Download Logic for Gottlob Frege and Bertrand Russell:

Philosophy of Language: Wittgenstein
Prof. Boedeker
Fall 2003
Wittgenstein’s Tractatus Logico-Philosophicus: an “underview”
Propositions are the semantic content of thought, and propositions stand essentially in inferential
relations to each other.
“The” logico-philosophical question: how are inferential relations essential to thought?
I. Frege: formal logic can answer this question by developing a logical notation (Begriffsschrift)
that allows for the expression of two things:
1. all propositions (i.e., everything true or false);
and
2. general logical laws governing all inferential relations among propositions. These
general logical laws are the axioms, or “primitive propositions”, of a deductive system.
Frege’s account of logical laws: they are analytic (i.e., truths not restricted to a particular
domain of knowledge) and a priori (i.e., general laws mentioning no particular facts);
i.e., general laws that are true of everything. (cf. Grundlagen, p. 4)
II. Russell’s supplement to Frege’s account of logical laws:
1. Logical laws are not only entirely general (with qualifications), but also necessary, i.e.,
true in all possible worlds, cannot be false.
2. Russell’s account of the “primitive propositions” of (pure) logic:
The primitive propositions of logic are true of “logical facts”, consisting of relations
holding among “logical objects”, including the objects referred to by the logical
constants (“and”, “or”, “not”, “if…, then___”). Logical objects are universals, much like
Plato’s Forms.
3. The concrete applications of logical laws are necessarily true because they also have
logical objects as their constituents.
III. Wittgenstein’s view of logical truth and proof:
A. Wittgenstein’s alternative account of the logical constants:
1. Wittgenstein expands Frege’s so-called “context principle”: “never to ask for the
meaning of a word in isolation, but only in the context of a proposition.”
Wittgenstein expands Frege’s context principle to hold two things:
a. The sense of a proposition is just the conditions under which it is true or false (cf.
4.022). (Only propositions have a sense.)
b. The meaning of an expression is just its contribution to the sense of the propositions
in which it occurs (cf. 3.31).
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2. Applying Wittgenstein’s version of the context principle to the logical constants
reveals:
a. Unlike proper names, the meaning of a logical constant in the context of a proposition
is not to refer to an object.
b. Rather, the logical constants are just signs for the performance of truth-operations on
propositions. A truth-operation takes a set of propositions as its base, and forms a
new proposition whose sense (i.e., truth- and falsity-conditions) depend on that of the
bases (5.234-5.2341, 5.3-5.32).
c. The proposition resulting from performing a truth-operation on a set of propositions is
a truth-function of those propositions that are the bases of the truth-operation. The
truth-value of the resulting proposition depends entirely on that of the bases.
3. Wittgenstein’s criticism of Russell’s account of the logical constants:
“My fundamental thought is that the ‘logical constants’ do not represent, that there can be
no representatives of the logic of facts” (4.0312).
“there are no ‘logical objects’ or ‘logical constants’ (in Frege’s or Russell’s sense)” (5.4).
4. The truth-functional structure can be displayed in a truth-table. A truth-table shows the
proposition’s truth-value for exactly all possible combinations of truth-values to its
component (or base) propositions.
The rows of a truth-table must therefore display exactly all possible combinations of
truth-value assignments to its component propositions.
(Note that this formulation – and this stage of my reconstruction of the argument – does
not yet require that the component (or base) propositions be elementary.)
Examples of truth-tables:
not-p
p and q
p p
p, q
p&q
T F
TT
T
F T
TF
F
FT
F
FF
F
p, q
TT
TF
FT
FF
p or q
pq
T
T
T
F
p, q
TT
TF
FT
FF
If p, then q
pq
T
F
T
T
B. Wittgenstein on logical necessity (and logical truth):
1. Wittgenstein’s criticism of Russell’s account of logical necessity:
Because there are no logical objects, there are no logical facts. Thus Russell’s account of
logical necessity is faulty.
2. Wittgenstein’s alternative account of logical necessity:
Logically true propositions are just tautologies, i.e., propositions whose truth-functional
structure makes them true for every possible combination of truth-values of their base
propositions.
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3. Examples of tautologies:
Either it’s raining or it’s not raining.
In logical symbolism:
r: It’s raining
r (r  ~r)
T T T FT
F F T TF
If both it’s raining and its raining implies that the streets get wet, then the streets get wet.
In logical symbolism:
r: it’s raining
s: the streets get wet
r, s
[(r  s) & r]  s
TT
TTT TT TT
TF
TFF FT TF
FT
FTT TT TT
FF
FTF FF TF
C. Wittgenstein on logical proof:
1. To prove that a proposition is logically true is to display it in a truth-table so that it
shows that it is a tautology.
“It is the peculiar mark of logical propositions that one can recognize that they are true
from the symbol alone, and this fact contains in itself the whole philosophy of logic”
(6.113).
2. This dispenses of the need for conceiving of logic as a deductive system with primitive
propositions, or axioms.
“It is always possible to construe logic in such a way that every proposition is its own
proof” (6.1265).
“All the propositions of logic are of equal status: it is not the case that some of them are
essentially primitive propositions and others essentially derived propositions” (6.127).
3. This also dispenses of Frege’s and Russell’s notion that general propositions have any
priority in logic.
“The mark of a logical proposition is not general validity… An ungeneralized
proposition can be tautological just as well as a generalized one” (6.1231).
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D. Wittgenstein’s answer to “the” logico-philosophical question:
1. All thoughts can be expressed in a proposition, and can be displayed in a truth-table.
2. A truth-table display of a proposition shows the inferential relations between the
proposition and all possible combinations of truth-values of its bases.
3. There are inferential relations among propositions because of the truthoperations performed on these propositions’ bases.
4. The inferential relations between any two propositions are also shown by a comparison
of the two propositions’ truth-tables (5.11-5.132, 5.14).
5. If proposition “t” entails proposition “u”, then it must be possible to construct a truthtable that shows that the conditional “t  u” is a tautology (6.1264). (In such a case, all
the truth-grounds of “t” are also truth-grounds of “u” [5.11-5.12]; “t” says more than “u”
[5.14].)
E. Wittgenstein’s philosophy of logic: some very basic commitments of the Tractatus:
1. Every proposition has a modal status, i.e., is either a tautology (thus necessary), a
contradiction (thus impossible), or “has sense” (thus contingent) (4.464, 5.525).
2. A proposition’s modal status is determined by its truth-functional structure, which is
the result of truth-operations performed on its component propositions.
3. A proposition’s truth-functional structure, and thus its modal status, can be shown in a
truth-table.
4. (A proposition q, as displayed in a given truth-table t, presupposes p) if and only if (p
is necessary for the combinations of truth-value assignments to q’s component
propositions displayed in t to be just the possible ones).
5. If presupposition p is a proposition, then truth-table t is a proof that q has the modal
status displayed in t only if p is also shown to be true.
6. This raises the possibility of an infinite regress of justification, which Wittgenstein
proposes to stop with the following requirement:
7. Every proposition must be able to be expressed in a truth-table (or equivalent
notation) so that it shows its truth-functional structure without presupposing the
truth of any proposition (including itself or its negation).*
8. This is related to Wittgenstein’s view of the purpose of a Begriffsschrift: a notation is
perspicuous (i.e., “is governed by logical grammar – by logical syntax” [3.325]) just in
case it can express all and only propositions, such that they show their truth-functional
structure without presupposing the truth of any proposition.
Cora Diamond would call this “metaphysical… in an extremely exiguous sense (it lays down philosophical
requirements on language and does not look at the phenomena of language)” (RS 20).
*
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IV. Some consequences of Wittgenstein’s philosophy of logic:
A. Every proposition must have a final level of analysis, in which all singular terms (i.e., terms
purporting to refer to one and only one object) occurring in each of the component propositions
are logically simple names, and in which the values of all the variables are logically simple
names.
1. Analysis on the model of physical complexity:
Take the proposition, “The broom is in the corner, but the handle is not attached to the
straw”, i.e., “b is in the corner, but h is not attached to s”. Now assume that b is a
complex object composed just of h’s being attached to s. In that case, the truth-table for
the proposition would be:
Cb, hAs
Cb & ~hAs
T T
T F F T
T F
T T T F
F T
F F F T
F F
F F T F
In order to rule out the second row as an impossibility, the truth-table display of the
proposition presupposes that
(Cb  hAs) ,
which is clearly a proposition if “Cb & ~hAs” is. And this would violate Wittgenstein’s
logico-philosophical assumption given in III E 7 above.
To remedy this situation, we can subject “Cb & ~hAs” to logical analysis, with the help
of the definition:
The broom is in the corner = the handle is in the corner, the straw is in the corner,
and the handle is attached to the straw
(in logical symbols:) Cb =def. ([Ch & Cs] & hAs)
The analyzed proposition thus becomes:
([Ch & Cs] & hAs) & ~hAs
T T T T T F F T
F F T F T F F T
T F F F T F F T
F F F F T F F T
T T T F F F T F
F F T F F F T F
T F F F F F T F
F F F F F F T F ,
which clearly shows the proposition to be a contradiction.
If “h” and “s” are logically simple names, then the truth-table of the proposition
expressed at this level of analysis does not make the kinds of presuppositions of the truthtable of the unanalyzed proposition.
Wittgenstein’s picture is that any proposition containing singular terms for complex
objects must be subject to logical analysis until all its singular terms become logically
simple names, serving just to refer directly to a single simple object.
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2. Analysis using Russell’s theory of definite descriptions:
Take a proposition, such as “The King of France in 2000 is not a King of France in
2000”, that contains a definite description. Using the following scheme of abbreviation:
k =def. the King of France in 2000
Kx = def. x is a King of France in 2000
,
the truth-table for the proposition would be:
Kk
Kk
T
FT
F
TF
.
In order to rule out the second row as an impossibility, the truth-table display of the
proposition presupposes “Kk”, which is clearly a proposition if “Kk” is. And this would
violate Wittgenstein’s logico-philosophical assumption given in III E 7 above.
To remedy this situation, we can subject “Kk” to logical analysis, with the help of the
following contextual definition of “k”:
(x)(Kx  x=k)
.
The analyzed proposition thus becomes:
(x)(Kx & (y)(Ky  y=x) & Kx)
which, since it includes “(x)(Kx & … Kx)”, is clearly a contradiction. This is shown
in the truth-table for any finite expansion of the proposition:
{(Ka & [(Ka  a=a) & (Kb  b=a) & (Kc  c=a) & …] & Ka) 
(Kb & [(Ka  a=b) & (Kb  b=b) & (Kc  c=b) & …] & Kb) 
(Kc & [(Ka  a=c) & (Kb  b=c) & (Kc  c=c) & …] & Kc)  …}
If the values of “x” and “y” are logically simple names, then the truth-table of the
proposition expressed at this level of analysis does not make the kinds of presuppositions
that the truth-table of the unanalyzed proposition does.
Wittgenstein’s picture is that any proposition containing singular terms that are disguised
definite descriptions must be subject to logical analysis until all its singular terms become
logically simple names, serving just to refer directly to a single simple object.
3. Putting together (1) and (2):
Any proposition containing singular terms that are disguised definite descriptions must be
subject to logical analysis until all its singular terms become logically simple names,
serving just to refer directly to a single simple object; and such that the values of its
variables are logically simple names. This is the Tractarian doctrine that all propositions
have a final logical analysis (3.2-3.261).
4. The concomitant metaphysics of logical atomism (1-2.063) – that the world consists of
states of affairs that are determinate (2.031) combinations (2.01), or configurations
(2.0272), of logically simple objects – is a necessary condition, or “presupposition”
(6.124), for propositions to have sense (i.e., to avoid being nonsensical). And a
proposition has a modal status only if it is not nonsensical. On the assumption that a
proposition has its modal status necessarily, it follows that simple objects subsist
necessarily (cf. 2.024-2.0271).
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B. The saying/showing distinction: “What can be shown, cannot be said” (4.1212). Every truthtable expression of a proposition has presuppositions. Every proposition must be able to be
logically analyzed so that its truth-table shows its truth-functional structure presupposing only
non-propositional “nonsense”, i.e., things that cannot be stated. Such presuppositions, however,
must “show themselves” to someone if they are to understand the truth-table.
C. One thing that can be shown but not said is logical form (4.12, 4.121).
1. The logical form of a set of objects is the range of possible situations in which they
occur (cf. 2.014, 2.0141).
2. The logical form of a set of n (independent) states of affairs is the range of 2n
“possibilities of existence and non-existence” for those states of affairs (4.27).
Propositions have logical form in common with reality (2.18, 2.2, 4.12); propositions “mirror”,
“show”, “display” the logical form of reality (4.121). This is because each row of a truth-table
represents a possible combination of truth-value assignments to the component propositions, and
all rows represent just the possible combinations of such truth-value assignments (cf. III A 4
above). Since logical form is presupposed by any truth-table display of a proposition, it cannot
be expressed in a proposition. The attempt to express logical form must thus be “nonsense”.
D. Statements whose logical expression would require modal terms are nonsense. This is
because modal terms allow the formation of statements that, were they propositions, could show
their truth-functional structure in a truth-table only by presupposing either themselves or their
negations. And this would violate Wittgenstein’s logico-philosophical assumption given in III E
7 above.
(Note that this formulation makes use of the following terminological distinction:
Proposition: something either true or false.
Statement: something purported, rightly or wrongly, to be a proposition.)
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Take, for example, “a=a” and “a≠b”. Were any of these statements a proposition, they could
show their modal status in a truth-table only by presupposing themselves.
If the statement “a=a” were a true proposition, we would have the following would-be truthtables:
a=a
a=a
a≠b
T
T
F
F
F
T
In this case, the truth-table of the statement “a=a” would presuppose the statement itself.
On the other hand, if the statement “a≠b” were a false proposition, we would have the following
would-be truth-tables:
a=a
a=a
a≠b
T
T
F
F
F
T
In this case, the truth-tables of the statement “a≠b” would also presuppose the statement itself.
E. One paradoxical consequence of Wittgenstein’s basic logico-philosophical views: tautologies
and contradictions are nonsensical (unsinnig). Thus there are no senseless (sinnlos) propositions;
all propositions are sinnvoll. This is because tautologies are presupposed by the truth-table
expression of any proposition. And since contradictions are simply the negations of tautologies,
they must be nonsensical as well.
In the Tractatus, Wittgenstein seems not to have acknowledged this as a consequence of his
views. (He does, however, write: “Tautologies and contradictions are the limiting cases – indeed
the dissolution – of the combination of signs” [4.466].)
Nevertheless, G.E. Moore reports that by 1932-33 Wittgenstein
expressly raised the questions “What is meant by the decision that a sentence makes or
does not make sense?” and “What is the criterion of making sense?” having said that, in
order to answer these questions, he must “plunge into something terrible”, and that he
must do this in order to “put straight” what he had just been saying, which, he said, he
had not “put correctly”. In trying to answer these questions or this question (for I think
he was using the two expressions to mean the same) he said many things, including the
statement that he had himself been “misled” by the expression “sense”; and he went on to
say that his present view was that “‘sense’ was correlative to ‘proposition’” (meaning,
apparently, here by “proposition” what he had formerly called “proposition in the
narrower sense”, i.e., “experiential proposition” [= sinnvoller Satz], thus excluding, e.g.,
mathematical “propositions” [= sinnlose Sätze])… (Philosophical Occasions, p. 66)
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