Hoffmann admits that
deductions or proofs are the core of logic: while concepts and
propositions might be the result we strive for in logic, deductions are
the primary logical means, by which these results are gained. Thus,
it is no wonder that he spends dozens of pages for a division of
types of deduction – especially as he thinks that the usual method
of dividing deductions is quite faulty.
Hoffmann's main
criticism of the traditional Aristotelian logic is its overt reliance
on syllogistic. True, we might be able to transform all deductions in
syllogisms, but this loses the peculiarity of different deductions
and loses sight of the different conditions in which different types
of deduction apply.
The simplest form of
deduction is purely verbal: it changes something contingent in a
proposition, without affecting the relations between ideas. Such a
change might affect only a mode of cogitation, such as when we start
from a proposition ”work is means for earning money” and conclude
”earning money is the purpose of working”. Similar verbal changes
occur when some irrelevant abstractions are removed or added, such as
when we from proposition ”burning biomass is a way to produce
energy” conclude ”if we burn biomass, we produce energy”. Such
verbal deductions might appear rather useless, but Hoffmann notes
that they are often important ingredients in more difficult
deductions.
Slightly more
complex are deductions involving opposition in the sense that they
deduce from a link between ideas X and Y a link between the
non-existence of Y and the non-existence of X. This might seem like a
verbal deduction, but the involvement of opposition, instead of an
affinity of ideas, gives this type of deduction a distinct look.
Hoffmann also delineates various types of this sort of deduction,
which include disjunctive deduction (”Soul is either mortal or
immortal, it is not mortal, thus, it is immortal”), deduction of
immediate opposition involving predicate (”All created things are
finite, therefore, none of them are infinite”), deduction of
immediate opposition involving copula (”It is true that snow is
white, therefore, it is false that snow is black”) and deduction of
immediate opposition involving subject (”movement is change of
place, thus, rest is non-change of place”).
Another quite simple
type of proposition not following syllogistic formula is conversion,
which can be simple or not involve change of quantity (”Some cats
are grey animals, thus, some grey animals are cats”) or accidental
or involve change of quantity (”All cats are animals, hence, some
animals are cats”). Together with a suitable deduction of
opposition, conversion can be used to form contrapositions.
Taking look at three
types of deductions delineated thus far – verbal deduction,
opposition and conversion – we note that two of them share a
commonality. While deduction of opposition works through some clash
of ideas – these ideas cannot be connected together – both verbal
deductions and conversions work through ideas sharing some common
element, that is, through subordination. In case of verbal deductions
and conversions this common element is something peculiar – meaning
of words in case of one, and relation between certain propositions in
case of other. In addition, one might also make deductions, which are
based on nothing else but bare subordination – if A is somehow
linked to B and B is somehow linked to C, then A is also somehow
linked to C. This fourth type of deduction is once again not
syllogistic, Hoffmann says, because the link in question need not be
that between species and genus.
It goes without
saying that although all deductions are not syllogisms, Hoffmann
allows still that all syllogisms are deductions. Syllogisms are also
deductions based on subordination or common elements between ideas,
but here the subordination is of a particular type – because A is a
logical part of B and B is a logical part of C, then A is a logical
part of C, where A being a logical part of B means that A is species
or individual under genus B.
Syllogism is then a
deduction based on the notion of logical parthood. There are also
other deduction types based on part/whole -relations in general. In
some of these, one deduces from a feature of part or parts to a
feature of whole. One can, firstly, deduce that something
characterising all parts characterises also the whole (if all parts of
human body are made of flesh, then the whole human body is made of
flesh), secondly, that something characterising no part does not
characterise the whole (if no part of animal is unhealthy, then the
whole animal is not unhealthy), and thirdly, that something
characterising a part characterises also the whole (if hand of a
person is injured, then we could say that the whole person is injured). Hoffmann
notes that all these deductions work only in some special contexts –
for instance, although individual units don't have any number,
collection of units does have.
Understandably,
Hoffmann also thinks there are deductions moving from wholes to
parts. An important specimen involves causal notions – what made a
whole makes also the parts. Here the whole must really be caused by
this something in a proper fashion – parents can be said to have
generated their child, but because they haven't actually generated
the whole child, we cannot say that they would have created her soul.
Another possibility is to deduce from the notion of species as a
whole that some of its features are at least possible features of the
genus (if birds do actually fly, then animals in general might be
capable of flight) or to conclude from something affecting the whole
that a part is also affected (if the whole house is painted red, then
also the roof is so painted).
We are now in a
position to give a more detailed division of deduction types. All the
types of deduction thus far discovered have been based on either
opposition or subordination. Those based on subordination had several
subtypes, one of them being the general type, based on nothing more
than mere subordination or existence of some link between ideas. More
particular types of deductions based on subordination included verbal
deduction, based on the nominal meaning of words, and several types
based on some sort of part/whole -relation. This leaves only the
conversion uncounted, and it could be described as being based on the
logical relations between subjects and predicates. This description
suggests another type of deduction, based on some further, non-logical
relation – for instance, if we know that Philip is a father, we
know he must have a child.
Of the three groups
of particular deductions of subordination (verbal deductions,
deductions based on logical or non-logical relations and deductions
based on logical or non-logical part/whole -relations), the third
group contains still some further subtypes. We have seen logical
part/whole -relations used in syllogisms, while deductions from parts
to whole and vice versa used what Hoffmann calls non-integral
part/whole-relations, in which parts can be separated from the whole
and other parts. This still leaves the possibility of deductions
involving integral part/whole-relations, in effect, magnitudes. One
type of such deductions involves comparisons – if we know that
Caesar achieved same results with less soldiers being killed than with Alexander, then we can conclude Caesar was a better general than
Alexander. In such deductions we use the known order of the
magnitudes of certain qualities as a standard for deciding the order
of the magnitudes of other qualities – furthermore, we require some
justification or reason connecting the standard to the case to be
decided.
While in comparative
deduction we do not know the exact quantities, in mathematical
deductions we do. Mathematical deductions come in many varieties,
simple deductions relying on some easy calculation (if a person makes
one sin in an hour and is awake seventeen hours in a day, he will
make 365 x 17 sins in a year), but more complex depend on
intricate relations between various quantities. Most interesting type
of mathematical deductions are those, in which some quantities (three
sides of triangle) determine some other quantities in a stronger
sense (such as the sum of the three angles): Hoffmann calls them
mathematical deductions a priori. In these cases, it is not just a
matter of quantities in some relation, but quantities having causal
effects - therefore, these deductions belong to a completely different type.
All the deductions
thus far have mostly been what Hoffmann calls existential, that is,
they depend on static features and relations of things or ideas. The
only exception was the group of mathematical deductions a priori,
which Hoffmann counts as a form of causal deductions, which are based
on necessary links leading from causes to effects. Hoffman delineates
a number of subtypes of causal deductions: simple causal deductions,
which move through one causal link from cause to effect, complex
affirmative causal deductions, which use a combination of causal
links to get from a distant cause to its effect, negative causal
deductions, which show the impossibility of getting to some effect
from a cause, imperfect causal deductions, which move from effects to
causes or by analogy from similarity of causes to similarity of
effects, and causal deductions of opposition, which determine the
effects of opposed causes. An important point to emphasise is
Hoffmann's insistence that causal deductions have different
conditions of application than mere existential deduction. For
instance, one cannot just assume a general existential proposition,
like law of inertia, to explain some effects, if one is not clear on
the actual causal mechanism leading to these effects – or then at
least this is not deduction, but a weaker type of argumentation.
A special kind of
causal deduction, which Hoffmann raises to a status of independent
type, is formed of practical deductions, which either
attempt to show that some action is means for a purpose or then argue
that some element of the supposed means prevents the fulfillment of
the purpose. What makes practical deductions separate from other
causal deductions is a normative element – in practical deductions
we are often interested to show also that some means are good or even
best for achieving some goal.
This concludes
Hoffmann's discussion of types of deduction. To summarise, his
division of types of deduction is as follows:
1. Existential
deductions
A) Deductions of
opposition
B) Deductions of
subordination
AA) General
deductions of subordination
BB) Particular
deductions of subordination
a) Verbal deductions
b) Deductions based
on relations
i) Conversions
ii) Relative
deductions
c) Deductions based
on part/whole -relationships
i) Syllogisms
ii) Deductions based
on non-integral part/whole -relationships
aa) Deductions from
parts to wholes
bb) Deductions from
wholes to parts
iii) Deductions
based on integral part/whole -relationships
aa) Comparative
deductions
bb) Mathematical
deductions
2. Causal deductions
A) Causal deductions
as such
B) Practical
deductions
What is interesting
in this division is Hoffmann's attempt to make the traditional theory
of syllogisms less formal and make logic into a general scientific
methodology, through which also peculiarities of causal reasoning
could be handled. We shall see more of Hoffmann's attempts to give
more methodological substance to logic in later posts.
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