Logic: an Art of Thinking, a Branch of Mathematics

1. Traditional logic

1.1 The Aristotelian framework

Aristotle does not use the word “logic” in his logical and ontological writings gathered in the Prior and Posterior Analytics. In his own words, he deals with “demonstrative analytical behavior (reasoning, discourse)”, which corresponds “to the current meaning of the term logic.” (Kotarbinski [1964], p. 5; Woods 2014). The 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 separating demonstration from dialectical and rhetorical syllogisms” (In Aristotle, SA, I, 2, 15-25; Note 3 p. 8). The scientific syllogism produces categorical knowledge, the dialectical syllogism produces probable, that is criticized, knowledge where no categorical knowledge is available, and the rhetorical syllogism produces persuasive representations. The position of persuasion in the rhetoric of Aristotle 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 of a theory of syllogism.

1.2 Neo-Thomist 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, i.e. 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:

(Maritain 1923, p. 1; capitals in the text)

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

The following definition stresses the normative value of “formal logic” 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)

S. 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 first defined logic not in the manner of Aristotle 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 covers a set of theses and techniques synthesizing proposals of Aristotelian, Stoic or Medieval origin. It consists in 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, of complex propositions on the basis of simple or complex propositions, and with the determination of valid formulas (logical laws, tautologies).

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

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

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

Identitya = 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 logics no longer regard these principles as laws of thought, but as possible axioms, among others.

The contemporary era saw the multiplication of “unconventional” logical formalisms, sometimes inspired by certain phenomena of ordinary language 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 considered the art of thinking correctly, that is, of combining propositions in such a way as to convey the truth of the premises to the conclusion, in a universe of shared and stable symbols and meanings. Logic provides the theory of rational discourse and of scientific argumentation by defining and determining the valid reasoning schemes.

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 of expansion and discoveries offered by mathematical models. Nonetheless, it certainly has its place in relation to ordinary thinking, anchored in ordinary language. It indeed illuminates the necessity to take into account the progressive and multi-dimensional construction of an argument, articulating words and concepts into judgments, and propositions into arguing 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”, then delimits it scope: “some men”, “all the men”.

Judgment: the mind constructs a proposition, affirming or denying something about this delimited 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 this concept in a statement; and arguing.

— Naming: Speaking of something clearly delimited. The concept is anchored in language by a term according its quantity, S. Proposition.

— Predicating: Saying something about this delimited concept, that is constructing a proposition (a linguistic statement) by imposing a predicate on this term.

Arguing: Composing the statements orderly into the premises of a discourse so as to produce a new proposition, the conclusion, developed exclusively from the premises which are already known. Argumentation on the discursive level thus corresponds to reasoning on the cognitive level.

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

2.2 Logic as the art of reasoning and the emergence of scientific method

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

From the point of view of argumentation, this evolution began in the Renaissance, and can be traced back to Ramus (Ong 1958), for whom judgment, logic and method must be considered as stand-alone operations we would call epistemic or cognitive, independent from rhetoric and language. The mutation appears clearly if 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]) to 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 argument is never considered, as shown by the case of analogy, which is reduced to mathematical proportion.

2.3 Mathematization of logic

Logic is by its nature formal, it is interested not in the content (in substance, in the particular objects) of reasoning, but in the form. In contemporary times it has been axiomatized and mathematized. The publication of Frege’s Begriffschrift, “Concept Writing” in 1879 set the point from which logic cannot be seen 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 built at will. And this plurality of logics withdraws its privileges to classical logic, which is now merely 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 renounce its reflexive and critical function over 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 questioned, in the 1950s and 1970s, by the Natural, Non-formal and Substantial logics. Classical logic can indeed also be appended to this list.

2.4 Neo-Thomism: Resistance to the formalization trend

In 1879, the year when Frege published the Begriffschrift, Pope Leo XIII established Thomas Aquinas and his interpretation of Aristotelianism as a quasi-official philosophy of the Catholic Church in the Aeterni Patris Encyclical. This decision was certainly unfortunate, insofar as it promoted an outdated vision of logic. Nonetheless, it has brought about a powerful trend of research and teaching on classical logic as a method of thought and as an analytic frame for natural language cognition. Substantial developments relating to classical logic constructions and interesting considerations on arguments schemes and sophisms can be found in textbooks for the Neo-Thomist philosophical curriculum at a higher level.

Under various agendas, Maritain’s Logic (1923), Tricot (1928), Chenique (1975) reflect this continuing interest in classical logic. This trend may be compared and contrasted with the so-called revivals of rhetoric that developed from the fifties onward.

3. Pragmatic logic and argumentative calculations

In a quite different tradition, that of the philosophy or ordinary language, Toulmin was the first to suggest that the formalization movement in logic required an accompaniment and counterpart able to address “logical practices”, ([1958], p. 6), mobilizing “substantial” and “field-dependent” argument (id., p. 125; p. 15). He sought a logic which would be a “generalized jurisprudence” (id., p. 7), whose primary purpose would be “justificatory” (id., p. 6).

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

Unlike other theories of argumentation, and perhaps in opposition to the utter rejection of logic by the New Rhetoric, Informal Logic and Natural Logic have retained the word logic in their name, perhaps to stress the fact that, beyond their specific difference 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 severely regimented ordinary language, whilst the speaker of natural language is a virtuoso of contextualization, implicitness and polysemy. These characteristics are constitutive of the efficiency, dynamism and adaptability of natural language in ordinary life circumstances and the possibilities of strategic management of the worlds of action and interaction. Nevertheless, these observations do not imply any rejection of logic: the practice of ordinary discourse necessitates logical competences, just as it necessitates some arithmetical capacities: “It takes about two hours to reach the refuge, night will falls in about one hour, we will arrive at the refuge in the dark; that is risky”; “some mushrooms are edible, not all: you can’t cook any mushroom like that, that is risky”.

4. Entries concerning classical logic

— Predicate Logic: S. Proposition; Syllogism

— Propositional Logic: S. Connectives