Although the idea of inversion is well known and has been used in many different contexts we will start from preliminaries because of its pivotal role in the thesis. This is also
essential for making explicit the sense in which the idea is seen as an essentiallogical relation
for a possible logic of construction.
The fundamental nature of the operation negation is well known to philosophers and logicians. Deductive logic is impossible without negation. We say this on the basis of
the well known fact of logic that with ∼ (negation) and either ∧ (and) or ∨ (or) we can
get a functionally complete system of logic. However, one might say that logical reductions
to one primitive operation, such as alternative denial and joint denial, have shown that a
functionally complete system of logic can be obtained with only one operation (only one
connective, to be precise).9 These claims are interesting when we look at the matter from
9
a formal mode alone, i.e., viewing them as only symbols without interpretation—the only
interpretation being the truth tables.10
However, when we look at the manner in which we start making sense of the above mentioned primitive connectives, we tend to see them as
species of denial or negation. Though the names suggested by Quine, alternative denial and
joint denial, are to be formally viewed only as names of connectives defined in a specific
manner, all applications of them suggest that the primary mode of denial or negation is
implicit.
Take for example, the case of joint denial↓. The sentence (φ↓ψ) is true when both
φ and ψ are false. Therefore, it is suggested that (φ↓ ψ) be read as ‘Neither φ norψ’. In
other words, it is equivalent to the sentence (∼ φ∨ ∼ ψ). Though formally the reduction
to one primitive is successfully achieved, the only cases where such primitives can be applied
are cases where an equivalent form of the sentence wouldnecessarily contain negation. Since
all well-formed-formulae where the joint denial↓occurs can be translated into an alternative
form where ∼ and another connective (such as ∧ or ∨) necessarily occur, we can conclude
that it is impossible to construct a deductive system without any primitive connective that either explicitly or implicitly involves negation.
There is another fundamental reason to regard that deductive reasoning is funda- mentally dependent on negation. Of the three principles of logic—the principle of identity, the principle of contradiction and the principle of excluded middle—the latter two employ
explicitly the operation negation. Nothing is an assertion unless we deny at the same time
the negation of the assertion.
Can we conceive of any alternative logics that are not based on the above principles? Is negation the only species of opposition? Aren’t there other ways of opposition that we regularly employ in our thought? Since negation is fundamental to any assertion, no logic of assertion can ever be conceived without it. Are we capable of thinking without employing any assertions? If the assertive mode of thinking excludes nothing, then it is legitimate to say that we can’t think without making any assertions. But fortunately our thinking abilities are not limited to the assertive mode alone. We have a mode of thinking (we may also say we have
a special mode of inference) that is neither inductive nor deductive.11 This alternative logic
is a logic of construction (synthesis), and is therefore necessarily ampliative. The structures
alternative denial andjoint denial. Also Copi 1979,Elements of Symbolic Logicpp. 281-282.
10
In fact the discovery story of these connectives will be a good instance of constructive thinking.
11
We have already argued above (§4.8 page 107) how inductive logic should be viewed as a logic of abstrac- tion, and that it is based onthe principle of excluded extremes.
168 Chapter 6. Inversion
that are constructed belong to the logical category of concepts. Since the outcome of the logic being developed is a concept, and the concept being a structure, the logic can be called
constructive abstractionto distinguish it frominductive abstraction (§4.8 page 107). However, since we are specifically going to talk about the possibility of articulating a logical mode of
constructive inference that is based on the logical relation, inversion, we will call itinversive
abstraction.
Though modern logic has done remarkably better than traditional logic with respect to relations, the synthetic role of certain relations have not come to light because of the predominant tendency to view logic only as a tool of analysis. Dealing with logic always in a propositional or assertive mode has led to a state where even talking about the possible patterns of non-assertive modes of thinking means to certain thinkers ‘illogical’.
The inference called inversive abstraction is based on a species of logical opposition
called inversion, just as deductive logic is based on a species of opposition called negation.
The modest objective is to convince the reader that there exists the possibility of formulating
at least one more mode of ampliative inference that is not inductive and being ampliative certainly not deductive. We will present a tentative and non-rigorous formulation of what is being visualized.
It is of some interest to note that the notion of inversive abstraction is not too
different from what Hermann Weyl called constructive cognition, orconstructive abstraction
(See below§6.3 page 169). We are attempting to enrich the notion by necessarily linking it
with a logical relation of inversion—towards a methodology of ampliative logic. Weyl also contrasts it with inductive abstraction. He discusses the example of the formation of the concept of mass by Galileo who defines it as follows: Two bodies have the same mass if, at equal velocities, they possess equal momenta. This definition is arrived at by mental (creative) and experimental construction and is not inductively arrived at. In this case experimental manipulations are made, unlike in the realm of numbers where intellectual manipulations are made. These experimental manipulations make numerical determination of characters possible. Historically this was a turning point, because, before Galileo only geometrical
characters are known to be amenable to numerical determination.12
In the process mass became the dynamic coefficient according to which inertia resists the deflecting force. Motion
12
Describing this Weyl says: “This is a step of great importance. After matter was stripped of all sensory qualities, it seemed as first as though only geometrical properties could be attributed to it. In this respect Descartes was wholly consistent. But it now [after Galileo] appears that other numerical characteristics of bodies can be gathered from the laws to which changes of motion in a reaction are submitted. Thus the sphere of properly mechanical and physical concepts is opened up beyond geometry and kinematics.” (Weyl H. 1949, Philosophy of Mathematics and Natural Science p. 148.)
according to Galileo, depends on the struggle of two [opposing] tendencies, inertia and force, force that deflects the body from the path dictated by inertia.
This conceptualization of mass is markedly different from the Aristotelian way of ascending from particulars to universals, where only the really existing objects are concerned, for it is inductive.
In the mathematical-physical or ‘functional’ formation of concepts, on the other hand, no abstraction takes place, but we make certain individual features variable that are capable of continuous gradation, ... , and the concept does not extend
to all actual, but all possible objects thus obtainable.13
Therefore induction, as already noted in the above chapter, cannot explain the genesis of
notions such as mass. For such concepts we needconstructive abstraction. Weyl’s character-
ization of it is as follows:
1. We ascribe to that which is given certain characters which are not manifest
in phenomena but are arrived at as the result of certain mental operations. It is
essential that the performance of these operations is held universally possible and
that their result is held to be uniquely determined by the given. But it is
not essential that the operations which define the character be actually carried out.
2. By the introduction of symbols the assertions are split so that one part of
the operations is shifted to the symbols and therebymade independent of the
given and its continued existence. There by the free manipulation of concepts is contrasted with their application, ideas become detached from reality and acquire a relative independence.
3. Characters are not individually exhibited as they actually occur, but their
symbols are projected on the background of anordered manifold of possibil-
itieswhich can be generated by a fixed process and is open into infinity.14
The themes that we are presently developing are more or less contained in the above points on abstraction. In the context where the mental operations are performed, scientists are hardly concerned about the application, and thus they are in a nonassertive mode of thinking.
Another point to take note of is regarding the role of the given. (See the underlined portions
in the above quotation.) Though in constructive abstraction scientists begin from the given, they eventually get “detached” to acquire “independence”. This becomes essential in order to transcend the limitations of inductive knowledge. The idea is not only to understand the given, but also to understand the “manifold of possibilities”. Thus:
All knowledge, while it starts with intuitive description, tends toward symbolic
construction.15
13
Ibid,p.150.
14
Ibid,p.37-38. Boldface is ours, italics are original.
15
170 Chapter 6. Inversion
Weyl further says, citing Dilthey, that the scientific imagination of man is regulated by the strict methods which subject the possibilities that lie in mathematical thinking to experience,
experiment, and confirmation by facts.16
Thus we ‘inherited’ a lot from Weyl’s insightful thoughts. In order to further this line of thought, however, it is insufficient to prove the strict methodological (logical) character of construction. We suggest that inversion, being a logical relation and—most importantly—being a constructive relation, contains the secret of a logic of construction. In what follows we present in what sense inversion plays a crucial role in this context.
We will use the term ‘inversion’ for a special kind of relation where the two terms that are oppositely related are opposite, or inverse, by virtue of a third term. In other words,
the notion of opposition that is involved here is relative to a third term. Metaphorically
speaking, here we not only have two opposite poles, but also a center. The three terms
involved will be called atriad; the structure thus formed will be calledinverse structure; and
the mode of thinking that leads to such a structure will be called inverse thinking. Since it
is by virtue of the third term that the specification of the inverse relation is made possible, the inverse structure will be identified by the name of the third term, and the third term of
the structure will be called the identity element.
For example,−2 and +2 are inversely structured with respect to the identity element
0; 2 and 1/2 are inversely structured with respect to the identity element 1. These examples,
and the use of inversion in mathematics is well known. However, the point of the thesis is to demonstrate that it is the same inverse structure that gives shape to all scientific knowledge. Many examples from natural sciences will be presented below.
Both polar thinking and inverse thinking yields structures that put together the opposite terms. The nature of opposition involved in these special cases is such that the opposites are not viewed as contradicting one another, but are parts of the same structure. Since the opposite terms necessarily belong to one single structure they are both applicable together at the same time to that structure. The terms that are related by negation are not
applicable to the same thing at the same time, as stated inthe principle of non-contradiction:
nothing can be both P and ∼ P at the same time. The terms that are related by either
inversion or polarity are applicable to the same object at the same time. We will consider
this principle sufficiently fundamental; it therefore needs to be added to the list ofprinciples
of thought. We will call itthe principle of included extremes, for the opposites are included without contradiction in a structure.
16
We will illustrate the fundamental significance of the principle of included extremes in the genesis, development and the structure of scientific knowledge.