"We are less certain than ever about the ultimate foundations of logic and mathematics"
(Hermann Weil),
"Logic has no existence independent
of mathematics" (Tobias Dantzig).
"Logic fills the world; the limits of the world are also its limits" (Wittgenstein).
The Logics
Mathematical Logic
Historically, mathematics and logic have been distinct disciplines. But modernly logic has become more mathematical and mathematics more logical. Today it is impossible to draw a dividing line between the two.
Mathematics formalized logic to create so-called "mathematical logic". Its main proponents were Boole, Peano, Frege and Russell:
Boole created the algebra of logic, binary logic, based on two truth values (True, False), or their equivalent binary values (0, 1).
Frege created predicate logic and introduced quantifiers.
Peano contributed to its formalization by devising new symbols for set theory and for quantifiers. He also argued that logic should be based on implication. For Peano, mathematical logic was not only the mathematization of logic but also the logic of mathematics.
Russell made contributions to mathematical logic by attempting to formalize mathematics through logic in his work (in collaboration with Whitehead) Principia Mathematica.
The new logics.
Until practically the end of the 19th century there was only one logic, the classical Aristotelian logic. This logic was considered to be the only one that could exist, so this subject was considered complete and closed. Nowadays there is a profusion of different logics (besides the classical one): modal, fuzzy, linear, intuitionistic, paraconsistent, paracomplete, equational, etc. . These logics have been created so that they could be applied to specific domains (computer science, artificial intelligence, cognition, etc.), since classical logic is not sufficiently generic and flexible.
Faced with this situation of confusion, it becomes necessary to go towards the foundation of logic, in the abstract, to search for its essence and to see if all these logics can be derived from a fundamental logic, which we can call "generic", "abstract" or "universal".
For this, what is needed is not a symbolic formalization, but a philosophical logic, a logic based on primitive concepts. Formalism must be discarded, because it is a mere game of manipulation of symbols, by means of certain rules, a blind mechanism, without semantics.
Characteristics of this universal logic must be:
That it contemplates the logic of deduction and the logic of decision.
That it be independent of the syntax of the formal language used.
That it allows expressing other types of logic (derived logics), so that in each domain the most appropriate logic is applied.
That it integrates and connects with the fundamental concepts of mathematics.
The development of non-classical logics began on May 18, 1910, when the logician and philosopher Nikolai Vasiliev published the article "On Partial Judgments, Triangle of Opposition, Law of the Excluding Room", in which he expounded the principles of an "imaginary logic", inspired by Lobatchevsky's non-Euclidean geometry (also initially qualified by its author as "imaginary"). In the same way that this geometry relaxes Euclid's V postulate (the postulate of parallels), in imaginary logic the principle of contradiction and the law of the exclusive third are relaxed to admit three kinds of judgments: affirmative, negative and indifferent. This new logic, like imaginary geometry, was also consistent (or paraconsistent, i.e., with a certain tolerance of contradiction).
This new logic was justified for several reasons:
Because classical logic was insufficient to deal with complex problems.
To create new paradigms, new ways of looking at things, and even to create new imaginary worlds. Classical logic is a limited instrument of knowledge, valid for a limited class of worlds (classical worlds). Imaginary worlds obey laws of imaginary logics.
To be able to apply the experimental method, as in the natural sciences.
Because of the so-called "logical relativism". People from different cultures may have different logics. What is irrational in one culture may be rational in another.
Vasiliev did not make a formal development of his ideas. A.I. Arruda [1977] formalized some of Vasiliev's ideas.
There is a great variety of specialized logics, without a common root, i.e. without a basic or fundamental logic from which to derive all other logics (see Addendum).
Limitations of formal logic.
The limitations of formal logic (or mathematical logic) are the following:
Logical operators are not generic, since they operate or act exclusively on logical (True, False), or Boolean (0, 1) values or variables. They do not act on any type of arguments.
The notation of predicates is not very expressive. For example, the sequence of symbols Px indicates that element x has predicate P. In addition, there is a disconnection between syntax and semantics since there is no explicit operator. And it is not generic, since it is restrictive with respect to the types of elements and predicates. For example, how to express a predicate of a function, a rule, an object, etc.?
The connective "logical implication" x→y is defined in Boolean logic in an unnatural and not very conceptual way as ¬x∨y, the so-called "material implication". It is more appropriate to ground Boolean logic from the concept of "condition" (if x, then y).
There are two types of logic: the logic of deduction (or inference) and the logic of decision (choosing one or more options from a set of possible options). But these two types of logic should be two types of application of the same logical mechanism or resource: the condition.
The concept of "truth"
The concept of "truth" is linked to logic and should be something essentially "logical". However, it is a controversial and ambiguous concept.
It is based on the true-false duality. In this sense, it is the most superficial concept of all. But truth is something transcendent, something associated with consciousness. Truth, like consciousness, is something that cannot be defined because it is supposed to be beyond the mind. According to Frege, truth is a concept impossible to define.
Truth is usually founded on the so-called "principle of correspondence" between the internal (what is conceived at the mental level, and whose semantics could be expressed by a proposition ) and the external (reality). But this principle is diffuse and difficult to define. Frege saw this correspondence as a simple "truth function", which assigns to each statement a truth value. For example, v(Frege is German) = V (truth), where v is the truth function.
There are propositions that are constructions and that it makes no sense to assign them a truth value (V or F), for example, an algebraic expression (such as x+3y) or the definition of a function (such as f(x) = 3x+2). Therefore, the concept of truth value (or truth function) is not universal; it is a concept unique to logic. A particular type of statements or expressions to which it makes sense to assign a truth value are usually called "logical propositions".
There are things that we do not know whether or not they are true, especially when infinity is involved. For example, we do not know whether or not certain mathematical entities exist, or whether they have a certain property. Brower and Heyting (the two main figures of the intuitionist school) claimed the impossibility of knowing whether the sequence 0123456789 appears in the decimal development (infinity) of π (although it was later located in 1997 by Yasumada Kanada and Daisuke Takahashi, of the University of Tokyo, after computing 17 trillion decimals). However, this was only an example, which can be complicated by choosing a much longer sequence. Therefore, the principle of the excluded third party (something is either true or false) does not always hold.
Truth should be considered as a magnitude to be able to speak of "quantity", degree of truth or truth factor (a number between 0 and 1). For example, 0.7*V and where it is verified that 0.7*V ≡ 0.3*F. In general, f*V ≡ (1−f)*F
In formal axiomatic systems, a proposition is true if it is provable. Thus truth becomes relative to the formal axiomatic system. Only consistency matters, which is the absence of contradiction, that is, that a proposition and its contrary cannot be deduced.
According to Gödel's incompleteness theorem, in formal axiomatic systems that include the arithmetic of natural numbers, there are undecidable propositions, i.e., that it cannot be proved (with axioms and rules of inference) whether they are true or false.
It is better to replace the concept "truth" by that of "existence", which is much more general and more concrete, less fuzzy and more manageable:
"Existence" understood, not in the physical world, but in formal, abstract or mathematical space.
"Existence" of an expression in general, which can be a relation, a set, a function, etc.
"Existence" refers to something that a system can verify internally by itself, i.e. it is self-sufficient, without the need to resort to external elements.
Logical paradoxes.
Logical paradoxes have traditionally been considered a source of problems for mathematics, to the point that they have affected its very foundation.
The most famous paradox is what we call today "Russell's paradox": "Does the set of sets that do not belong to themselves belong to themselves?". Put another way: "The set whose elements are all the sets that are not elements of themselves, is it or is it not an element of itself?". The answer to this question is both true and false. This paradox arose out of Frege's attempts to formalize Cantor's set theory, and which Russell communicated to Frege in 1902. In general, Russell's paradox states that it is not possible to assume that every property determines a set of entities having that property, since some properties lead to contradiction.
Paradoxes arise from the confluence of two circumstances: self-references and true-false dualism. But paradoxes disappear when truth or falsity is considered as a mere attribute of a mathematical entity, and self-references become, as we shall see, fractal expressions (expressions in which the same pattern is repeated indefinitely). The concept of fractal is of great importance, since it is linked to the structure of the mind, consciousness and reality itself.
Paradoxes are in fact expressions of consciousness, since two opposite concepts meet or converge. And not only should we not reject them, but we should accept them because they are precisely the expressions that have the greatest power. Paradoxes do not take us away from the truth and the fundamentals, but quite the contrary.
Formal axiomatic systems.
Mathematics, since Euclid, uses the axiomatic method. A formal axiomatic system is composed of:
A finite set of symbols for constructing formulas.
A formal grammar, a mechanism for constructing valid formulas.
A set of axioms.
A set of rules of inference.
Theorems that can be proved from axioms and rules of inference.
A formal axiomatic system must fulfill three properties:
Consistency. It must not be possible to prove a formula and its negation.
Completeness. All valid formulas (true under any interpretation) must be provable.
Decidability. There must be an effective method for deciding whether a formula is true or false.
But the axiomatic method presents problems:
It is limited by Gödel's incompleteness theorem.
Sometimes it is difficult to prove the consistency of axioms.
There is no clear distinction between axioms and definitions.
It is only reasoning oriented. It lacks the operational and descriptive components.
Different rules of inference are used. There is no consensus on the rules to apply.
In the latter aspect, 3 rules are usually used:
Union rule:
If A and B are theorems, then A∧B (logical conjunction) is also.
Rule of separation (modus ponens):
If A and A→B (logical implication) are theorems, then B is a theorem.
Substitution rule:
If A is a theorem in which p1, ... , pn and B1, ... , Bn appear. ,Bn are sentences, then substituting in Ap1=B1, ... , pn=Bn, is a theorem.
In Principia Mathematica these three rules were used.
In propositional logic, the rules of separation and substitution are used. The rule of union is not a rule of inference but a constructive rule.
Robinson's resolution principle uses only one rule: From A∨B and A∨¬B follows A.
But the only "natural" rule to apply is the rule of separation, as already intuited by Peano and also defended by Russell in "The Principles of Mathematics": "When our minds are fixed on inference, it seems natural to take 'implication' as the fundamental primitive relation."
Addendum
The different logics.
The following list of logics is not necessarily exhaustive:
Arruda, A.I. On the imaginary logic of N.A. Vasile’v. En A.I. Arruda, N.C.A. da Costa & R.B. Chuaqui, eds. Non-Classical Logics, Model Theory, and Computability, North-Holland, Ámsterdam, 1977.
Deaño, Alfredo. Introducción a la lógica formal. Alianza editorial, 1975.
Ferrater Mora, José; Leblanc, Hughes. Lógica Matemática. Fondo de Cultura Económica, 1962.
