Reduction and Elimination in Philosophy and the Sciences

The Logic of Sensorial Propositions
Luca Modenese, Padova, Italy
1. Introduction
Propositions 6.375 and 6.3751 of the Tractatus state respectively
that there’s only a logical necessity and explain
it with an example based on colours and the logical impossibility
of having two different colours in the same place
because of the logical structure of colour itself.
Considering an object named ‘a’, it seems difficult to
accept that if a has two colours at the same time this
generates a genuine contradiction (despite of
Wittgenstein’s statement), because of the absurd situation
described by the first line of the truth table presented in
Table 1: that needs a syntax consideration about the use
of the language of colours to be rejected, and Wittgenstein
himself in Some Remarks on Logical Form spoke of
“exclusion” in contrast to contradiction analyzing such a
type of logical product.
Truth table of the presumed logical contradiction indicated in proposition 6.3751
In my opinion it is not possible to conclude anything about
logical necessity from a table like this, because it’s not
clear how to manage its first line by a logical point of view,
being the “exclusion” not a significant logical operation.
This difficulty is connected to a not satisfactory
logical inference theory based on tautologies that derive by
independent atomic propositions. I think anyway that it’s
possible to overcome this problematic situation using the
same principles accepted by Wittgenstein in the first phase
of his philosophical development.
This paper will try to find a new point of view to
consider Table 1 in order to make it clearly logically
significant.
2. Sensorial Spaces
It seems reasonable to call propositions like ‘This table is
white’ or ‘This wall is rough’ or ‘This food is salty’ sensorial
propositions. These propositions describe objects using
sense data and can be analyzed using 5 main classes
(each related to a sense) and their combinations. This
main classification is clear if a sensorial proposition is defined
as a statement of a sense impression or as a logical
product of statements of sense impressions 1 . ‘This table is
rough’ can be expressed in a form that matches the defini-
1 It will be not considered, at this level of the analysis, the influence if the
logical nature of the subject of the proposition.
230
tion of sense propositions if transformed in: ‘This colour
spot with the shape that I use to call ‘table’ is in my field of
vision and, when I move the pink shape that I use to call
‘hand’ near it, I feel a sensation of roughness’.
The distinction of the contributions of the single
senses in a sensorial proposition is possible also in more
subtle contexts. For instance, a proposition like ‘This
pullover is comfortable’ can be analyzed using fuzzy logic
to define in a quantitative manner the grades of each
sense involved.
The class of the characteristic properties related to a
percipient sense can be indicated with Wittgenstein’s
metaphor of space.
When a proposition states a certain property of a
material subject, the coordinates of a point in the space of
that sense will be fixed by it. This specification will exclude
other properties of that space because ‘a particle […]
cannot be in two places at the same time; that is to say,
particles that are in different places at the same time
cannot be identical.’ (6.3751). This proposition of the
Tractatus describes the main property of the structure of
sensorial spaces. Is this structure logical or empirically
derived? It depends on what we intend for logic: if we kept
in mind proposition 5.552 (‘The "experience" which we
need to understand logic is not that such and such is the
case, but that something is; but that is no experience.
Logic precedes every experience that something is so. It is
before the How, not before the What’) it is probably to be
considered pure logical.
3. Logic Of Sense Proposition
The logical form of the sensorial propositions will be found
using the method described in proposition 3.315 of the
Tractatus. Taken a certain proposition ‘Ra’, it is possible to
rewrite it in the following form, where all the properties of
the sensorial space are involved:
RSa∧ ~ TSa∧
~ USa
∧ etc
This form is allowed by proposition 6.3751 and propositions
3.311-3.313 (see also Anscombe 1963, chap. 6,
about expressions and symbols). The subscript ‘s’ stands
for a generic sensorial space. In details, the written logical
product is an attempt to stress with a different propositional
sign the structure of the sensorial space involved by the
proposition analyzed.
The expression above can be expressed also as:
RS a ~ RSa
∧ RS
∀ with ∈ S ∧ R ≠ R
RS S
where R could indicate every property of the space S
S
different from R.
The determination of the logical form of a sensorial
proposition follows three steps: 1) the generalization (i.e.
the substitution with a variable) of the subject, 2) the
generalization of the sense properties indicated in each
component of the logical product of the sensorial
proposition, 3) the generalization of the space in which the