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
ii) Relative deductions
c) Deductions based on part/whole -relationships
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.