individual variable of L is free.
D2.5. An open sentence of L is a sentence of L which is not closed.
D2.6. (a) Let P be a closed sentence of L. Then P is the instance of P in L.
(b) Let P be an open sentence of L; let W1, W2, … , and Wn be the n (n≥1) individual variables of L which are free in P; and let P* be like P except for containing, for each i from 1 to n, occurrences of an individual constant of L at all the places where P contains free occurrences of Wi. Then P* is an instance of P in L.
Among the expressions pronounced sentences in D2.2(a) are to be found, for example, ‘Cervantes is a Spanish writer,’ ‘Mary just came back from w,’ or ‘w lies between x and y,’ which I throw for the occasion into the form ‘Is a Spanish writer (Cervantes),’ ‘Just came back from(Mary, w),’ and ‘Lies between and(w, x, y),’ and so on. Among those pronounced sentences in D2.2(b) are to be found, for example, ‘The Morning Star = the Morning Star,’ ‘w = the 35th President of the U.S.A.,’ or ‘w = x,’ expressions commonly called identities. Among those pronounced sentences in D2.2(c) are to be found, for example, ‘~ (Philadelphia = the capital of Pennsylvania),’ ‘~((Has read(w,x)) ⊃ (Remembers(w,x))),’ or ‘~ ((∀w)(Floats on water(w))),’ expressions commonly called negations. Among those pronounced sentences in D2.2(d) are to be found, for example, ‘(Votes Democratic (Harry)) ⊃ (Votes Republican (John)),’ ‘((∀w)(w = w)) ⊃ (x = x),’ or ‘(Likes(Ann,w)) ⊃ ((∀x)((Is a friend of(x, w)) ⊃ (Likes(Ann,x)))),’ expressions commonly called conditionals. Among those pronounced sentences in D2.2(e) are to be found, for example, ‘(∀w)(w = w),’ ‘(∀w)((Is a swan(w)) D (Is white(w))),’ or ‘(∀w)((∀x) ((Is the father of (w, x)) ⊃ (~ (Is the father of (x, w))))),’ expressions commonly called universal sentences.
The parentheses that officially go around P in a negation ~ (P), around P and Q in a conditional (P) ⊃ (Q), and around P in a universal sentence (∀w)(P) are indispensable when P in the first case, P or Q in the second, and P in the third are conditionals. Without them, indeed, we could no longer tell ~ (P ⊃ Q) from ~ P ⊃ Q, (P ⊃ Q) ⊃ R from P⊃(Q ⊃ B), nor (∀w)(P ⊃ Q) from (∀w)P ⊃ Q. Otherwise, the parentheses in question may be dispensed with, and I shall frequently do so.
A few examples should shed light on the distinctions drawn in D2.3. Consider the three sentences:
and
The first two occurrences of ‘w’ in (1) are in a subsequence of (1) which opens with ‘(∀w)’—the subsequence ‘(∀w)(Is colored(w))’— and hence are bound in (1) ; the third occurrence of ‘w’ in (1), on the other hand, is free in (1). All three occurrences of ‘w’ in (2) are in a subsequence of (2) which opens with ‘((∀w))’—the subsequence (2)— and hence are bound in (2). Finally, both occurrences of V in (3) are free in (3). As for ‘w’ itself, it is both bound and free in (1), bound (and bound only) in (2), and free (and free only) in (3). Occurrences of individual constants, and individual constants themselves, are pronounced free in D2.3 for the sake of convenience.
According to D2.4, ‘Is west of (Toronto, Montreal) ⊃ ~ Is west of (Montreal, Toronto),’ ‘(∀x)(Is west of (Toronto, x) ⊃ ~ Is west of (x,Toronto)),’ and ‘(∀w)(∀x;)(Is west of (w, x) ⊃ ~ Is west of (x, w))’ are closed sentences, no individual variable being free in any one of the three sentences. According to D2.4–5, on the other hand, ‘Is west of(w, x) ⊃ ~ Is west of(x, w),’ ‘Is west of(w,Montreal) ⊃ ~ Is west of (Montreal, w),’ and ‘(∀w)(Is west of (w, x) ⊃ ~ Is west of (x,w)Y are open sentences, ‘w’ and ‘x’ being free in the first sentence, ‘w’ in the second, and ‘x’ in the third.
There are clearly two ways of turning an open sentence into a closed one. The first is to put the sentence in parentheses and preface the resulting expression with one universal quantifier, that is, one expression of the form (∀W) per individual variable W which is free in the sentence. We can, for example, turn the open sentence ‘Weighs as much as(w, x) ⊃ Weighs as much as(x, w)’ into a closed one by putting it in parentheses and prefacing the resulting expression with ‘(∀w)(∀x)’; the outcome will read: ‘(∀w)(∀x)(Weighs as much as(w, x) D Weighs as much as(x,w)).’ The other way is to substitute (occurrences of) individual constants for the free occurrences of the various individual variables which are free in the sentence. We can turn the open sentence ‘Excels at (Jack,w) ⊃ ~ (∀w) ~ Excels at(Jack,w),’ for example, into a closed one by substituting ‘poker’ (or any other individual constant we please) for the initial occurrence of ‘w’ in the sentence; the outcome will read: ‘Excels at(Jack, poker) ⊃ ~ (∀w) ~ Excels at (Jack, w).’
The outcomes of thus substituting individual constants of L for the free occurrences of the individual variables of L that are free in an open sentence of L are what I call the instances of the sentence in L (D2.6). An open sentence of L in which n individual variables of L, say W1, W2, … , and Wn, are free has in a sublanguage LN of L∞ (or, as I shall often put it, in LN) Nn instances. Note for proof that (1) W1 can be substituted for in N different ways; (2) for each one of the N ways in which W1 can be substituted for, W2 can be substituted for in N different ways; (3) for each one of the N² ways in which W1 and W2 can be substituted for, W3 can be substituted for in N different ways; and so on. In L∞ the same sentence has 0n instances and hence, 0n being equal to 0, 0 instances.¹² It proves convenient to extend the classical notion of an instance and allow, as I did in D2.6(a), a closed sentence of L to serve as its own instance in L.
Before closing this section, I graft onto L five so-called defined signs. The first three are sentence connectives, namely, ‘&,’ ‘v,’ and ‘≡,’ to be respectively read ‘and,’ ‘or’ (more explicitly, ‘and,’ ‘or’), and ‘if and only if.’ The fourth one is a quantifier letter, namely, ‘ ,’ to be read ‘For some’ or ‘There exists at least one · · · such that,’ The fifth one is a predicate, namely, ‘D,’ which, once supplied with two or more individual constants or variables, yields sentences like ‘D(w,x),’ ‘D(w,x,y),’ and so on. These sentences may be read ‘w and x are distinct from each other,’ ‘w, x, and y are distinct from one