Reduction and Elimination in Philosophy and the Sciences
Reduction and Elimination in Philosophy and the Sciences
Reduction and Elimination in Philosophy and the Sciences
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The Logic of Sensorial Propositions<br />
Luca Modenese, Padova, Italy<br />
1. Introduction<br />
Propositions 6.375 <strong>and</strong> 6.3751 of <strong>the</strong> Tractatus state respectively<br />
that <strong>the</strong>re’s only a logical necessity <strong>and</strong> expla<strong>in</strong><br />
it with an example based on colours <strong>and</strong> <strong>the</strong> logical impossibility<br />
of hav<strong>in</strong>g two different colours <strong>in</strong> <strong>the</strong> same place<br />
because of <strong>the</strong> logical structure of colour itself.<br />
Consider<strong>in</strong>g an object named ‘a’, it seems difficult to<br />
accept that if a has two colours at <strong>the</strong> same time this<br />
generates a genu<strong>in</strong>e contradiction (despite of<br />
Wittgenste<strong>in</strong>’s statement), because of <strong>the</strong> absurd situation<br />
described by <strong>the</strong> first l<strong>in</strong>e of <strong>the</strong> truth table presented <strong>in</strong><br />
Table 1: that needs a syntax consideration about <strong>the</strong> use<br />
of <strong>the</strong> language of colours to be rejected, <strong>and</strong> Wittgenste<strong>in</strong><br />
himself <strong>in</strong> Some Remarks on Logical Form spoke of<br />
“exclusion” <strong>in</strong> contrast to contradiction analyz<strong>in</strong>g such a<br />
type of logical product.<br />
Truth table of <strong>the</strong> presumed logical contradiction <strong>in</strong>dicated <strong>in</strong> proposition 6.3751<br />
In my op<strong>in</strong>ion it is not possible to conclude anyth<strong>in</strong>g about<br />
logical necessity from a table like this, because it’s not<br />
clear how to manage its first l<strong>in</strong>e by a logical po<strong>in</strong>t of view,<br />
be<strong>in</strong>g <strong>the</strong> “exclusion” not a significant logical operation.<br />
This difficulty is connected to a not satisfactory<br />
logical <strong>in</strong>ference <strong>the</strong>ory based on tautologies that derive by<br />
<strong>in</strong>dependent atomic propositions. I th<strong>in</strong>k anyway that it’s<br />
possible to overcome this problematic situation us<strong>in</strong>g <strong>the</strong><br />
same pr<strong>in</strong>ciples accepted by Wittgenste<strong>in</strong> <strong>in</strong> <strong>the</strong> first phase<br />
of his philosophical development.<br />
This paper will try to f<strong>in</strong>d a new po<strong>in</strong>t of view to<br />
consider Table 1 <strong>in</strong> order to make it clearly logically<br />
significant.<br />
2. Sensorial Spaces<br />
It seems reasonable to call propositions like ‘This table is<br />
white’ or ‘This wall is rough’ or ‘This food is salty’ sensorial<br />
propositions. These propositions describe objects us<strong>in</strong>g<br />
sense data <strong>and</strong> can be analyzed us<strong>in</strong>g 5 ma<strong>in</strong> classes<br />
(each related to a sense) <strong>and</strong> <strong>the</strong>ir comb<strong>in</strong>ations. This<br />
ma<strong>in</strong> classification is clear if a sensorial proposition is def<strong>in</strong>ed<br />
as a statement of a sense impression or as a logical<br />
product of statements of sense impressions 1 . ‘This table is<br />
rough’ can be expressed <strong>in</strong> a form that matches <strong>the</strong> def<strong>in</strong>i-<br />
1 It will be not considered, at this level of <strong>the</strong> analysis, <strong>the</strong> <strong>in</strong>fluence if <strong>the</strong><br />
logical nature of <strong>the</strong> subject of <strong>the</strong> proposition.<br />
230<br />
tion of sense propositions if transformed <strong>in</strong>: ‘This colour<br />
spot with <strong>the</strong> shape that I use to call ‘table’ is <strong>in</strong> my field of<br />
vision <strong>and</strong>, when I move <strong>the</strong> p<strong>in</strong>k shape that I use to call<br />
‘h<strong>and</strong>’ near it, I feel a sensation of roughness’.<br />
The dist<strong>in</strong>ction of <strong>the</strong> contributions of <strong>the</strong> s<strong>in</strong>gle<br />
senses <strong>in</strong> a sensorial proposition is possible also <strong>in</strong> more<br />
subtle contexts. For <strong>in</strong>stance, a proposition like ‘This<br />
pullover is comfortable’ can be analyzed us<strong>in</strong>g fuzzy logic<br />
to def<strong>in</strong>e <strong>in</strong> a quantitative manner <strong>the</strong> grades of each<br />
sense <strong>in</strong>volved.<br />
The class of <strong>the</strong> characteristic properties related to a<br />
percipient sense can be <strong>in</strong>dicated with Wittgenste<strong>in</strong>’s<br />
metaphor of space.<br />
When a proposition states a certa<strong>in</strong> property of a<br />
material subject, <strong>the</strong> coord<strong>in</strong>ates of a po<strong>in</strong>t <strong>in</strong> <strong>the</strong> space of<br />
that sense will be fixed by it. This specification will exclude<br />
o<strong>the</strong>r properties of that space because ‘a particle […]<br />
cannot be <strong>in</strong> two places at <strong>the</strong> same time; that is to say,<br />
particles that are <strong>in</strong> different places at <strong>the</strong> same time<br />
cannot be identical.’ (6.3751). This proposition of <strong>the</strong><br />
Tractatus describes <strong>the</strong> ma<strong>in</strong> property of <strong>the</strong> structure of<br />
sensorial spaces. Is this structure logical or empirically<br />
derived? It depends on what we <strong>in</strong>tend for logic: if we kept<br />
<strong>in</strong> m<strong>in</strong>d proposition 5.552 (‘The "experience" which we<br />
need to underst<strong>and</strong> logic is not that such <strong>and</strong> such is <strong>the</strong><br />
case, but that someth<strong>in</strong>g is; but that is no experience.<br />
Logic precedes every experience that someth<strong>in</strong>g is so. It is<br />
before <strong>the</strong> How, not before <strong>the</strong> What’) it is probably to be<br />
considered pure logical.<br />
3. Logic Of Sense Proposition<br />
The logical form of <strong>the</strong> sensorial propositions will be found<br />
us<strong>in</strong>g <strong>the</strong> method described <strong>in</strong> proposition 3.315 of <strong>the</strong><br />
Tractatus. Taken a certa<strong>in</strong> proposition ‘Ra’, it is possible to<br />
rewrite it <strong>in</strong> <strong>the</strong> follow<strong>in</strong>g form, where all <strong>the</strong> properties of<br />
<strong>the</strong> sensorial space are <strong>in</strong>volved:<br />
RSa∧ ~ TSa∧<br />
~ USa<br />
∧ etc<br />
This form is allowed by proposition 6.3751 <strong>and</strong> propositions<br />
3.311-3.313 (see also Anscombe 1963, chap. 6,<br />
about expressions <strong>and</strong> symbols). The subscript ‘s’ st<strong>and</strong>s<br />
for a generic sensorial space. In details, <strong>the</strong> written logical<br />
product is an attempt to stress with a different propositional<br />
sign <strong>the</strong> structure of <strong>the</strong> sensorial space <strong>in</strong>volved by <strong>the</strong><br />
proposition analyzed.<br />
The expression above can be expressed also as:<br />
RS a ~ RSa<br />
∧ RS<br />
∀ with ∈ S ∧ R ≠ R<br />
RS S<br />
where R could <strong>in</strong>dicate every property of <strong>the</strong> space S<br />
S<br />
different from R.<br />
The determ<strong>in</strong>ation of <strong>the</strong> logical form of a sensorial<br />
proposition follows three steps: 1) <strong>the</strong> generalization (i.e.<br />
<strong>the</strong> substitution with a variable) of <strong>the</strong> subject, 2) <strong>the</strong><br />
generalization of <strong>the</strong> sense properties <strong>in</strong>dicated <strong>in</strong> each<br />
component of <strong>the</strong> logical product of <strong>the</strong> sensorial<br />
proposition, 3) <strong>the</strong> generalization of <strong>the</strong> space <strong>in</strong> which <strong>the</strong>