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). The Posterior Analytics define scientific knowledge as follows:

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). A scientific syllogism produces categorical knowledge, a dialectical syllogism produces probable, i.e. criticized, knowledge, when categorical knowledge is unavailable, and a rhetorical syllogism produces persuasive representations. The role of persuasion in Aristotle’s rhetoric should be understood within this framework.

Traditional logical theory is based on 1) an analysis of propositions as subject-predicate constructions,  2) a definition of the relations between the four forms of a general proposition and 3) 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, or “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 adopted for example 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, Ch. 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. One of Aristotle’s revolutionary contributions is the introduction of 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 also known as predicate calculus, and the theory of the syllogism.

—The logic of unanalyzed propositions or propositional calculus, deals with 1) constructing complex propositions from simple or complex propositions, using logical connectives, and 2) determining of valid formulas (logical laws or tautologies).

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

— Principle of non-contradiction, “non-(P and non-P)”; a proposition cannot be both true and false, that is « non-(P and non-P) ».

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

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

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

The contemporary era has seen the multiplication of “unconventional” logical formalisms, some of which draw from phenomena of ordinary language that were not considered 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 until the end of the nineteenth century, classical logic was regarded as the art of correct thinking. In other words, it was the art of combining propositions in a way that convey the truth of the premises to the conclusion, within 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 reasoning schemes.

The theory of the three operations of the mind originates with Maritain (1937, §2–3). This approach had long been abandoned by logicians, who were legitimately motivated by the fantastic potential for expansion and discovery offered by mathematical models. Nevertheless, this approach certainly has its place in relation to ordinary thinking, anchored in ordinary language. It illuminates the need to consider the progressive and multidimensional construction of an argument, the articulation of words and concepts into judgments, and of propositions into arguments and reasoning. Such a model is quite compatible with Grize’s (1996) definition of schematization as defined in Natural Logic.

(i) Argumentation as a mental process

As a mental process, argumentation is defined as the third « operation of the mind », proceeding from apprehension, to judgment and reasoning.

— Apprehension: The mind grasps a concept, such as“man”, and then limits its scope: “some men”, “all 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 either true or false. 

— Reasoning: The mind links the judgments without losing any truth, in order to develop new truths based 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 according to its quantity, see proposition.

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

— Arguing: Arranging the propositions in an orderly discourse, so as to produce a new proposition, the conclusion, developed exclusively from the premises already known. Thus, argumentation on the discursive level corresponds to reasoning at the cognitive level.

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

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

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

From the point of view of argumentation, this development began in the Renaissance, and can be traced back to Ramus (Ong 1958). According to Ramus, judgment, logic and method are independent operations that we would call epistemic or cognitive, and they are independent of rhetoric and language. This mutation becomes clear when comparing 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]) and Condillac’s Treatise on the Art of Reasoning ([1796]). In the latter work, the language of the “art of reasoning” is not that of syllogism, but rather of geometry. Rhetorical argumentation is never considered; for example, analogy is reduced to mathematical proportion.

2.3 The Mathematization of Logic

Logic is  formal by nature, it is not interested in the content of argumentation its substance, its the particular objects, but rather in its form. In contemporary times, logic has been axiomatized and mathematized. Frege’s publication of Begriffsschrift, “Concept Writing” in 1879 marked the point at which logic could no longer be regarded as an “art of thinking”, but rather as an “art of calculating”, that is, as a branch of mathematics. At the beginning of the 20th 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. This plurality of logics removes classical logic’s privileges, reducing it to one system among others. Like them, logic is a simple formal architecture whose validity depends only on its internal coherence. (Id., pp. 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 serve as a model for rationally argued discourse or dialectical exchange. Logic is now a mathematical discipline, that 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’s interpretation of Aristotelianism as the quasi-official philosophy of the Catholic Church in the encyclical Aeterni Patris. This decision was unfortunate, because it promoted an outdated view of logic. Nevertheless, it has produced a substantial research and teaching on classical logic as a method of reasoning and as an analytical framework for natural language cognition. Substantial developments in classical logic as well as interesting reflections on argumentation schemes and sophisms can be found in textbooks for the higher philosophical curriculum of neo-Thomism.

In France, Maritain’s Logic (1923), Tricot (1928), and Chenique (1975)–each with a different focus–reflect this ongoing interest in classical logic. This trend can be compared to the so-called revivals of rhetoric that have emerged since the 1950s.

3. Pragmatic Logic and Argumentative Reasoning

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

The logico-pragmatic movement which includes non-formal, substantial, natural, and generally dialogical logics, distances itself from axiomatized formalisms in order to consider 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’s total rejection of logic, Informal Logic and Natural Logic have retained the word logic in their names. This may be to emphasize the fact that, despite their specific differences, they do belong to a common genre, see argumentation studies; demonstration; proof.

These pragmatic logics must combine with ordinary language and subjectivity. Classical logic has its roots in highly regimented ordinary language, whereas speakers of natural language are virtuosic in their use of contextualization, implicitness and polysemy. These characteristics are constitutive of the efficiency, dynamism and adaptability of natural language in everyday circumstances, as well as the resources for strategically managing  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, which is risky.
Some mushrooms are edible, but not all. You can’t cook and eat just any mushroom, that’s risky.

4. Entries concerning classical logic

— Predicate logic: see proposition; syllogism.
— Propositional logic: see connective.