Logic

Scheme of the modus puts, a fundamental inference rule of propositional logic.

Logic is a branch of philosophy of an interdisciplinary nature, understood as the formal science that studies the principles of demonstration and valid inference, fallacies, paradoxes and the notion of true.

Logic is divided into several categories depending on its field of study. Philosophical logic studies the concept and definition, the statement or proposition and the argumentation using the methods and results of modern logic for the study of philosophical problems. Mathematical logic studies inference through formal systems such as propositional logic, first-order logic, and modal logic. Informal logic focuses on the linguistic development of reasoning and its fallacies. Computational logic is the application of mathematical logic to computer science.

The origins of logic date back to ancient times, with independent outbreaks in China, India, and Greece. Since then, logic has traditionally been considered a branch of philosophy, but in the 20th century logic has become primarily mathematical logic, and is therefore now also considered part of mathematics, and even an independent formal science.

There is no universal agreement on the exact definition or limits of logic. However, the scope of logic (broadly interpreted) includes:

• The classification of arguments.
• Systematic analysis of logical forms.
• The systematic study of the validity of deductive inferences.
• The strength of inductive inferences.
• The study of defective arguments, such as fallacies.
• The study of logical paradoxes.
• The study of the syntax and semantics of formal languages.
• The study of the concepts of meaning, denotation and truth.

Historically, logic has been studied primarily in philosophy since ancient times, in mathematics since the mid-19th century, and in computer science since the mid-20th century. More recently, logic has also been studied in linguistics and in cognitive science. In general, logic remains a strongly interdisciplinary area of study.

Etymology and meanings

The word «logic» derives from the ancient Greek λογική logikḗ, which means «endowed with reason, intellectual, dialectical, argumentative» and which in turn comes from λόγος (lógos), "word, thought, idea, argument, reason or principle".

In everyday language, expressions such as «logic» or «logical thinking» also provide a meaning around a comparative «lateral thinking», making the contents of the statement coherent with a context, either of discourse or of a theory of science, or simply with the beliefs or evidence transmitted by the cultural tradition.

In the same way there is the sociological and cultural concept of logic such as «sports logic», which in general, we could consider as «everyday logic» - also known as «common sense logic».

In these areas, «logic» usually has a linguistic reference in pragmatics.

An argument in this sense has its "logic" when it is convincing, reasonable and clear; in short, when it fulfills an efficiency function. The ability to think and express such an argument corresponds to rhetoric, whose relation to truth is a probable relation.

Themes

Inference

Validity

Fallacies

Paradoxes

This section is a Paradox extract.[chuckles]edit]

The impossible cube is a paradoxical object.

A paradox (from Latin paradox, ‘the opposite of common opinion’) or antilogy is a logically contradictory or opposed idea to what is considered true to the general opinion. It is also considered paradox a proposition in false appearance or that violates the common sense, but does not imply a logical contradiction, in contrast to a sophisma that only appears to be a valid reasoning. Some paradoxes are seemingly valid arguments, starting from real appearance premises, but leading to contradictions or situations contrary to common sense. In rhetoric, it is a figure of thought that consists in employing expressions or phrases that imply contradiction. Paradoxes are stimulus for reflection and philosophers are often used to reveal the complexity of reality. The paradox also demonstrates the limitations of human understanding; the identification of paradoxes based on concepts that seem simple and reasonable has prompted important advances in science, philosophy and mathematics.

Truth

This section is an extract of Truth.[chuckles]edit]

Truth and Falseness by Alfred Stevens, 1857-66.

Time saving the Truth of Falseness and Envy, fabric of François Lemoyne, 1737.

The truth is the coincidence between an affirmation and the facts, or the reality to which such an affirmation refers or fidelity to an idea. The term is used in a technical sense in various fields such as science, logic, mathematics and philosophy.

The use of the word Truth It also encompasses honesty, good faith and human sincerity in general; also the agreement of knowledge with the things that are affirmed as realities: the facts or the thing in particular; and, finally, the relation of the facts or things in their entirety in the constitution of the All, the Universe.

Things are true when they are “fiable”, faithful because they fulfill what they offer.

The term does not have a single definition in which the majority of scholars and theories about truth are agreed to continue to be widely discussed. There are different positions on issues such as:

• What constitutes the truth.
• How we can identify and define it.
• If the human being has innate knowledge or can only acquire it.
• Whether there are revelations or truth can be attained only through experience, understanding and reason.
• If the truth is subjective or objective.
• If the truth is relative or absolute.
• And to what extent can each of these properties be affirmed.

This article seeks to introduce the main interpretations and perspectives, both historical and current, on this concept.

The question for the truth is and has been the subject of debate among theologians, philosophers and logics over the centuries considering itself a subject concerning the soul and the study of a call rational psychology within the field of philosophy.

It is currently a subject of scientific research and philosophical foundation:

• Scientific research of cognitive function introduces new perspectives on evidence-based knowledge as epistemologically true belief with valid justification.
• It interferes with linguistics because language is an expression of the truth itself.
• It interferes with philosophical anthropology, since it seems evident that humans prefer truth to falsehood, error or lie and prefer certainty to doubt.
• It is interesting to history, because appreciation for the truth and condemnation of lies or error varies in intensity according to the times and cultures, for both the concept of truth and its valuation is not always the same throughout history and according to different cultures.
• It interferes with science as such in its claim of valid knowledge.

The importance of this concept is that it is rooted in the heart of any alleged personal, social and cultural. Hence his complexity.

Branches

Mathematical logic

Computational Logic

This section is a computational Logic extract.[chuckles]edit]

Visual logic diagram

Computer logic is the same mathematical logic applied to the context of computer sciences. Its use is fundamental in several levels: in computer circuits, in logical programming and in the analysis and optimization (of temporal and spatial resources) of algorithms.

Logic extends to the heart of computing as it emerges as a discipline: Alan Turing's work on Entscheidungsproblem followed by the work of Kurt Gödel on incomplete theorems. The notion of the general-purpose computer that emerged from this work was of great importance to the computer machine designers in the 1940s.

In the 1950s and 1960s, research predicted that when human knowledge could be expressed using logic with mathematical notations, it would be possible to create a machine capable of reasoning or artificial intelligence. This was more difficult than expected because of the complexity of human reasoning. In the programming logic, a program consists of a collection of axioms and rules. The logical programming systems (such as Prolog) calculate the consequences of axioms and rules organized to respond to a consultation.

Today, logic is widely applied in the fields of artificial intelligence and computer science, and these fields provide a rich source of problems in formal and informal logic. The theory of argumentation is a good example of how logic is being applied to artificial intelligence. The ACM computer classification system, in particular, considers:

• Section F.3 in Logicals and Meanings of Programs and F.4 in Mathematical Logic and formal Languages as part of the theory of computer science: this work covers the formal semantics of programming languages as well as the work of formal methods as Hoare's logic.
• Boolean logic as a foundation on computer hardware, particularly the section of the B.2 system in the arithmetic and logical structure, related to AND, NOT and OR operators.
• Many fundamental logical formalisms are essential to section I.2 on artificial intelligence, for example modal logic and default logic in formalisms and methods of representation of knowledge, Horn clauses in logical programming and description logic.

In addition, computers can be used as tools for logic. For example, in symbolic logic and mathematical logic, human testing can be assisted by computers. Using the automated test of the theorem, machines can find and test tests as well as work with the tests too long to write on hand.

Philosophical Logic

Informal Logic

Syllogistic logic

A representation of the fifteenth century of the Opposition Table of the Judgments that expresses the fundamental dualities of the siloge.

The Organon was Aristotle's body of work on logic, the First Analytics constituting the first explicit work on formal logic, introducing syllogistics. Syllogistic logic, also known as term logic, are the analysis of judgments in propositions consisting of two terms that are related by one of a fixed number of relations, and the expression of inferences by means of syllogisms consisting of two propositions. that share a common term as a premise, and a conclusion that is a proposition involving the two unrelated terms of the premises.

Aristotle's work was considered in classical times and from medieval times in Europe and the Middle East as the very image of a fully elaborated system. However, it was not the only one: the Stoics proposed a system of propositional logic that was studied by medieval logicians. Also the problem of multiple generality was recognized in medieval times. However, the problems of syllogistic logic were not seen as needing revolutionary solutions.

Today, some scholars claim that Aristotle's system is generally seen as having little more than historical value (although there is some current interest in extending the logic of terms), considered obsolete by the advent of propositional logic and the calculus of predicates. Others use Aristotle in argumentation theory to help develop and critically challenge the "argumentation schemes" used in artificial intelligence and legal arguments.

Propositional Logic

A propositional calculation or logic (also a sentencing calculation) is a formal system in which formulas that represent propositions can be formed by combining atomic propositions (usually represented with p, q, etc.) using logical connectives (And,→ → , ,≡ ≡ ,♥ ♥ ,{displaystyle And, rightarrowlorequivsim} etc.); these propositions and connectives are the only elements of a standard propositional calculation. Unlike the logic of preached or the siloge logic, where individual subjects and preached (who have no truth values) are the smallest unity, the propositional logic takes full propositions with truth values as its most basic component. Quantifiers (e.g., paratordors{displaystyle paratodos} or existe{displaystyle exists}) are included in the extended propositional calculation, but only quantify on complete propositions, not on individual subjects or preached. A given propositional logic is a formal test system with rules that establish which well-formed formulas of a given language are "theorems" showing them from axioms that are assumed without proof.

Modal Logic

In language, modality deals with the phenomenon that subparts of a sentence can have their semantics modified by special verbs or modal particles. For example, "Let's go to the games can be modified to give "We should go to the games', and " We can go to the games and maybe "We will go to the games. In a more abstract way, we could say that the modality affects the circumstances in which we take a statement for granted. Modality confusion is known as modal fallacy.

Aristotle's logic is largely concerned with the theory of non-modalized logic. Although, there are passages in his work, such as the famous naval battle argument in On Interpretation § 9, which are now seen as anticipating modal logic and its connection to potentiality and time, the first formal system of modal logic was developed by Avicenna, who eventually developed a theory of "modalized temporality" syllogistic.

Although the study of necessity and possibility remained important to philosophers, few logical innovations occurred until the landmark investigations of C. I. Lewis in 1918, who formulated a family of rival axiomatizations of random modalities. His work unleashed a torrent of new work on the subject, expanding the types of modality treated to include deontic logic and epistemic logic. Arthur Prior's seminal work applied the same formal language to treat temporal logic and paved the way for the union of the two themes. Saul Kripke discovered (contemporaneously with Prior) his theory of Kripkean semantics, which revolutionized the formal technology available to modal logicians and gave a new graph-theoretic way of looking at modality that has prompted many applications in computational linguistics and computer science, such as dynamic logic.

Predicate Logic

Gottlob Frege's Begriffschrift introduced the notion of quantifier in a graphical notation, which here represents the judgement that Русский Русский x.F(x){displaystyle forall x.F(x)} is true.

The logic of preaching is the generic term for symbolic formal systems as the logic of first order, the logic of second order, the logic of many orders and the infinite logic. It provides an account of quantifiers generally enough to express a broad set of arguments that occur in natural language. For example, Bertrand Russell's famous barber paradox, "there is a man who shaves all and only men who do not shave themselves," can be formalized by the sentence. (consuming consuming x)(Hoover(x)∧ ∧ (Русский Русский and)(man(and)→ → (shaving(x,and)▪ ▪ ¬ ¬ afeitan(and,and)){displaystyle (exists x)({text{hobre}}(x)wedge (forall y)({text{man}}(y)rightarrow ({text{afeita}}}}(x,y)leftrightarrow neg {text{afeitan}(y,y)}}, using the non-logical preaching man(x){displaystyle {text{man}(x)}} to indicate x is a man, and the non-logic relationship shaving(x,and){displaystyle {text{afeita}(x,y)} to indicate x shave and; all the other symbols of the formulas are logical, and express universal and existential quantifiers, conjunction, implication, negation and biconditional.

Whereas Aristotelian syllogistic logic specifies a small number of forms that the relevant part of the implied judgments can take, predicate logic allows sentences to be parsed into subject and argument in several additional ways, allowing predicate logic to solve the problem of multiple generality that had perplexed medieval logicians.

The development of predicate logic is often attributed to Gottlob Frege, who is also credited as one of the founders of analytic philosophy, but the most widely used formulation of predicate logic today is first-order logic presented in Principles of Mathematical Logic by David Hilbert and Wilhelm Ackermann in 1928. The analytic generality of predicate logic enabled the formalization of mathematics, prompted research in set theory, and enabled the development of Alfred's approach Tarski on model theory. It provides the foundation of modern mathematical logic.

Frege's original system of predicate logic was second-order, rather than first-order. Second Order Logic is defended most prominently (against criticism by Willard Van Orman Quine and others) by George Boolos and Stewart Shapiro.

Non-classical logic

The logics discussed above are all "bivalent" or "two-valued"; that is, they are more naturally understood as the division of propositions into true and false. Non-classical logic systems are those that reject various rules of classical logic.

Hegel developed his own dialectical logic that extended Kant's transcendental logic but also brought it back to earth ensuring that neither in heaven nor on earth, neither in the world of the mind nor in the world of nature, there is nowhere an "or" abstract as the one that sustains the understanding. Everything that exists is concrete, with difference and opposition in itself.

In 1910, Nicolai A. Vasiliev expanded the law of the excluded middle and the law of contradiction to propose the law of the excluded fourth and contradiction-tolerant logic. At the turn of the century XX, Jan Łukasiewicz investigated the expansion of the traditional true/false values to include a third value, "possible" (or indeterminate, a hypothesis) thus inventing ternary logic, the first plurivalent logic in the Western tradition. A minor modification of ternary logic was later introduced in a sibling ternary logic model proposed by Stephen Cole Kleene. Kleene's system differs from Łukasiewicz's logic with respect to a result of implication. The first assumes that the implication operator between two hypotheses produces a hypothesis.

Since then, logics such as fuzzy logic have been devised with an infinite number of "degrees of truth", represented by a real number between 0 and 1.

Intuitionistic Logic was proposed by L.E.J. Brouwer as the correct logic to reason about mathematics, based on his rejection of the principle of the excluded middle as part of his intuitionism. Brouwer refused formalization in mathematics, but his student Arend Heyting studied intuitionistic logic formally, as did Gerhard Gentzen. Intuitionistic logic is of great interest to computer scientists, as it is an intuitionistic logic and sees many applications, such as extracting verified programs from tests and influencing the design of programming languages through formula-like correspondence. -guys.

Modal logic is not truly conditional, so it has often been proposed as non-classical logic. However, modal logic is normally formalized with the excluded middle principle, and its relational semantics are bivalent, so this inclusion is moot.

History

Disputes

Is logic empirical?

What is the epistemological status of the laws of logic? What type of argument is appropriate to criticize the supposed principles of logic? In an influential article entitled "Is Logic Empirical?" Hilary Putnam, building on a suggestion by W. V. Quine, argued that in general the facts of propositional logic have a similar epistemological status to that of the facts. about the physical universe, for example as the laws of mechanics or general relativity, and in particular that what physicists have learned about quantum mechanics provides a compelling case for abandoning certain familiar principles of classical logic: if we want to be realistic about the physical phenomena described by quantum theory, then we must abandon the principle of distributivity, substituting classical logic for the quantum logic proposed by Garrett Birkhoff and John von Neumann.

Another work of the same name by Michael Dummett argues that Putnam's desire for realism requires the law of distributivity. The distributivity of logic is essential for the realist to understand how propositions are true of the world in the same way who has argued that it is the principle of bivalence. Thus, the question "Is logic empirical?" can be seen as a natural response to the fundamental controversy in metaphysics over realism versus anti-realism.

Implication: strict or material

The notion of implication formalized in classical logic does not translate comfortably into natural language by means of "if... then...", due to a series of problems called paradoxes of implication material.

The first class of paradoxes involves counterfactuals, such as If the moon is made of green cheese, then 2+2=5, which are puzzling because natural language does not support the explosion principle. The elimination of this class of paradoxes was the reason for C. I. Lewis's formulation of strict implication, which eventually led to more radically revisionist logics such as relevant logic.

The second class of paradoxes involves redundant premises, falsely suggesting that we know the successor because of antecedent: thus, "if that man is chosen, granny will die" it is materially true since granny is mortal, regardless of the man's choice prospects. Such sentences violate the Gricean maxim of relevance, and can be modeled by logics that reject the Monotonicity principle of implication, such as the logic of relevance.

Tolerate the impossible

Georg Wilhelm Friedrich Hegel was deeply critical of any simplified notion of the Principle of Non-Contradiction. He based himself on Gottfried Wilhelm Leibniz's idea that this law of logic also requires a sufficient foundation to specify from what point of view (or time) it is said that something cannot be contradicted. A building, for example, moves and does not move; The terrain for the first is our solar system and for the second the earth. In the Hegelian dialectic, the law of non-contradiction, of identity, is itself based on difference and, therefore, is not independently affirmable.

Closely related to issues raised by implication paradoxes is the suggestion that logic must tolerate inconsistency. Relevant Logic and Paraconsistent Logic are the most important approaches here, although the concerns are different: a key consequence of classical Logic and some of its rivals, such as Intuitionistic Logic, is that they respect the principle of explosion, which means that logic collapses if it is capable of deriving a contradiction. Graham Priest, the leading proponent of dialetheism, has argued for paraconsistency on the grounds that there are, in fact, true contradictions.

Rejection of logical truth

The philosophical vein of various types of skepticism contains many types of doubt and rejection of the various bases on which logic rests, such as the idea of logical form, correct inference, or meaning, sometimes leading to the conclusion that there are no logical truths. This contrasts with the usual points of view in philosophical skepticism, where logic directs the skeptical inquiry to doubt received knowledge, as in the work of Sextus Empiricus.

Friedrich Nietzsche provides a strong example of the rejection of the usual basis of logic: his radical rejection of idealization led him to reject truth as a "...mobile army of metaphors, metonyms and anthropomorphisms-in abstract... metaphors that are worn and without sensual power; coins that have lost their images and now matter only as metal, no longer as coins". His rejection of truth did not lead him to reject the idea of inference or logic altogether, but instead suggested that "logic [came] into existence in man's head [out of] illogicality, whose realm originally must have been vast. Countless beings who made inferences in a manner different from our own perished". Thus, there is the idea that logical inference has a utility as a tool for human survival, but that its existence does not support the existence of truth, nor does it have a reality beyond the instrumental: "Logic is also based on assumptions that do not correspond to anything in the real world".

This position held by Nietzsche, however, has been subjected to extreme scrutiny for various reasons. Some philosophers, such as Jürgen Habermas, claim that his position is self-refuting and accuse Nietzsche of not even having a coherent perspective, let alone a theory of knowledge. Georg Lukács, in his book The Destruction of Reason, states that, "If we were to study Nietzsche's statements in this area from a logical-philosophical angle, we would find ourselves with a dizzying chaos of the most lurid, arbitrary and violently incompatible statements. & # 34; Bertrand Russell described Nietzsche's irrational assertions with & # 34; he is fond of expressing himself paradoxically and with a view to shocking conventional readers & # 34; in his book A History of Western Philosophy.

Additional bibliography

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• Deaño, Alfredo (1974). Introduction to formal logic 2. The logic of preaching. Madrid: Alianza Editorial.
• Deaño, Alfredo (1980). The conceptions of logic. Madrid: Taurusl.
• Ferrater Mora, J. (1984). Philosophy Dictionary (4 tomos). Barcelona. Alliance Dictionaries. ISBN 84-206-5299-7.
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• Padilla Gálvez, J. (2017). Right. Open Disputes. Valencia: Tirant Lo Blanch. ISBN 978-84-17069-58-2.
• Priest, Graham (2008). An introduction to non-classical logic: From if to is (2nd edition). New York: Cambridge University Press. ISBN 978-0521854337. Archived from the original on August 23, 2017. Consultation on 22 August 2017.
• Sacristán, Manuel (1976). Introduction to logic and formal analysis. Barcelona: Ariel. ISBN 84-344-3901-8.
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• Belnap, N. (1977). "A useful four-valued logic." In Dunn & Eppstein, Modern uses of multiple-valued logic. Reidel: Boston.
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• Haack, Susan (1996). Deviant Logic, Fuzzy Logic: Beyond the FormalismUniversity of Chicago Press.
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• Hodges, W. (2001). Logic. An introduction to Elementary LogicPenguin Books.
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• Hughes, R.I.G. (1993, ed.). A Philosophical Companion to First-Order Logic. Hackett Publishing.
• Kline, Morris (1972). Mathematical Thought From Ancient to Modern Times. Oxford University Press. ISBN 978-0-19-506135-2.
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