On Ramsey's `Silly Delusion' regarding Tractatus 5.53 Kai F. Wehmeier
I supply a semantics and a tableaux calculus for a rstorder logic based on Hintikka's strongly exclusive interpretation of the variables, and prove that the calculus is sound and complete with respect to the semantics. abstract.
1 Introduction. In Tractatus 5.53, Wittgenstein, with characteristic brevity, proposes a notational convention that is intended to make the identity symbol super uous in logical languages:
Identity of the object I express by identity of the sign and not by means of a sign of identity. Dierence of the objects by dierence of the signs. As it stands, the convention is ambiguous.
Hintikka (1956) suggests two
ways in which it might be interpreted; he refers to these as the `weakly' and `strongly exclusive interpretations of the variables,' respectively:
•
weakly exclusive:
From the range of a bound variable all objects are
excluded that are values of variables occurring free within the scope of the associated quanti er.
•
strongly exclusive:
From the range of a bound variable all objects
are excluded that are values of variables in whose scope the subformula governed by the associated quanti er occurs (where the scope of a bound variable is the scope of the quanti er that binds it, and the scope of a free variable is the entire formula in which it occurs free). In 5.531 through 5.5321, Wittgenstein proceeds to show how the contents of various formulas in Russellian notation can be rendered in a language employing the convention of 5.53. explains that Russell's
In particular, in 5.5321 Wittgenstein
∀x(F x → x = a) may be rewritten as (∃xF x → F a)∧
342
Kai F. Wehmeier
¬∃x∃y(F x ∧ F y).
As Hintikka (1956), p.230, observes, this example shows
that Wittgenstein cannot have had the strongly exclusive interpretation in mind.
For, under the strongly exclusive reading, the Wittgensteinian
formula says that, if anything other than there are no two objects
F.
x
and
y,
a
is
F,
a is also F , and that a, and both satisfying
then
both distinct from
This would be true in a two-element domain in which both elements
satisfy the predicate Under
the
F.
moniker
`Wittgensteinian
predicate
logic',
I
have
dis-
cussed the weakly exclusive interpretation in Wehmeier (2004) and in Wehmeier (2008); it shall not detain us further. While not Wittgenstein's intended convention, Hintikka's strongly exclusive reading is interesting in its own right, but also because Frank Ramsey appears to have misinterpreted Tractatus 5.53 as proposing such a reading. On November 11, 1923, Ramsey writes to Wittgenstein:
Have you noticed the di culty in expressing without = what Russell expresses by
∃x(F x ∧ x 6= a)? 1
After Wittgenstein supplies the formula
(F a → ∃x∃y(F x ∧ F y)) ∧ (¬F a → ∃xF x) , Ramsey sheepishly replies on December 27, 1923:
I didn't think there was a real di culty about ∃x(F x ∧ x 6= a), i.e., that it was an objection to your theory of identity, but I didn't see how to express it, because I was under the silly delusion that if an x and an a occurred in the same proposition the x could not take the value a. 2 Two points about this exchange seem remarkable. First, no matter whether one adopts the strongly or the weakly exclusive interpretation, Ramsey's troublesome formula can be rewritten simply as
∃x ((F a ∨ ¬F a) → F x), so Ramsey should have been able to nd an appropriate rendition even while under his `silly delusion.' Second, the `silly delusion' appears to be exactly Hintikka's strongly exclusive interpretation: No matter where in a formula a free variable
a occurs,
its value is excluded from the range of all bound variables occurring in the same formula. In the following, I will thus refer to a rst-order logical system using the strongly exclusive interpretation of the variables as Ramsey
predicate logic or simply R-logic . In what follows, I develop a model-theoretic semantics for R-logic, show its mutual interpretability with ordinary rst-order logic with identity
=
(FOL ), and provide a tableaux calculus, for which proofs of soundness and completeness are sketched.
1 Wittgenstein (1995), p.191. 2 Wittgenstein (1995), p.194.
On Ramsey's `Silly Delusion' regarding Tractatus 5.53
343
2 The Syntax and Semantics of R-Logic. The language
L
of rst-order logic (FOL), as understood here, has the
following primitive symbols. 1. the propositional connectives 2. the quanti er symbols
∀
and
¬, ∧, ∨, → ∃
3. countably many bound individual variables 4. countably many free individual variables 5. for each positive integer
n,
x, y, x1 , x2 , . . .
a, b, a1 , a2 , . . .
countably many
n-ary
predicate symbols
P n , Qn , Rn , P1n , P2n , . . . L-formulas are strings of the form Ra1 . . . an , where R is an n-ary ai are free variables. If A and B are L-formulas, then so are ¬A, A ∧ B , A ∨ B , and A → B . If F [a] is an L-formula in which x does not occur, then ∀xF [x] and ∃xF [x] are also L-formulas. The 3 language of R-logic is L. = = The language L of rst-order logic with identity (FOL ) arises from L by the addition of a designated binary predicate symbol =. Its formulas Atomic
predicate symbol and the
are constructed just like those of R-logic, except that there is the additional binary predicate symbol
=
available.
A sentence is a formula without occurrences of free variables.
U
U together with an n-ary relation n-ary predicate symbol R of L. When considering U as a = structure for FOL , the additional predicate symbol = is to be interpreted as true identity on U . A U -assignment is a mapping ν from the individual variables into U . A structure
RU
over
U
is a non-empty domain
for each
Before I give the o cial de nition of the semantics for R-logic, let me illustrate one of its peculiarities. From the range of a bound variable we are to omit the values of all variables in whose scope the associated quanti er occurs, where the scope of a free variable is the entire formula in which it
P a∧∀xP x. The bound x occurs within the scope of the free variable a, and hence its range must not include the value assigned to a. If, on the other hand, we consider ∀xP x not as a subformula of a larger context, but as a formula in its own right, the bound variable x does not occur within the scope of any other
occurs free. By way of example, consider the formula variable
variable, and should hence range over the entire domain. This means that when we break down
P a∧∀xP x into components for the purpose of semantic
3 In matters of syntax, we are largely following Schütte (1977), especially in the use of typographically distinguished free variables and of the theory of nominal forms.
344
Kai F. Wehmeier
evaluation, we must memorize the fact that the right conjunct originally occurred within the scope of
a,
or else we will interpret its bound variable
incorrectly. In R-logic, the range of a bound variable thus depends on the context of the associated quanti er's occurrence within the formula, and it is this feature of R-logic that necessitates a somewhat more cumbersome semantic apparatus than those of either
FOL=
or W-logic.
U be a structure with non-empty domain U . We regard every element u of U as a constant symbol with denotation u (i.e., we use elements Now let
from the domain as autonymous constants, again following Schütte 1977). The
L(U)-sentences
arise from the
rence of a free variable for
L= (U)-sentences).
a
L-formulas
by replacing every occur-
with a constant symbol
u∈U
(and analogously
Alternatively, we may describe the
L(U)-sentences L by the
as the closed formulas of the rst-order language obtained from
u, as an additional primitive symbol, for u ∈ U (and similarly for the L= (U)-sentences). If ν is a U = ν assignment, and A is an L-formula (or an L -formula), then A is the result of replacing every individual variable a in A by the individual constant ν(a). addition of an individual constant each element
The semantics to be de ned below is new in the sense that Hintikka (1956) only provides rules by means of which the quanti ers of R-logic can be successively eliminated in favor of standard quanti ers and the identity symbol, but does not specify an independent semantics for R-logic. We rst de ne the relation
U , relative to a nite set V
of R-truth of an L(U)-sentence in a structure U (these will represent the objects
of elements of
excluded from the range of bound variables). 1.
U hRu1 . . . un , V i
2.
U h¬A, V i
3.
U hA ∧ B, V i
i
i
hu1 , . . . , un i ∈ RU
U 6 hA, V i i
U hA, V i
and
U hB, V i,
and similarly for the
other Boolean connectives 4.
U h∀xF [x], V i
i for every
u ∈ U \ V , U hF [u], V ∪ {u}i
5.
U h∃xF [x], V i
i for some
u ∈ U \ V , U hF [u], V ∪ {u}i
L(U)-sentence A, we de ne U A to mean U hA, V i, where U that occur as constants in A. Since every Lsentence is an L(U)-sentence, we have thus de ned R-truth of L-sentences in structures U generally: An L-sentence A is R-true in U i U hA, ∅i; It is R-valid if it is R-true in every structure U . If S is a set of L-sentences, and F an L-sentence, we say that F is an R-consequence of S if every structure U in which every element of S is R-true also makes F R-true. Given an Given an
V
is the set of elements of
On Ramsey's `Silly Delusion' regarding Tractatus 5.53
345
L-formula A and a variable assignment ν , we say that A is R-true in U ν = under ν if U A . Standard Tarskian truth |= of L (U)-sentences in U is de ned as usual.
3 Interpreting R-Logic in FOL= Our rst aim is to show that to every
L= -sentence ψ(A)
L-sentence A there corresponds an U , U A i U |= ψ(A), i.e.,
such that, for all structures
that R-logic is interpretable in
FOL= .
Hintikka (1956), pp.229-231, achieves this by providing rules for rewriting R-logic quanti ers by means of standard quanti ers and identity; A given
L-formula
is successively transformed, by application of these rules, into
formulas containing both Ramseyan and standard quanti ers, as well as identity, until all Ramseyan quanti ers have disappeared.
Our recursive
translation avoids passing through such mixed formulas and must thus, for reasons explained in the preceding section, make an additional book-keeping eort with respect to free variables. We rst de ne a binary translation function
L-formula A
and a nite set
1. For atomic formulas
V
ψ,
taking as arguments an
of free variables.
A, ψ(A, V ) := A.
2.
ψ(¬A, V ) := ¬ψ(A, V )
3.
ψ(A∧B, V ) := ψ(A, V )∧ψ(B, V ), and similarly for the other Boolean connectives
4.
V ψ(∀xF [x], V ) := ∀x b∈V x 6= b → ψ(F [x], V ∪ {x}) , where ψ(F [x], V ∪ {x}) is short for ψ(F [a], V ∪ {a})xa , i.e., the result of replacing a with x in ψ(F [a], V ∪ {a}), where a is a fresh (neither in ∀xF [x] nor in V ) free variable.
5.
ψ(∃xF [x], V ) := ∃x
V
b∈V
x 6= b ∧ ψ(F [x], V ∪ {x})
By abuse of language, we also de ne a unary translation function
L-formula A, ψ(A) := ψ(A, FV(A)), where FV(A) set of free variables occurring in A. We write ν[V ] for the image of V the mapping ν , i.e., ν[V ] = {ν(a) : a ∈ V }. follows: For any
U be a structure, let V be a nite set of U -assignment, and let A be an L-formula. Then
Lemma 1 Let
be a
U hAν , ν[V ]i ⇔ U |= ψ(A, V )ν .
ψ
as
is the under
free variables, let
ν
346
Kai F. Wehmeier
The proof of the lemma proceeds by induction on the formula
A.
The atomic
case is trivial, as are the inductive cases involving the Boolean connectives. It remains to consider the quanti er cases; We discuss only the universal
A is of the form ∀xF [x]. Then U h∀xF [x]ν , ν[V ]i a u ∈ U \ ν[V ], U hF [u]ν , ν[V ] ∪ {u}i, i.e., U hF [a]νu , νua [V ∪ a {a}]i, where νu is just like ν except that it maps a to u, a being a fresh quanti er. Suppose that
i for every
free variable.
By the induction hypothesis, this is the case i for every a
u ∈ U \ ν[V ], U |= ψ(F [a], V ∪ {a})νu . Since a is not in V , this holds i ν a V a 6= b → ψ(F [a], V ∪ {a}) u . This, however, is for all u ∈ U , U |= b∈V V x ν equivalent to U |= ∀x b∈V x 6= b → ψ(F [a], V ∪ {a})a , q.e.d. Corollary 1 (Interpretability of R-logic in
L-formulas A, all U Aν ⇔ U |= ψ(A)ν .
1. For all
2. For all
L-sentences A
structures
U,
and all
and all structures
FOL= ) U -assignments ν ,
U , U A ⇔ U |= ψ(A).
V is FV(A), and part 2 follows L-sentences A, Aν is the same expression as
Part 1 is the special case of lemma 1 where from part 1 by noting that for
A.
4 Interpreting FOL= in R-Logic In this section, we will establish a converse to corollary 1, that is, we will
FOL=
show that
can be interpreted in R-logic.
There are two main dierences between our translation and the one proposed by Hintikka (1956). First, in the quanti er case, Hintikka's translation procedure requires a prior rewriting of the formula to be translated into a particular normal form (schema (6) in Hintikka 1956, pp.231-232), which our translation does without.
Second, the need for book-keeping of free
variables involved in our translation manifests itself in Hintikka's procedure through the requirement that, for the purpose of translating a formula
F,
all rewritings of quanti ers are to be imagined as taking place inside the
F
context of
(so that variables free in
F
but absent in quanti ed subformu-
las are still recognized as constraining the range of the subformulas' bound variables). We rst de ne a binary translation function sentences
A
and nite sets
V
φ,
taking as arguments
L= -
of free variables, as follows:
a and b are distinct free variables, φ(a = b, V ) := ⊥(a, b), ⊥(a, b) is some propositional contradiction in precisely the two free variables a and b (say Rab∧¬Rab, for some binary relation symbol R).
1. Where where
On Ramsey's `Silly Delusion' regarding Tractatus 5.53 2.
φ(a = a, V ) := >(a),
where
>(a) a
in precisely the one free variable predicate symbol
347
is some propositional tautology
P a ∨ ¬P a,
(say
for some unary
P ).
3.
φ(A, V ) := A
4.
φ(¬A, V ) := ¬φ(A, V )
5.
φ(A ∧ B, V ) := φ(A, V ) ∧ φ(B, V ), and similarly for the other Boolean
for all other atomic formulas
A.
connectives. 6.
V φ(∀xF [x], V ) := ∀xφ(F [x], V ∪ {x}) ∧ b∈V φ(F [b], V ), where φ(F [x], V ∪ {x}) is short for φ(F [a], V ∪ {a})xa , a being a fresh free variable.
7.
φ(∃xF [x], V ) := ∃xφ(F [x], V ∪ {x}) ∨
W
b∈V
φ stand for φ(A) := φ(A, FV(A)).
By abuse of language, we also let tion de ned as follows:
A FV(A),
Lemma 2 Let
be an
including
let
1-1
on
V.
U
L= -formula,
let
V
φ(F [b], V )
the unary translation func-
be a nite set of free variables
be a structure, and let
ν
be a
U -assignment
that is
Then
U |= Aν ⇔ U φ(A, V )ν . The proof proceeds by induction on the formula
A.
Only two cases are
of interest.
a = b, with distinct free variables a and {a, b}. Also U 6 ⊥(a, b)ν . Second, suppose A is of the form ∀xF [x] (the existential case is analν ν ogous). We have U |= ∀xF [x] i for every u ∈ U , U |= F [u] , i.e., for a νu all u ∈ U , U |= F [a] , where a is a fresh free variable. This in turn is First, suppose
b. U 6|= (a = b)ν ,
A
is of the form
because
ν
is
1-1
on
equivalent to: (*) For every a νu .
a
u ∈ U \ ν[V ], U |= F [a]νu ;
and for every
u ∈ ν[V ],
U |= F [a]
By the induction hypothesis, the rst conjunct in (*) holds i for every a
u ∈ U \ν[V ], U φ(F [a], V ∪{a})νu , i.e., i U φ(∀xF [x], V )ν . The second ν conjunct of (*) holds i for every b ∈ V , U |= F [b] , and hence by induction V ν hypothesis i U b∈V φ(F [b], V ) . Putting things together, we see that ν (*) is equivalent to U φ(∀xF [x], V ) , q.e.d.
348
Kai F. Wehmeier
Corollary 2 (Interpretability of
FOL=
in R-Logic)
L= -formulas A, all structures U , and 1-1 on FV(A), U |= Aν i U φ(A)ν .
1. For all are
2. For all
L= -sentences A
and all structures
all
U -assignments ν
U , U |= A
i
that
U φ(A).
Part 1 is an immediate consequence of the lemma, and part 2 follows from part 1 by noting that every variable assignment is set of free variables, and that for sentences
A, Aν
1-1
on the empty
is the same expression as
A.
5 A Tableaux Calculus for R-Logic Before we embark on the presentation of our tableaux calculus, it is perhaps appropriate to discuss brie y the calculus suggested by in Hintikka (1956) pp.235-236, for the system sketched there for the strongly exclusive interpretation (i.e., R-logic) appears to be unsound. In order to see this, it is helpful to review some of Hintikka's apparatus, since his calculus is rather unlike any of those in use today. tive rules are framed in terms of a metalogical relation among called equivalence, which we shall write
⇔.
The deduc-
L-formulas
This relation is by stipulation
L-formulas equivalent in the sense of ⇔ are considered inter⇒ is de ned as follows: A ⇒ B i A ⇔ A ∧ B . An L-formula A is said to be provable if ¬A ⇒ A and refutable if A ⇒ ¬A. Hintikka's rule (11)(c) (p.235) tells us that ∀xP x ⇒ P y , i.e., that ∀xP x ⇔ ∀xP x ∧ P y . So the L-formulas on the left and on the right are transitive, and
changeable in all contexts. A consequence relation
interchangeable irrespective of context; in particular, then,
(∗) ∀xP x ∧ ¬P y ⇔ (∀xP x ∧ P y) ∧ ¬P y. (∗) is tautologically equivalent to the L-formula (∀xP x ∧ ¬P y) ∧ ¬(∀xP x ∧ ¬P y) and also contains the same free variables
The right hand side of
as the latter; hence by Hintikka's rule (9), we obtain
(∗∗) From
(∗)
(∀xP x ∧ P y) ∧ ¬P y ⇔ (∀xP x ∧ ¬P y) ∧ ¬(∀xP x ∧ ¬P y). and
(∗∗)
it follows by the transitivity of
⇔
that
∀xP x ∧ ¬P y ⇔ (∀xP x ∧ ¬P y) ∧ ¬(∀xP x ∧ ¬P y) from which, by de nition of
⇒,
we obtain
∀xP x ∧ ¬P y ⇒ ¬(∀xP x ∧ ¬P y),
On Ramsey's `Silly Delusion' regarding Tractatus 5.53 i.e., that
∀xP x∧¬P y
is refutable. This
says that everything other than
y
is
P,
349
L-formula, however, is satis able (it and y itself isn't P ). An analogous
result holds for provability instead of refutability: In Hintikka's calculus, the 4
L-formula ∀xP x → P y
is provable , yet it is not R-valid on any reasonable
de nition of validity for R-logic (it fails to hold, for instance, in any oneelement domain where
P
is interpreted by the empty set, and
y
is assigned
the sole member of the domain as its value). If the preceding argument is correct and Hintikka's calculus is indeed unsound for R-logic, then our tableaux system constitutes the rst sound and complete proof procedure for R-logic. While the tableaux calculus introduced below is in many ways similar to the Boolos-Burgess calculus of Boolos (1984) for
FOL
and to the W-
procedure of Wehmeier (2008) for Wittgensteinian predicate logic, there is one obvious dierence. The need for keeping tabs on free variables, which manifested itself already in the semantics for R-logic, also necessitates the
hA, V i of L-formulas A and nite sets V L-formulas. Note that in the context of tableaux, it is always understood that hA, V i is a pair consisting of an L-formula A and a set V of free variables, whereas in the context of the semantics, A is taken to be an L(U)-formula and V a subset of the domain of U . Given a U -assignment ν , there is of course an obvious correspondence
labeling of tableaux with pairs
of free variables, rather than just
between the two notions. The tableaux rules can be given schematically as follows (add as many of the usual propositional rules as you like). 1. Closure:
h¬P, V i hP, W i × where
P
is any atomic
L-formula.
2. Double Negation:
h¬¬A, V i hA, V i 4 One can show generally that, if A is refutable, ¬A is provable. We've just seen that ∀xP x ∧ ¬P y is refutable, hence its negation, which is tautologically equivalent to ∀xP x → P y , is provable.
350
Kai F. Wehmeier
3. Conditional:
hA → B, V i h¬A, V i|hB, V i 4. Negated Conditional:
h¬(A → B), V i hA, V i h¬B, V i 5. Negated Universal:
h¬∀xF [x], V i h∃x¬F [x], V i 6. Negated Existential:
h¬∃xF [x], V i h∀x¬F [x], V i 7. Universal Instantiation:
h∀xF [x], V i hF [a], V ∪ {a}i where
a
is a free variable that does not occur in
(a) there is a pair that
a ∈ W,
(b) for all pairs
hB, W i
F , a 6∈ V ,
and either
on the branch under consideration such
or
hB, W i
on the branch under consideration,
W = ∅.5
8. Existential Instantiation:
h∃xF [x], V i hF [b1 ], V ∪ {b1 }i| . . . |hF [bk ], V ∪ {bk }i|hF [a], V ∪ {a}i 5 It is possible that there is no free variable a not occurring in F and V even if case (2) doesn't obtain (viz., if all the elements of free-variable sets on the branch are also in V ). In such a situation, universal instantiation is not applicable.
On Ramsey's `Silly Delusion' regarding Tractatus 5.53
351
S is the union of all sets W such that hB, W i occurs on the branch under consideration, then {b1 , . . . , bk } = S \ V , and the free variable a is new to the branch under consideration (i.e., for any hB, W i on the branch, a does not occur in B and a 6∈ W ). with the following proviso: If some pair
We say that a branch in a tableau is closed if it contains pairs
h¬P, W i, where P
is atomic and
V
and
W
A
if all its branches are closed. A tableau for
h¬A, FV(A)i.
A closed tableau for
A
hP, V i and
are arbitrary. A tableau is closed is a tableau beginning with
is also called a proof of
A.
6 Soundness Let
U
be a structure, let
variables, let
S
A
L-formula, let V be a nite set of free hB, W i where B is an L-formula and W a
be an
be a set of pairs
nite set of free variables. We make the following de nitions. 1. The pair is
1-1
on
hA, V i is R-satis able V , U hAν , ν[V ]i.
L-formula A
2. The
in
U
is R-satis able in
if for some
U
if
U -assignment ν
hA, FV(A)i
that
is.
S is simultaneously R-satis able if there is a U -assignment ν S 1-1 on {W : for some L-formula B, hB, W i ∈ S} and such U hB ν , ν[W ]i for every hB, W i ∈ S .
3. The set that is that 4. Let
T
L-formulas. Then T {hB, FV(B)i : B ∈ T } is.
be a set of
if the set
is simultaneously R-satis able
5. A branch in a tableau is simultaneously R-satis able if the set of all pairs
hB, W i
occurring on it is.
Our aim in this section is to prove the following
U be a structure, let V be a nite set A be an L-formula such that hA, V i is R-satis able in U . Then every tableau starting with hA, V i has at least one branch that is simultaneously R-satis able in U . Lemma 3 (Soundness Lemma) Let
of free variables, and let
A is a senh¬A, ∅i is not R-
The soundness theorem follows readily from the lemma: If
h¬A, ∅i structures U .
tence and some tableau begun with satis able, hence
U A
for all
closes, then
The proof of the soundness lemma proceeds by way of induction along the generation of tableaux.
Since
the induction basis is trivial.
hA, V i
is R-satis able in
U
by assumption,
Applications of the propositional rules and
352
Kai F. Wehmeier
the negated quanti er rules are easily handled by means of the induction hypothesis. Let us thus focus on the quanti er instantiation rules. First, the universal case.
Suppose given a simultaneously R-satis able
T ∗ arise from T by extending B to B ∗ according to the universal instantiation rule. That is, there is a pair h∀xF [x], V i on B , and B ∗ results from B by writing the pair hF [a], V ∪ {a}i below its last node. Here, a does not occur in F , a 6∈ V , and one of the
branch
B
in a tableau
T,
and let
following two cases obtains:
hB, W i on B , a ∈ W . In this case, assuming that U -assignment that simultaneously R-satis es B , ν will also ∗ simultaneously R-satisfy B : Since h∀xF [x], V i occurs on B , we know ν that U hF [u] , ν[V ]∪{u}i for every u ∈ U \ν[V ]; But because a 6∈ V , we know that ν(a) is such a u. Also, because ν is 1-1 on the union of all free-variable sets occurring on B , and no additional free variables ∗ occur on B , the requisite injectivity condition is still ful lled.
1. For some pair
ν
is the
B are empty. In this case, too, ν B ∗ . We already know that it satis es ν every pair on B . But since h∀xF [x], ∅i is on B , U hF [u] , {u}i for every u ∈ U , in particular, for ν(a). The injectivity condition is trivial.
2. All free-variable sets occurring on itself simultaneously R-satis es
B , simultaneously R-satis ed ν , contains the entry h∃xF [x], V i and has been extended into k + 1 branches B1 , . . . , Bk+1 according to the existential instantiation rule. We know the fresh free variable a occurring for the rst time in the last entry on Bk+1 , i.e., in hF [a], V ∪ {a}i, is not an element of V . We also know that since ν R-satis es h∃xF [x], V i, there is an element u ∈ U \ ν[V ] ν such that U hF [u] , ν[V ] ∪ {u}i. Fix such an element u. We distinguish Second, the existential case. Say the branch
by the assignment
two cases:
bi that occurs in one of the free-variable sets V , ν(bi ) = u. I claim that ν then simultaneously R-satis es Bi : Clearly ν R-satis es hF [bi ], V ∪ {bi }i in U ; It simultaneously R-satis es B by induction hypothesis, and the injectivity
1. For some free variable on
B,
but not in
condition also follows from the induction hypothesis, as no new variables occur on
νua
Bi .
Bk+1 in U : It hF [a], V ∪{a}i, and it equally clearly simultaneously a R-satis es B , because a is new to B . Further, νu is 1-1 on the union of all free-variable sets on B and {a} because a is new to B and u is not the ν -image of any free variable occurring in any of the free-variable sets on B .
2. Otherwise,
simultaneously R-satis es the new branch
clearly R-satis es
On Ramsey's `Silly Delusion' regarding Tractatus 5.53
353
7 Completeness The completeness proof follows the usual strategy of constructing a model
U
for an open branch
B
in a `systematic' tableau, where the domain of
the set of free variables occurring on
B.
U
is
In analogy with the Hintikka sets
of Smullyan (1968), we de ne Ramsey sets, or R-sets for short, as follows.
hA, V i, where A is an L-formula and V a nite U be a non-empty set of free variables such that S {FV(A) ∪ V : hA, V i ∈ S} ⊆ U . Then S is an R-set for U if, whenever Let
S
be a set of pairs
set of free variables. Let
1.
hA, V i, hB, W i ∈ S
2.
h¬¬A, V i ∈ S ,
3.
hA ∧ B, V i ∈ S ,
4.
h¬(A ∧ B), V i ∈ S ,
and
then
A
B
is atomic, then
is not
¬A;
hA, V i ∈ S ;
then
hA, V i, hB, V i ∈ S ;
then
h¬A, V i ∈ S
or
h¬B, V i ∈ S ;
and similarly
for the other Boolean connectives; 5.
h∀xF [x], V i ∈ S ,
6.
h¬∀xF [x], V i ∈ S ,
7.
h∃xF [x], V i ∈ S ,
8.
h¬∃xF [x], V i ∈ S ,
then for every then
h∃x¬F [x], V i ∈ S ;
then for some then
b ∈ U \ V , hF [b], V ∪ {b}i ∈ S ;
b ∈ U \ V , hF [b], V ∪ {b}i ∈ S ;
h∀x¬F [x], V i ∈ S .
Lemma 4 (Hintikka's Lemma for R-logic) If
S
is simultaneously R-satis able in some structure
S is an R-set for U , then U with domain U .
S be an R-set for U . We turn U into a structure U by n-ary relation symbol P , the relation PU as follows: ha1 , . . . , an i ∈ PU :⇔ for some V, hP a1 . . . an , V i ∈ S . Let ν be the U -assignment that maps every variable a in U to itself, and maps every other variable to an arbitrarily selected element of U . Note that, since ν is the identity function on all relevant free variables, in what follows ν we needn't distinguish between L-formulas A and L(U)-sentences A , nor between nite sets V of free variables and their images ν[V ] under ν . Toward a proof, let
de ning, for any given
We claim that (a) whenever
hA, V i ∈ S ,
(b) whenever
h¬A, V i ∈ S ,
then
U hA, V i,
then
and
U 6 hA, V i.
354
Kai F. Wehmeier
L-formula A.
A, U hP a1 . . . an , V i. Then, by de nition of U , for some W , hP a1 . . . an , W i ∈ S . Since S is an R-set, it follows that h¬P a1 . . . an , V i 6∈ S . If A is of the form ¬B , then part (a) follows from the induction hypothesis, part (b). For part (b), if h¬¬B, V i ∈ S , then since S is an R-set, also hB, V i ∈ S , and so by the induction hypothesis, part (a), U hB, V i, i.e., U 6 h¬B, V i. The other Boolean connectives can be handled analogously, The proof proceeds by induction on the
U.
part (a) follows from the de nition of
For atomic
For part (b), suppose
using the requisite induction hypotheses.
A is ∀xF [x]. For part (a), assume h∀xF [x], V i ∈ S . Since S U , we know that for all b ∈ U \V , hF [b], V ∪{b}i ∈ S . Thus by part (a) of the induction hypothesis, for all b ∈ U \ V , U hF [b], V ∪ {b}i, i.e., U h∀xF [x], V i. For part (b), suppose h¬∀xF [x], V i ∈ S . Since S is an R-set for U , h∃x¬F [x], V i ∈ S , and thus for some b ∈ U \ V , h¬F [b], V ∪ {b}i ∈ S . Part (b) of the induction hypothesis then implies that for some b ∈ U \ V , U 6 hF [b], V ∪ {b}i; hence U 6 h∀xF [x], V i. The Now suppose
is an R-set for
existential case is entirely analogous. Q.e.d. Next we describe a systematic procedure for generating tableaux.
For
the purpose, we will assume that the free variables come ordered in an
ω -sequence. To systematically generate tableaux, iterate the following steps as long as possible. 1. Pick, if possible, the highest and leftmost pair checkmark next to it and such that
A
hA, V i
that has no
is non-atomic.
2. If the previous step is inapplicable, pick the highest and leftmost pair
hA, V i, with A of the form ∀xF [x], that has the least number of checkmarks, and to which universal instantiation can be applied with respect to at least one branch passing through it. 3. If both previous steps are inapplicable, the systematic procedure terminates. 4. Apply the applicable rule to the chosen pair form
∀xF [x],
hA, V i.
If
A
is of the
pick the instantiating free variable so that it is the rst
eligible free variable
a
such that the resulting instance
yet occur on the branch at hand. branch, pick the rst free variable. 5. Put a check mark next to the pair
F [a]
does not
If no free variables occur on the
hA, V i.
On Ramsey's `Silly Delusion' regarding Tractatus 5.53
355
We now de ne a systematic tableau to be either 1. the result of the systematic procedure, if it terminates after nitely many iterations, or 2. if the systematic procedure does not terminate after any nite number of iterations, the limit (set-theoretic union) of the results of all nite stages of the systematic procedure.
Lemma 5 Every non-closed branch
for
UB :=
S
{W : for
some
B
L-formula
of a systematic tableau is an R-set
B,
hB, W i
occurs on
B}.6
For a proof, note that condition (1) on R-sets is satis ed because
B
is an
open branch. The other conditions, with the possible exception of (5), are obviously satis ed by construction. Now consider condition (5). Suppose that
b
h∀xF [x], V i occurs on B and that b ∈ UB \ V . The free variable UB through some application of either universal or existential
entered
instantiation. The systematic procedure ensures that, at some point after the rst occurrence of
h∀xF [x], V i
b
on
B,
universal instantiation will be applied to
with respect to the free variable
also occur on
B,
b.
Hence
hF [b], V ∪ {b}i
will
which concludes the proof of the lemma.
The completeness theorem now follows as usual:
Theorem 1 If a sentence
A
is R-valid, then there exists a proof of
a closed tableau beginning with
For if there are no closed tableaux for tableaux for
A.
A
(i.e.,
h¬A, ∅i). A, then there are no closed systematic A is nite, it follows immedi-
If the systematic tableau for
ately that it must contain a non-closed branch. If the systematic tableau for
A
is in nite, then by König's lemma, it contains an in nite branch, which
is clearly non-closed. In either case, by the previous lemma, the systematic tableau for
A
contains a branch
B
that is an R-set for
lemma for R-logic, there is thus a structure R-true. Hence
h¬A, ∅i
is satis able, and
A
U
UB .
By Hintikka's
in which all pairs on
B
are
is not R-valid.
6 Note that U cannot be empty: Let the initial node be hA, FV(A)i. If A isn't an B L-sentence, FV(A) and hence UB isn't empty. Otherwise, since L contains no individual constants, A must contain a quanti er. Thus, after nitely many steps, the systematic
procedure will attempt to apply either universal or existential instantiation in each open branch. Now suppose all free-variable sets were empty. In that case, (i) existential instantiation does not result in branching and introduces a fresh free variable into the branch, and (ii) universal instantiation is applicable (condition (b) of the rule being ful lled) and similarly introduces a fresh free variable into the branch.
356
Kai F. Wehmeier
BIBLIOGRAPHY
Boolos, G. (1984). Trees and Finite Satis ability: Proof of a Conjecture of Burgess, Notre Dame Journal of Formal Logic, vol. 25, no. 3, pp.193-197. Hintikka, J. (1956). Identity, Variables, and Impredicative De nitions, Symbolic Logic, vol. 21, pp.225-245. Schütte, K. (1977).
Proof Theory,
Smullyan, R.M. (1968).
Journal of
Springer-Verlag, New York.
First-Order Logic,
Springer-Verlag, New York.
Wehmeier, K.F. (2004). Wittgensteinian Predicate Logic, vol. 45, no. 1, pp.1-11.
Formal Logic,
Notre Dame Journal of
Wehmeier, K.F. (2008). Wittgensteinian Tableaux, Identity, and Co-Denotation, Erkenntnis, vol. 69, no. 3, pp.363-376. Wittgenstein, L. (1922). Tractatus Routledge and Kegan Paul, London. Wittgenstein, L. (1995).
Logico-Philosophicus,
transl. C. K. Ogden,
Cambridge Letters: correspondence with Russell, Keynes, Moore, Ramsey, and Sraa, ed. B. McGuinness and G. H. von Wright, Blackwell,
Oxford.
![wittgenstein - tractatus logico philosophicus [portugues].pdf ...](https://p.pdfkul.com/img/300x300/wittgenstein-tractatus-logico-philosophicus-portug_5a566e211723dd9d71e7ccc6.jpg)
![wittgenstein - tractatus logico philosophicus [portugues].pdf ...](https://p.pdfkul.com/img/300x300/wittgenstein-tractatus-logico-philosophicus-portug_59cda7651723dd26d6e22540.jpg)
