LOGIC:
Art of Thinking, Branch of Mathematics

 

1. Traditional Logic

1.1 The Aristotelian Framework

Aristotle does not use the word “logic” in his logical and ontological writings collected in the Prior and Posterior Analytics. In his own words, he is concerned with “demonstrative analytical behavior (reasoning, discourse)”, which corresponds “to the current meaning of the term logic.” (Kotarbinski [1964], p. 5; Woods 2014). Posterior Analytics defines scientific knowledge:

We attain knowledge through demonstration […] I call demonstration a scientific syllogism. (Post. An., I, 2; Owen, p. 247)

It follows that “it is necessary that demonstrative science should be from things true, first, immediate, more known than, prior to, and the causes of the conclusion” (ibid).

In a note added to this passage, Tricot points out that “syllogism is the genre (“producer of science”) common to demonstrative, dialectical and rhetorical syllogisms; scientific is the specific difference that distinguishes demonstration from dialectical and rhetorical syllogisms” (In Aristotle, SA, I, 2, 15-25; Note 3 p. 8). Scientific syllogism produces categorical knowledge, the dialectical syllogism produces probable, i.e. criticized, knowledge, where categorical knowledge is not available, and the rhetorical syllogism produces persuasive representations. The place of persuasion in Aristotle’s rhetoric should be understood within this framework.

Traditional logical theory is based on an analysis of propositions as subject-predicate constructions, on a definition of the relations between the four forms of a general proposition and on a theory of syllogism.

1.2 Neo-Thomistic logic

In the Middle Ages, Thomas Aquinas took up the Aristotelian definition of logic and defined it in relation to the reflexivity of the act of reasoning, that is “its ability to reflect upon itself”:

An art is needed to direct the act of reasoning, so that by it a man when performing the act of reasoning might proceed in an orderly and easy manner and without error. And this art is logic, that is, the science of reason. (Com. Post. An., “Foreword”)

This definition is taken up by the Neo-Thomist tradition, especially by Maritain, who defines logic as:

The art THAT DIRECT THE VERY ACT OF REASON.
(Maritain 1923, p. 1; capitals in the original)

This definition is taken up by Chenique in his Elements of Classical Logic (1975).

The following definition emphasizes the normative value of “formal logic”, which is defined as:

A science that determines the correct (or valid) forms of reasoning.”
(Dopp 1967, p. 11, italics in the original)

1.3 Logic and inference

In mathematics, logic is defined as :

The discipline that deals with correct inference. (Vax 1982, Logic)

Logic is concerned with the principles of valid inference. (Kneale and Kneale, [1962], p. 1)

See inference. Logic is the study of the valid forms of deduction:

Logic has the important function of saying what follows from what. (Kleene, 1967, Chap. 1, §1)

1.4 Logic is a science

Logic, like any science has as its business the pursuit of truth. (Quine, 1959, p. xi)

The Stoics were the first to define logic not in Aristotelian terms as an organon, an instrument (in the service of the sciences), but as a science.

1.5 Classical logic

Classical logic (or traditional logic, according to Prior 1967) is by nature a formal logic: it is one of the revolutionary merits of Aristotle to have introduced a systematic use of variables. Classical logic includes a set of theses and techniques that synthesize proposals of Aristotelian, Stoic or Medieval origin. It consists of two parts:

— The logic of analyzed propositions or predicate calculus, and the theory of the syllogism.

—The logic of unanalyzed propositions or propositional calculus, which deals with the construction, using logical connectives, and with of complex propositions from simple or complex propositions, and with the determination of valid formulas (logical laws, tautologies).

Classical logic is based on a set of principles, that are considered to be laws of thought and rational discourse:

— Non-contradiction, “non-(P and non-P)”; a proposition cannot be both true and false.

— Excluded middle (excluded third), “either (P or non-P)”; a proposition must be true or false.

— Identity “a = a”, and its practical consequences, such as the principle of indiscernibility and intersubstitutability of the identicals, and the unicity and stability of meaning of the logical symbols in the same universe of discourse (same reasoning).

Contemporary logic no longer considers these principles as laws of thought, but as possible axioms, among others.

The contemporary era has seen the multiplication of “unconventional” logical formalisms, sometimes inspired by certain phenomena of ordinary language that are not taken into account by classical logic, such as time or modality.

2. Logic: An Art of Thinking, a Branch of Mathematics

2.1 The Three Operations of the Mind

From Aristotle to the end of the nineteenth century, classical logic was regarded as the art of correct thinking, that is, of combining propositions in such a way as to convey the truth of the premises to the conclusion, in a universe of common and stable symbols and meanings. Logic provides the theory of rational discourse and of scientific argumentation by defining and determining the valid schemes of reasoning.

The theory of the three operations of the mind comes from Maritain (1937, §2-3). For a long time, such an approach was abandoned by logicians, who were legitimately motivated by the fantastic potential for expansion and discovery offered by mathematical models. Nevertheless, it certainly has its place in relation to ordinary thinking, anchored in ordinary language. Indeed it illuminates the need to consider the progressive and multidimensional construction of an argument, articulating words and concepts into judgments, and propositions into arguments and reasoning. Such a model is quite compatible with the idea of schematization as defined in Grize’s Natural Logic.

(i) Argumentation as a mental process

As a mental process, argumentation is defined as the third “operation of the mind”, apprehension, judgment and reasoning.

— Apprehension: The mind grasps a concept, “man”, and then limits it scope: “some men”, “all the men”.

— Judgment: The mind constructs a proposition, that affirms or denies something about this quantified concept: “some men are wise”. This judgment is categorical, it is true or false and nothing else.

— Reasoning: The mind concatenates the judgments without any loss of truth, so as to develop new truths on the basis of known truths.

(ii) Argumentation as a discursive process

In the discursive process, argumentation is defined as the third of the three basic linguistic operations: naming the concept; predicating something of that concept in a statement; and arguing.

— Naming: Speaking of something clearly delimited. The concept is anchored in language by a term corresponding to its quantity, see proposition.

— Predicating: To say something about this delimited concept, that is to construct a proposition (a linguistic statement) by imposing a predicate on this term.

— Arguing: To arrange the propositions of the discourse in an orderly way so as to produce a new proposition, the conclusion, developed exclusively from the already known premises. Argumentation on the discursive level thus corresponds to reasoning at the cognitive level.

In Aristotelian logic, the rules of correct argumentation are given by the theory of syllogism, which distinguishes between valid syllogisms and paralogisms (vicious arguments, fallacies, sophisms).

2.2 Logic as the Art of Reasoning and the Emergence of the Scientific Method

In modern times, this view of logic as a theory of discursive reasoning and the assimilation of discursive reasoning to scientific reasoning has been destabilized by the emergence of the natural sciences and experimental reasoning, based on observation, measurement, prediction and experimentation, all regulated by mathematical calculation. In contemporary times, this development has been complemented by the integration of logic into mathematics. The rules of the scientific method include and go beyond logic.

From the point of view of argumentation, this development began in the Renaissance, and can be traced back to Ramus (Ong 1958), for whom judgment, logic and method must be considered as independent operations that we would call epistemic or cognitive, independent of rhetoric and language. The mutation becomes clear when one compares the Port-Royal Logic, in its full title: Logic, or, the Art of Thinking: Containing, Besides Common Rules, Several New Observations Appropriate for Forming Judgment of Arnauld and Nicole ([1662]) with Condillac’s Treatise on the Art of Reasoning ([1796]). In the latter work, the language of the “art of reasoning” is not syllogistically organized natural language, but geometry. Rhetorical argumentation is never considered, as shown by the case of analogy, which is reduced to mathematical proportion.

2.3 The Mathematization of Logic

Logic is by its nature formal, it is not interested in the content (in the substance, in the particular objects) of argumentation, but in the form. In contemporary times it has been axiomatized and mathematized. The publication of Frege’s Begriffsschrift, “Concept Writing” in 1879 marked the point at which logic could no longer be regarded as an “art of thinking”, but as an “art of calculating”, that is, as a branch of mathematics. At the beginning of the twentieth century, classical logic was overwhelmed by the “twilight of self-evidences” (Blanché 1970, p. 70):

We move from Logic to logics that can be constructed at will. And this plurality of logics withdraws its privileges to classical logic, which is now only one system among others, like them a simple formal architecture whose validity depends only on its internal coherence. (Id., p. 71-72)

To become an axiomatic exercise, logic had to relinquish its reflexive and critical function with respect to common thought and discourse. It could no longer provide the model of rationally argued discourse or dialectical exchange. Logic is now the mathematical discipline, which was challenged, in the 1950s and 1970s, by natural, non-formal and substantial logics. Classical logic should open this list.

2.4 Neo-Thomism: Resistance to the Formalization Tendency

In 1879, the year when Frege published the Begriffsschrift, Pope Leo XIII established Thomas Aquinas and his interpretation of Aristotelianism as the quasi-official philosophy of the Catholic Church in the encyclical Aeterni Patris. This decision was certainly unfortunate, in that it promoted an outdated view of logic. Nevertheless, it has led to a strong trend of research and teaching on classical logic as a method of reasoning and as an analytical framework for natural language cognition. Substantial developments on classical logic and interesting reflections on argumentation schemes and sophisms can be found in textbooks for the neo-Thomistic higher philosophical curriculum.

In France, Maritain’s Logic (1923), Tricot (1928), Chenique (1975) under different agendas, reflect this continuing interest in classical logic. This trend may be compared and contrasted with the so-called revivals of rhetoric that have developed since the fifties.

3. Pragmatic Logic and Argumentative Reasoning

In a quite different tradition, that of the philosophy or ordinary language, Toulmin was the first to suggest that the formalization movement in logic needed an accompaniment and counterpart capable of addressing “logical practices”, ([1958], p. 6), and mobilizing “substantial” and “field-dependent” argumentation (id., p. 125; p. 15). He saw logic as a “generalized jurisprudence” (id., p. 7), whose primary purpose was “justificatory” (ibid., p. 6).

The logico-pragmatic movement which includes non-formal, substantial, natural, and generally dialogical logics, distances itself from axiomatized formalisms in order to take into account the ecological conditions of argumentation. People argue in natural language, and in a given context; classical logic does not satisfy the second condition, but it does satisfy the first, at least for the limited aspects of language it can deal with.

Unlike other theories of argumentation, and perhaps in contrast to the New Rhetoric total rejection of logic, Informal Logic and Natural Logic have retained the word logic in their names, perhaps to emphasize the fact that, beyond their specific differences they do belong to a common genre, S. Argumentation Studies; Demonstration; Proof.

These pragmatic logics must combine with ordinary language and subjectivity. Classical logic has its roots in a highly regimented ordinary language, whereas the speaker of natural language is a virtuoso of contextualization, implicitness and polysemy. These features are constitutive of the efficiency, dynamism and adaptability of natural language in everyday life circumstances and of the possibilities for strategic management of the worlds of action and interaction. However, these observations do not imply a rejection of logic: the practice of ordinary discourse requires logical skills, just as it requires some arithmetic skills:

It will take about two hours to reach the shelter, in about one hour, it will be night, we will arrive at the shelter in the dark; that’s risky.
Some mushrooms are edible, not all, you can’t cook any mushroom like that, that’s risky.

4. Entries concerning classical logic

— Predicate logic: see proposition; syllogism.

— Propositional logic: see connective.