Draft of a paper which will be published in: Second Pisa Colloquium in Logic, Language and Epistemology, ed. E. Moriconi. Pisa: ETS. 

Does logic slowly pass away, or has it a future?

Carlo Cellucci

Abstract: The limitations of mathematical logic either as a tool for the foundations of mathematics, or

as a branch of mathematics, or as a tool for artificial intelligence, raise the need for a rethinking of logic.

In particular, they raise the need for a reconsideration of the many doors the Founding Fathers of

mathematical logic have closed historically. This paper examines three such doors, the view that logic

should be a logic of discovery, the view that logic arises from method, and the view that logic is not the

whole of reason, and on this basis proposes an alternative approach to logic.

1. Introduction

There was a time when mathematical logic was the undisputed paradigm of logic.

Thus, in 1931, Carnap stated that, while “traditional logic was totally incapable of

satisfying the requirement of richness of content, formal rigor and technical utility,”

mathematical logic is an “efficient instrument in the place of the old and useless

one” (Carnap 1971: 134). Again in 1931, Scholz stated that mathematical logic

“gives us the complete inferential rules which the development of the tremendously

exacting modern mathematics requires” (Scholz 1961: 67).

That very same year, however, Gödel published his first incompleteness

theorem, which suggested that, contrary to Carnap’s and Scholz’s claims,

mathematical logic neither satisfies the requirement of richness of content nor gives

us the complete inferential rules which modern mathematics requires. Moreover,

mathematical logic limits logic essentially to the study of deduction. This leads to a

restricted concept of reason and rationality, because it means that reasoning other

than deduction escapes logic.

All these facts raise doubts about the role of mathematical logic. Reflecting on

this role, van Benthem observes that mathematical logicians “show a curiously

defensive attitude, ill-fitting their status and achievements. Friendly proposals to

extend the classical agenda (we cannot keep singing hymns to Gödel and Tarski

forever – unless we are already in Heaven) are perceived as threats, to be received

with suspicion and sometimes even personal attacks. And innovative ‘agents

provocateurs’ like Lakatos” are “subjected to torrents of abuse” (van Benthem 2008:

40). I “have never been able to understand this defense of the status quo. In

particular, I have always found many outspoken critics of” mathematical “logic

(Blanshard, Perelman, Toulmin, Lakatos) extremely interesting and well-worth

reading, and a useful reminder of the many doors our Founding Fathers have closed

historically – doors that could be opened again now” (ibid.).

Indeed, while opening new doors, mathematical logic has closed many other

doors that not only could, but should be opened again now if logic is to remain alive

and vital. The paper examines three such doors, the view that logic should be a logic

of discovery, the view that logic arises from method, and the view that logic is not

the whole of reason. On this basis, it outlines an alternative approach to logic.

2. Logic and Discovery

The first door mathematical logic has closed is the view that logic should be a logic

of discovery, and hence should provide means to acquire knowledge.

Mathematical logic rejects this view. This is clear from Frege, the first

Founding Father of the subject, who states that “we can inquire, on the one hand,

how we have gradually arrived at a given proposition and, on the other, how we can

finally provide it with the most secure foundation” (Frege 1967: 5). The first

question, discovery, is merely subjective, because it “may have to be answered

differently for different persons,” only the second question, justification, “is more

definite” (ibid.). Therefore, logic must concern itself “not with the way in which”

mathematical propositions “are discovered but with the kind of ground on which

their” justification “rests” (Frege 1959: 23). Thus logic cannot be a logic of

discovery and must be a logic of justification. Indeed, the task of logic is to provide

a proposition with the most secure foundation, so as to place “the truth of a

proposition beyond all doubt” (ibid.: 2).

To this aim, logic must “set up laws according to which a judgment is justified

by others” (Frege 1979: 175). These are the laws of deduction, therefore logic must

concern itself with deduction. On the other hand, logic can “forgo expressing

anything that is without significance for the inferential sequence” (Frege 1967: 6).

That is, it can abstain from expressing “anything that it is not necessary for setting

up the laws of deduction” (Frege 1979: 5). Therefore, logic can concern itself only

with deduction. Thus “everything necessary for a correct inference is expressed in

full, but what is not necessary is generally not indicated” (Frege 1967: 12). As a

result, logic will fail to express all aspects of mathematics. But this is irrelevant,

because the task of logic is to provide a proposition with the most secure foundation.

Logic is suited to this task, and “one must not condemn it because it is not suited to

others” (ibid.: 6).

Hilbert even tries to show that logic can concern itself only with deduction by

trivializing the question of discovery. Indeed, on the one hand, he states that there is

no question of discovering the axioms because “the axioms can be taken quite

arbitrarily” (Hilbert 2004, 563). They are only subject to the condition that “the

application of the given axioms can never lead to contradictions” (Hilbert 1996a,

1093). On the other hand, he states that the question of discovering demonstrations

of mathematical propositions from given axioms is a purely mechanical business,

because of “the decidability of a mathematical question in a finite number of

operations” (Hilbert 1996b, 1113).

3. Aristotle’s View of Logic

The view of mathematical logic, that logic can be only a logic of justification,

contrasts with Aristotle’s view, that logic should primarily be a logic of discovery.

Syllogism can be seen in a twofold manner: as a means of obtaining

conclusions from given premises, so a means of justification, or as a means of

obtaining premises for given conclusions, so a means of discovery. Syllogism seen

as a means of obtaining conclusions from given premises is the conception the

deductivist view of syllogism attributes to Aristotle (see, for example, Boger 2004).

This view is based on the first 26 Chapters of the first book of Prior Analytics,

where Aristotle describes the morphology of syllogism.

But then, in Chapter 27, Aristotle states: “Now it is time to tell how we will

always find syllogisms on any given subject, and by what method we will find the

premises about each thing. For surely one ought not only to investigate how

syllogisms are constituted, but also to have the ability to produce them” (Aristotle,

Analytica Priora, A 27, 43 a 20–24). In order to produce them, one must indicate

“how to reach for premises concerning any problem proposed, in the case of any

discipline whatever” (ibid., B 1, 53 a 1–2.). That is, one must indicate, for any given

conclusion, how to reach for premises from which that conclusion can be deduced.

From this it is apparent that, for Aristotle, syllogism is primarily a means of

discovery. While, according to the deductivist view, syllogism is a means of

obtaining conclusions from given premises, for Aristotle it is a means of obtaining

premises for given conclusions, thus a means for solving problems. Therefore

Aristotle’s view of syllogism is a heuristic view. For this reason Aristotle says that,

while “arguments are made from premises,” the “things with which syllogisms are

concerned are problems” (Aristotle, Topica, A 4, 101 b 15–16). By this he means

that the thing with which syllogisms are concerned is solving problems.

Consistently with this view, in Chapters 27–31 of the first book of Prior

Analytics Aristotle describes a heuristic procedure for finding premises to solve

problems. The medievals called this procedure inventio medii [discovery of the

middle term] because it is essentially a procedure for finding the middle term of a

syllogism, given the conclusion. (For a description of such procedure, see Cellucci

2013: Chapter 7).

4. Aristotle’s Analytic-Synthetic Method

That, for Aristotle, syllogism is primarily a means of discovery depends on the fact

that he develops his theory of syllogism as a tool for the method of science, which

he identifies with the analytic-synthetic method. (Note that Aristotle’s analytic-

synthetic method is not to be confused with Pappus’ analytic-synthetic method, see

Cellucci 2013: Chapter 5). Inventio medii is one of the two procedure by means of

which premises are obtained in Aristotle’s analytic-synthetic method. The other one

is induction.

In order to state Aristotle’s analytic-synthetic method we need first recall some

of his views about science. For Aristotle, each thing belongs to some kind. Each

science is concerned with one and only one kind, because a “science is one if it is

concerned with one kind” (Aristotle, Analytica Posteriora, A 28, 87 a 38). For

example, arithmetic is concerned with units, geometry with points and lines. Each

science is based on principles that “must be in the same kind as the things

demonstrated” (ibid., A 28, 87 b 2–3). Principles are given once for all and must be

not only true but also known to be true, otherwise we could not “have scientific

knowledge of what follows from them, absolutely and properly” (ibid., A 3, 72 b

14).

Then, Aristotle’s analytic-synthetic method is the method according to which,

to solve a given problem of a certain kind, one must find premises of that kind from

which a solution to the problem can be deduced. The premises are obtained from the

conclusion “either by syllogism or by induction” (Aristotle, Topica, Θ 1, 155 b 35–

36). That is, they are obtained from the conclusion either by inventio medii or by

induction. The premises must be plausible, that is, the arguments for them must be

stronger than those against them. Therefore, in order to see whether a premise is

plausible, one must “examine the arguments for it and the arguments against it”

(Aristotle, Topica, Θ 14, 163 a 37–b 1).

If the premises thus obtained are not principles of the kind in question, one

must look for new premises of that kind from which the previous premises can be

deduced. The new premises are obtained from the previous premises either by

inventio medii or by induction and must be plausible. And so on, until one arrives at

premises which are principles of the kind in question. Then the process terminates.

This is analysis.

At this point one tries to see whether, inverting the order of the steps followed

in analysis, one may obtain a deduction of the conclusion from the principles of the

kind in question. This is synthesis. When synthesis is successful, this yields a

solution to the problem.

Then Aristotle’s analytic-synthetic method can be schematically represented as

follows.

A2

A1

B

Synthesis

Problem to solve

A

Analysis

Principle

Note that, as Kant points out, plausibility is not to be confused with probability.

Indeed, “plausibility is concerned with whether, in the cognition, there are more

grounds for the thing than against it” (Kant 1992, 331). Conversely, “probability is a

fraction, whose denominator is the number of all the possible cases, and whose

numerator contains the number of winning cases” (ibid., 328). Plausibility “rests

merely on the subject,” while “probability rests on the object” (ibid., 153).

Plausibility is not a mathematical concept, while “there is a mathematics of

probability” (ibid., 331). (On Kant’s distinction between plausibility and probability,

see Capozzi 2002: Chapter 7, Section 5, and Chapter 15).

5. A Paradigm Change in Logic

Since, as it has been said in Section 3, for Aristotle syllogism is primarily a means of

discovery, for him logic should be primarily a logic of discovery.

This shows how inadequate is the widespread view that mathematical logic is

an extension of Aristotle’s logic merely because it is capable of dealing with

relations, while Aristotle’s logic can only deal with properties. This view is put

forward, for example, by Carnap who states that “the new logic is distinguished

from the old” because the new logic is able to deal with “the theory of relational

sentences,” while “the only form of statements (sentences) in the old logic was the

predicative form: ‘Socrates is a man,’ ‘All (or some) Greeks are men’” (Carnap

1971: 137).

This view is inadequate, because the change from Aristotle’s logic to

mathematical logic is not merely a change from properties to relations but rather a

paradigm change. While, for Aristotle, logic is to be a logic of discovery, Frege

rejects the view that that logic should concern itself with discovery. He considers the

question of discovery as a merely subjective one, and restricts logic to justification

and the study of deduction. Thus Frege brings a paradigm change in logic with

respect to Aristotle. (A different interpretation of Frege is put forward by Macbeth

2005, 2014).

That mathematical logic brings a paradigm change in logic with respect to

Aristotle is something of which the Founding Fathers of mathematical logic not only

were fully aware, but even boasted about. Thus Russell states that “any person in the

present day who wishes to learn logic will be wasting his time if he reads Aristotle”

(Russell 1967: 202). With respect to Aristotle, mathematical logic “introduced the

same kind of advance into philosophy as Galileo introduced into physics” (Russell

2009: 48). This shows that the expectations of the Founding Fathers on

mathematical logic were very high indeed.

6. Logic and Method

Another door that mathematical logic has closed is the view that logic arises from

method.

Mathematical logic rejects this view, as it is clear from Frege, who states that

logic is not concerned “with the question of how men think,” which is subjective

and psychological, but rather “with the question of how they must think” (Frege

1979: 149). The laws of logic “prescribe universally the way in which one ought to

think if one is to think at all” (Frege 1964: 12). Since the laws of logic prescribe

universally the way in which one ought to think, logic is a precondition of method

and hence cannot arise from method.

But the claim that the laws of logic prescribe universally the way in which one

ought to think is unrealistic. As Dewey points out, thus logic “is elevated into the

supreme and legislative science only to fall into the trivial estate of keeper of such

statements as A is A” (Dewey 2004: 76). Moreover, the conclusion that logic is a

precondition of method, and hence cannot arise from method, contrasts with the fact

that, as again Dewey states, “as the methods of the sciences improve, corresponding

changes take place in logic” (Dewey 1938: 14).

A first example of the fact that, as the methods of the sciences improve,

corresponding changes take place in logic, is provided by Aristotle’s logic that, as

we have seen in Section 4, originated as a tool for Aristotle’s method of science,

Aristotle’s analytic-synthetic method. Another example is provided by Bacon’s and

Descartes’ attempts to develop a new logic as a tool for the method of the new

science originating from the Scientific Revolution.

Bacon wants to develop a new logic aimed at “the discovery of arts,” and “not

of inferences from principles” (Bacon 1961–1986: I, 135). The new logic will be

based on the true method, which consists of “two parts: the first one is about how to

educe and form axioms from experience” (ibid.: I, 235). This is the concern of

induction. The “second part is about how to deduce or derive new experiments from

axioms” (ibid.). This is the concern of deduction.

Descartes wants to develop a new logic which will teach “one to direct his

reason to discover the truths of which one is ignorant” (Descartes 1996: IX–2, 13–

14). The new logic will be based on the true method. We “shall comply with it

exactly if we will gradually reduce convoluted and obscure propositions to simpler

one” (ibid.: X, 379). Thus we will eventually arrive at the simplest of all

propositions, “which can be intuited first and per se, independently of any other

one” (ibid.: X, 383). This is analysis. Then, “from the intuition of the simplest of all”

propositions, “we will try to ascend through the same steps to a knowledge of all the

others” (ibid.: X, 379). For “all the others can be perceived only by deducing them

from those” (ibid.: X, 383). This is synthesis. Thus the method “is twofold, one by

analysis, the other by synthesis” (ibid.: VII, 155).

Admittedly, the new logics that Bacon and Descartes developed have not been

very successful but this is irrelevant here. What is relevant is that they originated as

a tool for the method of the new science.

7. Logic and Reason

A further door mathematical logic has closed is the view that logic is not the whole

of reason.

Mathematical logic rejects this view, as it is clear from Frege who states that,

“if beings were even found whose laws of thought flatly contradicted” the laws of

logic, “I should say: we have here a hitherto unknown type of madness” (Frege

1964: 14). Thus, according to Frege, beings are rational only insofar as they obey

the laws of logic, therefore logic is the whole of reason, or at least a necessary

condition for reason.

Now, that logic is a necessary condition of reason is unjustified, because it

excludes feelings and emotions from the sphere of reason and rationality, so that any

human act influenced by feelings and emotions will be termed as irrational. This

contrasts with the fact that, as Damasio points out, “certain aspects of the process of

emotion and feeling are indispensable for rationality” (Damasio 1995: xiii). For

example, patients suffering a damage to the ventromedial prefrontal cortex generally

preserve intellectual abilities but have an abnormality in their processes of emotion

so severe that they are unable to decide advantageously on matters pertaining to

their own lives. In fact, emotions help us to make appropriate decisions on matters

pertaining to our lives, and hence they are not opposite to rationality but

indispensable to it. They help us to make appropriate decisions also on matters

concerning knowledge acquisition, because they may affect us both in the choice of

problems and in the choice of hypotheses for solving problems (see Cellucci 2013:

Chapter 15).

8. The Failure of the Task of Mathematical Logic

As it has been stated in Section 5, by restricting logic to the question of justification,

Frege brought a paradigm change in logic with respect to Aristotle. For him, the task

of logic was to provide a proposition with the most secure foundation. In the

Thirties, however, Gödel’s incompleteness theorems showed that this task could not

be realized. By Gödel’s first incompleteness theorem, mathematics cannot consist of

deductions from given axioms, and, by Gödel’s second incompleteness theorem,

axioms cannot be shown to be consistent by absolutely reliable means.

Also Hilbert’s attempt to trivialize the question of discovery failed. On the one

hand, as it has been just stated, by Gödel’s second incompleteness theorem,

arbitrarily taken axioms cannot be shown to be consistent by absolutely reliable

means. On the other hand, by the undecidability theorem, for any consistent,

sufficiently strong, deductive theory, there is no mechanical procedure for deciding

whether or not a mathematical proposition can be demonstrated from the axioms of

the theory.

These failures, however, did not lead to a complete rethinking of logic. The

only effect they had on mathematical logic was that the latter was developed into a

more and more mathematical discipline, so much so that Curry states:

“Mathematical logic, then, is a branch of mathematics” (Curry 1977, 2). Of course,

this involved giving up the ambition to contribute to the philosophical understanding

of the foundations of mathematics. But the usefulness of logic as a mathematical

discipline was expected to compensate for this renunciation.

However, the attempt to develop mathematical logic into a more and more

mathematical discipline has not been very successful. As Wang points out, although

today, “as practiced, mathematical logic is but a special branch of mathematics,” it

“is not often regarded as a very central branch” (Wang 1974: 21).

9. Feferman’s Vindication of Mathematical Logic

Despite the failure of mathematical logic to realize the task of providing a

proposition with the most secure foundation, and despite the marginal value of

mathematical logic when developed into a branch of mathematics, some

mathematical logicians have attempted a vindication of the subject.

Thus Feferman states that mathematical logic “comes much closer to

explaining our everyday mathematical experience than physics does to explaining

our everyday physical experience” (Feferman 1998: 92). It “gives us a good

underlying analysis of the structure of completed proofs (no gaps, no unsure

assumptions or steps)” (ibid.). It explains “what constitutes the underlying content of

mathematics and what is its organizational and verificational structure” (ibid.). And

“though formal systems are not normally conceived to represent ‘slices’ of

mathematics in a ‘frozen’ state,” one “can use these systems to model” mathematical

“growth and change” (ibid.).

Feferman’s vindication, however, is unconvincing. Mathematical logic does not

come close to explaining our everyday mathematical experience, because real proofs

do not usually consist in deductions from axioms. As Hersh says, “the view that

mathematics is in essence derivations from axioms is backward. In fact, it’s wrong”

(Hersh 1997: 6). Therefore, mathematical logic does not give a good underlying

analysis of the structure of proofs. Moreover, by Gödel’s first incompleteness

theorem, mathematical logic is unable to explain the organizational structure of

mathematics, and by Gödel’s second incompleteness theorem is unable to explain

the verificational structure of mathematics.

It is also inappropriate to claim that one can use formal systems to model

mathematical growth and change. This claim is meant to suggests that although, by

Gödel’s first incompleteness theorem, mathematics cannot be exhausted by any

single formal system, it can be represented by a sequence of formal systems, and

such sequence can model mathematical growth and change. But then, as Curry

points out, mathematical proof would be “that sort of growing thing which the

intuitionists have postulated for certain infinite sets” (Curry 1977: 15). This is

incompatible with the axiomatic method, according to which proof is a fixed thing.

Now, Feferman states that, “once the axioms and basic concepts are granted, all else

in mathematics is obtained by logical argument” (Feferman 2003: 3). Thus,

according to Feferman, the method of mathematics is the axiomatic method. Since

the view that mathematical proof is a growing thing is incompatible with the

axiomatic method, it follows that mathematical growth and change cannot be

modeled by a sequence of formal systems.

10. Kowalski’s Proposal for Revitalizing Logic

The failure of mathematical logic to realize the task of providing a proposition with

the most secure foundation, and the marginal value of mathematical logic when

developed into a branch of mathematics, have also led to seek a new role for

mathematical logic, for example in Artificial Intelligence.

Thus Kowalski states that, for most of the latter half of the twentieth century,

mathematical logic “was the mainstream of Artificial Intelligence. But then it all

went wrong. Artificial Intelligence researchers, frustrated by the lack of progress,

blamed many of their problems on the logic-based approach,” and began to develop

other approaches, “that were designed to simulate directly the neurological

mechanisms of animal and human intelligence” (Kowalski 2001: 2). Thus “logic

seemed to be dying” (ibid.). The question is: “Is logic really dead or only just

sleeping?” (ibid.).

To save logic from death, Kowalski proposes to embed it in an observation-

thought-decision-action cycle: “Repeatedly, observe the world, think, decide what

actions to perform, act” (Kowalski 2011: 95). In this cycle, “below the logical level,

perceptual processes transform raw sensations into observations, and motor

processes transform conceptual representations of actions into raw physical activity”

(ibid.: 124). At the logical level, logic takes care of thinking and deciding what

actions to perform. According to Kowalski, embedding logic in an observation-

thought-decision-action cycle “provides a more realistic framework, not only for

logic as a descriptive theory of how humans actually think, but also for logic as a

prescriptive theory of how humans and computers can reason more effectively. With

such a more realistic framework, even if logic” might be “only half awake today,” it

“can at worst be only sleeping, to come back with renewed and more lasting vigour

in the near future” (Kowalski 2001: 3).

Kowalski’s proposal, however, is unconvincing. First, Kowalski assigns only

deduction to the logical level, neglecting induction and generally non-deductive

inference. Now, Gillies states that “the current development of computers and

artificial intelligence” is “destined to change science, and in such a way that

Baconian induction becomes a standard part of scientific procedure” (Gillies 1996:

69). Commenting on this statement, Kowalski says that he himself has “neglected

this aspect of Computational Logic” (Kowalski 2011: 228). But thus Kowalski

overlooks that “most inferences we make are not deductive” (McDermott 2001: 67).

In particular, inferences involved in discovery are not deductive because deductive

inferences are non-ampliative, while discovery essentially involves going beyond

the data.

Secondly, Kowalski assigns perceptual processes to a non-logical level. This

does not explain why the images in our mind are so rich while the stimuli from

which they are obtained are poor. The mind integrates these stimuli producing a

sensory experience full of color, sound, and texture, so perception is inference,

specifically non-deductive inference. In particular, according to von Helmholtz, “the

formation of our sense-perceptions” is the result of “inductive conclusions

unconsciously formed” (von Helmholtz 1962: III, 26–27).

11. Towards an Alternative Perspective on Logic

In view of the limitations of mathematical logic, it seems legitimate to say that, if

logic is to have a future, an alternative perspective on logic is necessary. In what

follows I will briefly outline such alternative perspective. My brief outline only

indicates a way to move ahead, it does not exclude that there could be different ways

that might also be fruitful, although at the moment I do not quite see them.

The alternative perspective I want to propose is partly implicit in Aristotle’s

view that logic should primarily be a logic of discovery. Aristotle’s approach,

however, requires a basic change, because Aristotle’s analytic-synthetic method is

incompatible with Gödel’s incompleteness theorems.

Indeed, as we have seen in Section 4, according to Aristotle principles must be

in the same kind as the things demonstrated. But, by Gödel’s first incompleteness

theorem, for any consistent sufficiently strong principles of a given kind, there are

truths of that kind which cannot be demonstrated from those principles, their

demonstration may require principles of other kinds. Therefore, Aristotle’s analytic-

synthetic method is incompatible with Gödel’s first incompleteness theorem.

Moreover, again as we have seen in Section 4, according to Aristotle the

principles by means of which a problem is to be solved must be known to be true.

But, by Gödel’s second incompleteness theorem, for any consistent, sufficiently

strong principles, the principles cannot be known to be true, because they cannot be

shown to be such by absolutely reliable means. Therefore, Aristotle’s analytic-

synthetic method is incompatible with Gödel’s second incompleteness theorem.

In view of this, the method of solving problems cannot be identified with

Aristotle’s analytic-synthetic method.

12. The Analytic Method

Fortunately, since antiquity, another method is known which is compatible with

Gödel’s incompleteness theorems, the analytic method.

The latter is the method according to which, to solve a problem, one looks for

some hypothesis that is a sufficient condition for solving it. The hypothesis is

obtained from the problem, and possibly other data already available, by some non-

deductive rule. The hypothesis need not be of the same kind as the problem, and

must be plausible in the sense explained in Section 4. But the hypothesis is in its

turn a problem that must be solved, and is solved in the same way. That is, one looks

for another hypothesis that is a sufficient condition for solving the problem posed by

the previous hypothesis, it is obtained from the latter, and possibly other data

already available, by some non-deductive rule, it need not be of the same kind as the

problem, and must be plausible. And so on, ad infinitum. Thus solving a problem is

a potentially infinite process.

Then the analytic method can be schematically represented as follows.

A2

A1

B

Analysis

Plausible Hypotheses

Problem to solve

In the analytic method there are no principles, everything is a hypothesis. The

problem and the other data already available are the only basis for solving the

problem.

The analytic method is compatible with Gödel’s first incompleteness theorem.

For in such method the solution to a problem is obtained from the problem, and

possibly other data already available, by means of hypotheses which are not

necessarily of the same kind as the problem. Since Gödel’s first incompleteness

theorem implies that solving a problem of a certain kind may require hypotheses of

other kinds, Gödel’s result even provides evidence for the analytic method.

The analytic method is also compatible with Gödel’s second incompleteness

theorem. For in such method the hypotheses for the solution to a problem, being

only plausible, are not absolutely certain. Since Gödel’s second incompleteness

theorem implies that no solution to a problem can be absolutely certain, Gödel’s

result even provides evidence for the analytic method.

In view of this, it seems reasonable to claim that the method of solving

problems can be identified with the analytic method.

13. Rules for Finding Hypotheses

The non-deductive rules by means of which hypotheses are obtained in the analytic

method are not a closed set, given once for all. Rather, they are an open set which

can always be extended as research develops. In any case, such rules will at least

include induction, analogy, generalization, specialization, metaphor, metonymy,

definition, and diagrams (see Cellucci 2013: Chapters 20 and 21).

Which kind of non-deductive rule is to be used to find a hypothesis for solving

a given problem depends on the problem, and possibly other data already available.

The problem and the other data, however, do not uniquely determine which non-

deductive rule is to be used. Different non-deductive rules may be used to solve one

and the very same problem. This is a basic feature of the analytic method. Moreover,

which non-deductive rules is to be used may be positively influenced by feelings

and emotions.

14. Science as an Open System

From what it has been said above it is clear that there is a basic difference between

the analytic method and Aristotle’s analytic-synthetic method.

In terms of Aristotle’s analytic-synthetic method, a science is a closed system.

For it is based on principles that are given once for all, and its development consists

in deducing conclusions from them. Since deduction is non-ampliative, a science is

implicitly contained in its principles.

On the contrary, in terms of the analytic method, a science is an open system.

For it initially consists only of the problem to be solved, and possibly other data

already available, and its development consists in obtaining more and more

hypotheses for solving the problem from the problem itself, and possibly other data

already available, by non-deductive rules. Since non-deductive rules are ampliative,

a science is not implicitly contained in the problem or in the other data already

available.

15. Analytic Method and Knowledge

In Section 12 it has been stated that it seems reasonable to claim that the method of

solving problems can be identified with the analytic method. Now, knowledge is the

result of solving problems. Then it seems reasonable to claim that knowledge is the

result of solving problems by the analytic method.

By knowledge I mean not only scientific knowledge but also everyday

knowledge, that is, the kind of knowledge which is necessary for the everyday

management of life, starting with survival. Indeed, there is continuity between

scientific and everyday knowledge. Scientific knowledge can be seen as an

extension of everyday knowledge. In particular, scientific knowledge can be seen as

an extension of the activities by which our remotest ancestors solved their survival

problem. Such activities, and those underlying scientific knowledge, are both

essentially based on the same method, the analytic method. Our hunter-gatherer

ancestors solved their survival problem by means of the analytic method, making

hypotheses about the location of predators or prey on the basis of crushed or bent

grass and vegetation, bent or broken branches or twigs, mud displaced from streams,

and so on.

That scientific knowledge is an extension of everyday knowledge is related to

the fact that they are both ways of dealing with the world. The world is an aleatory,

unstable and unsafe place, its dangers are irregular, and human beings can survive in

it only using everyday knowledge and scientific knowledge.

16. An Alternative Logic Paradigm

What has been said above suggests an alternative logic paradigm, based on the view

that logic is to provide means for acquiring knowledge. Logic can actually provide

such means because it can provide non-deductive rules by means of which one may

find hypotheses for solving problems in the analytic method. The alternative logic

paradigm also includes deductive rules, but their role is not a primary one because,

as it has been mentioned in section 10, most inferences we make are not deductive.

The alternative logic paradigm yields a new view of the relation of logic to

discovery. While, according to mathematical logic, logic cannot provide means of

discovery, according to the alternative logic paradigm, logic can provide means of

discovery. They consist of the analytic method and the non-deductive rules by

means of which hypotheses are obtained in such method.

Also, the alternative logic paradigm yields a new view of the relation of logic to

method. While, according to mathematical logic, logic does not originate from

method, according to the alternative logic paradigm, logic arises from method,

because it is developed to implement the analytic method.

Finally, the alternative logic paradigm yields a new view of the relation of logic

to reason. While, according to mathematical logic, logic is the whole of reason,

according to the alternative logic paradigm, logic is not the whole of reason, because

the choice of hypotheses for solving a problem is not uniquely determined by the

problem and the other data already available. It can be positively influenced by

feelings and emotions, so feelings and emotions are part of reason.

Acknowledgments

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