Classical Logic and Propositional Calculus

A principle that is still the base of today’s classical logic, which is the law of noncontradiction, was deeply discussed and demonstrated during the 4th century BC in Aristotle’s Metaphysics. The law of noncontradiction asserts that a proposition can not be true and false at the same time. More specifically, a digression upon truth can be found in the Γ (Gamma) section, also called the Fourth Book of Metaphysics. Although the formalization of logical notation arose in the second half of the 19th century thanks to George Boole, Aristotle’s effort to defeat the sophistic threat can be seen as the very first delineation of a core notion of formal logic: that of “truth value”. In fact, Aristotle’s attempt to preserve the truth from an ontological point of view, in contrast with Protagoras’ perspective, introduced the logical concept of negation. Not all formal logics follow the law of noncontradiction and not all philosophical perspectives agree with this ontological standpoint. Yet in this article, the classical logic (more in depth about the propositional calculus) will be introduced, and as written before, this kind of formalization originates from the above-mentioned law.

Figure 1: Bronze statue of Aristotle at the University of Freiburg.

The formal notation for classical logic is traditionally identified with the Boolean logic. George Boole (1815-1864) was a British mathematician, logician and philosopher. Despite his poor upbringing, Boole was able to self teach mathematics, philosophy, logic and many different languages. Boole’s contributions to algebra and logic were rewarded with a professorship in Cork, at Queen’s College. Boole's main project was the application of algebraic principles to logic, as illustrated by his work "An Investigation on the Laws of Thoughts" (1854). In this book, the logician extended the Aristotelian logic, giving it a mathematical connotation. Before this intertwining, formal logic was a mathematical subject, but thanks to Boole's work it then became a branch of philosophy. Both Boole and Aristotle recognized that the grammatical form of a sentence doesn't necessarily match with its logical form (Corcoran, 2008). In other words, the common sense that comes from the natural language can be misleading about the formal content of a statement, as it will be illustrated later on this article with an example. Boole started from Aristotle's law of noncontradiction and law of excluded middle to build a system in which these principles are mathematically connotated, as it'll be shown in the next paragraphs.

The three laws of thought are: the noncontradiction law, the excluded middle law and the principle of identity (Cambridge, 1999, p. 489). The excluded middle law, also called "tertium non datur" (translation from Latin: "a third is not given"), is a core notion of the Aristotelian logic. This law asserts that, given a 'p' proposition, there are just two possibilities: 'p' is true or 'p' is false. There are no third options. The principle of identity states that every object is identical to itself: A = A. As shown by the truth tables in the next paragraphs, all the three laws can be grasped from the formalization of the negation operator.

Figure 2: Queens College, Cork

The Boolean logic unfolds in variables which are either true or false. Boole wrote that logic is possible because there are these notions in our mind: general notions, the capacity to recognize a class of objects, and the ability to designate and discriminate them (Boole, 1993, p. 6). This makes logic a field that is deeply connected to our natural language (Boole, 1993, p. 10). Boole’s aim was also philosophical; he believed that the symbolization of logical propositions, in which the symbols function like our mental processes, would have benefited the philosophical language (Boole, 1993, p.6). This is true when considering the contemporary approach to ontology: today, in addition to the formal ontology (which is an abstract theory of the objectual universes), the other technical instrument of an ontologist is the theory of quantification, which originates from the propositional logic (Valore, 2008, p.33). While formal ontology sets the connections among entities, formal logic establishes the relations among truths (Valore, 2008, p. 20). Boole wanted to create a system in which every logic proposition would have been "exact and rigorous" (Boole, 1993, pp. 10-11) and from this start point, much more complex propositional structures would have been inferable. This conception is mirrored in the scholastic definition of propositional logic: a logic in which the assessment of truth of a complex predicate comes from a calculus that considers the truth of each atomic component of it (Boole, 1848, pp. 183-198).

Figure 3: Where Boole speaks of the three laws of thought in 1854

The Boolean logic can surely be described as a deductive system; in fact, every process consists of a deduction, and every syllogism is expressed through an equation. In order to understand better how the propositional calculus functions through formal notation, the truth tables will be introduced, so that also those who don’t know them will begin to have some necessary elementary notions. The truth tables are used in Boolean logic in order to extrapolate the truth value of atomic propositions, which then leads to the calculus of the truth value of more complex sentences. The truth tables are based on logical operators, and in this article there will be an overview on the four most elementary ones: ¬ (negation), Ʌ (conjunction), V (disjunction), → (implication).

At first sight, these connectives seem to retrace the natural language. For example, the operator of conjunction is translated into "and", the operator of disjunction represents "or", and the operator of implication indicates the expression "if... then". But although it is tempting to consider these connectives as a reproduction of everyday speech, their real function is to make the logical content of formal statements explicit (Valore, 2008, p. 40). This distinction between everyday speech and common sense on one hand, and formal logic on the other, will be shown later with a specific example for the case of implication. The most elementary truth table is that of negation. Talking in Boole’s terms of “equations” (as mentioned before from his own book), the operator of negation can be described with the following one:

p= ¬ ¬ p

The symbol ¬ indicates the negation, so the whole proposition means that the truth of p equals the double negation of p. This equation can be translated in the table form:

Figure 4: Truth table of negation, drawn by the author of this article with Latex.

It should be noted that this binary interpretation of a truth assessment can be described also by 1 in replacement of true, and 0 in substitution of false. This is usually the notation used in non-philosophical fields such as computer science, but sometimes it can be found even in ontological works (P. Valore, 2008, p. 35).

There’s a third way in which negation can be described through formal notation, and that is:

p Ʌ¬ p

The symbol Ʌ indicates the operator of conjunction. The conjunction of p and q is true only if both the propositions are true, so the formula that has been written above means that it’s either true that p is true or that its negation is true. This is a way to unfold the above mentioned law of noncontradiction through notation.

More in depth, the truth table of conjunction can be illustrated by the following figure:

Figure 5: Truth table of conjunction, drawn by the author of this article with LateX.

If p Ʌ q, or as the table shows, A Ʌ B, can be true only when the two propositions are both true, it’s consequential that A Ʌ B is a false statement in any other case.

After negation and conjunction, the third operator that will be introduced is that of disjunction, designated by the symbol V. Two disjuncted propositions are true if at least one of them is true.

The next figure will show how this translates in each case.

Figure 6: Truth table of disjunction, drawn by the author of this article with LateX.

The table is pretty intuitive: if it’s enough that either A or B is true to make the whole statement true, the only case in which the proposition A Ʌ B is false is when both A and B are false.

The fourth and last operator that will be taken in consideration in this chapter is the conditional, described by the symbol →. The proposition A→B can be read as “if A, then B”. A is the antecedent while B is the consequent. The way in which the operator functions is described by the next figure.

Figure 7: Truth table of implication, drawn by the author of this article with LateX.

As shown by the table, the only case in which this implication is false is when A is true and B is false. The truth table of implication shows how formal notation makes the logical form of the statements explicit, rather than retracing the natural language. In order to understand this, Valore suggests to consider this sentence: "If I am the Buddha, then I am not the Buddha" (Valore, 2008, p. 41). Reasoning with the common sense of our natural language, this looks like a contradiction: an individual is either the Buddha or not. But from the point of view of formal logic, this sentence can be true or false on a case by case basis. In other words: if the person who is talking is really the Buddha, then the whole statement is false because the consequent would be a negation of the antecedent (which as said, would be true), and as illustrated by the table, this is the only case in which the conditional is false: when the antecedent is true and the consequent is false. But if the person who is talking is not the Buddha, the whole sentence is true from a formal logical point of view, since the antecedent would be negated, and when this happens, the conditional is always true.

The propositional calculus is the floor upon which the classical logic is built, and understanding how it works is the first step towards the comprehension of this topic. As written at the beginning of this article, the classical logic is based on the law of noncontradiction, which then translates into a binary conception of truth. How could the truth value change its shape in different logical systems? Can the Boolean concept of truth assessment become more complex? And can it be extended by philosophical notions such as possibility and necessity, or such as probability and uncertainty? These questions will be unfolded throughout the series. The next episode of 101 will exhibit how propositional calculus develops into a logic of first order, through quantification. With propositional calculus and first order logic, the delineation of classical logic will be complete, so it'll be possible to explore alternative philosophical worlds in which the main principles of the Boolean logic (such as the law of noncontradiction) can be extended or even denied.

BIBLIOGRAPHICAL REFERENCES:

Audi, R. (Ed.). (1999). Laws of Thought. In Cambridge Dictionary of Philosophy (2nd ed., p. 489). essay, Cambridge University Press.

Boole, G. (1993). L'analisi Matematica della logica. Bollati Boringhieri. Boole, G. (1845). The Calculus of Logic. In The Cambridge and Dublin Mathematical Journal (Vol. 3). essay.

Broadbent, T. A. A. (1964). George Boole (1815-1864). The Mathematical Gazette, 48 (366), 373–378. https://doi.org/10.2307/3611693 Cassin, B. (1993). IL SENSO DI «GAMMA» LA STRATEGIA DI ARISTOTELE CONTRO I PRESOCRATICI IN “METAFISICA”, IV. Rivista Di Filosofia Neo-Scolastica, 85(2/4), 533–565. http://www.jstor.org/stable/43062995 Corcoran, J. (2003). Aristotle's Prior analytics and Boole's Laws of Thought. History and Philosophy of Logic, 24(4), 261–288. https://doi.org/10.1080/01445340310001604707 Valore, P. (2008). In L'inventario del mondo: Guida Allo Studio dell'ontologia (pp. 19–45). essay, UTET università.

VISUAL SOURCES:

Cover Image: "Three philosophy painting" by Samvel Marutyan. Retrieved from: https://www.saatchiart.com/art/Painting-Three-Philosophy/984121/3599162/view Figure 1: Image by Couleur from Pixabay. Online source: https://pixabay.com/photos/sculpture-bronze-figure-aristotle-2298848/

Figure 2: Image by wellness555 from Pixabay. Online source: https://pixabay.com/photos/cork-university-college-cork-2199757/ Figure 3: Cover Copy, An Investigation of the Laws of Thought: On Which Are Founded the Mathematical Theories of Logic and Probabilities by George Boole, 2009 (1854), Cambridge Library Collection - Mathematics. Online Source: https://www.cambridge.org/tr/academic/subjects/mathematics/historical-mathematical-texts/investigation-laws-thought-which-are-founded-mathematical-theories-logic-and-probabilities?format=PB Figure 4: Truth table of double negation, drawn by the author of this article with LateX. Online source for the creation: https://it.overleaf.com/ Figure 5: Truth table of conjuction, drawn by the author of this article with LateX. Online source for the creation: https://it.overleaf.com/

Figure 6: Truth table of disjuction, drawn by the author of this article with LateX. Online source for the creation: https://it.overleaf.com/ Figure 7: Truth table of implication, drawn by the author of this article with LateX. Online source for the creation: https://it.overleaf.com/