Applied logic
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What is applied logic?
Applied logic is the study of correct reasoning, encompassing both logical principles and nonlogical concepts such as knowledge, norms, and time. It evaluates reasoning and explores particular applications, including deductive, inductive, and interrogative reasoning. Applied logic is used in mathematics, computer technology, and empirical sciences. Fuzzy logic, originally developed by Lotfi Zadeh, is applied in computer science and relies on compositional methods, where the status of a complex expression depends on its components. Applied logic also concerns the use of logical methods in language, knowledge, and time. Hypothetical reasoning, an application of logic, studies conditional sentences and counterfactuals, using possible-worlds semantics and game-theoretic semantics to determine truth conditions and represent conditional connections.
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How does applied logic differ from basic logic?
Applied logic focuses on the practical application of reasoning to various fields, while basic logic deals with the fundamental principles of correct reasoning. Applied logic uses both logical principles and nonlogical concepts like knowledge, norms, and time to evaluate reasoning. It extends to language, computer science, and scientific methodologies.
Basic logic, in contrast, examines the laws of thought and valid arguments, independent of specific factual content. It focuses on logical constants such as quantifiers and propositional connectives. Applied logic uses these concepts of basic logic in a wide range of real-world scenarios.
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Where is applied logic used in mathematics and science?
Applied logic is the study of correct reasoning, with applications in mathematics, science, and various other fields.
In mathematics, formal systems employing axiomatic set theory are used, and most mathematical arguments are carried out in logical terms.
Outside of logic itself, the most general applications are to the study of language, knowledge, norms, and time. In computer science, fuzzy logic is used for computational methods, and artificial intelligence uses applied logic to create commercially viable smart systems.
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How is applied logic important in computer science?
Applied logic is crucial in computer science for various applications. It is used in the design of electronic circuits, using principles of Boolean algebra to conduct operations within the computer's control unit, arithmetic logic unit (ALU), and I/O controllers.
Fuzzy logic, a computational method that deals with imprecise concepts, is widely used by computer scientists. Logic is also applied to the study of language, knowledge, norms, and time. Moreover, denotational semantics, originally developed for logical systems, is now applied to computer languages.
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How does applied logic help people solve everyday problems?
Applied logic is the study of correct reasoning and its practical applications, incorporating both logical principles and nonlogical concepts such as knowledge and norms. It enables individuals to solve everyday problems by applying logical rules to derive conclusions from given information.
For instance, in conditional reasoning, one can draw conclusions based on "if...then" statements, helping to make informed decisions. Applied logic also extends to the study of language, knowledge, and even time, offering a versatile toolkit for navigating various aspects of daily life. Logic puzzles also promote ingenuity when solving problems.
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What role does applied logic play in developing artificial intelligence and technology?
Applied logic plays a crucial role in the development of artificial intelligence (AI) and technology by providing the foundation for right reasoning in machines. It is applied to the study of language, knowledge, norms, and time. AI programming languages, such as LISP and PROLOG, use applied logic. Fuzzy logic, a computational method, is also widely used by computer scientists.
Applied AI produces commercially viable smart systems, while artificial general intelligence (AGI) seeks to build machines that think like humans.
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Applied logic, the study of the practical art of right reasoning. This study takes different forms depending on the type of reasoning involved and on what the criteria of right reasoning are taken to be. The reasoning in question may turn on the principles of logic alone, or it may also involve nonlogical concepts. The study of the applications of logic thus has two parts—dealing on the one hand with general questions regarding the evaluation of reasoning and on the other hand with different particular applications and the problems that arise in them. Among the nonlogical concepts involved in reasoning are epistemic notions such as “knows that …,” “believes that …,” and “remembers that …” and normative (deontic) notions such as “it is obligatory that …,” “it is permitted that …,” and “it is prohibited that ….” Their logical behaviour is therefore a part of the subject matter of applied logic. Furthermore, right reasoning itself may be understood in a broad sense to comprehend not only deductive reasoning but also inductive reasoning and interrogative reasoning (the reasoning involved in seeking knowledge through questioning).
The evaluation of reasoning
Reasoning can be evaluated with respect to either correctness or efficiency. Rules governing correctness are called definitory rules, while those governing efficiency are sometimes called strategic rules. Violations of either kind of rule result in what are called fallacies.
Logical rules of inference are usually understood as definitory rules. Rules of inference do not state what inferences reasoners should draw in a given situation; they are instead permissive, in the sense that they show what inferences a reasoner can draw without committing a fallacy. Hence, following such rules guarantees only the correctness of a chain of reasoning, not its efficiency. In order to study good reasoning from the perspective of efficiency or success, strategic rules of reasoning must be considered. Strategies in general are studied systematically in the mathematical theory of games, which is therefore a useful tool in the evaluation of reasoning. Unlike typical definitory rules, which deal with individual steps one by one, the strategic evaluation of reasoning deals with sequences of steps and ultimately with entire chains of reasoning.
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Strategic rules should not be confused with heuristic rules. Although rules of both kinds deal with principles of good reasoning, heuristic rules tend to be merely suggestive rather than precise. In contrast, strategic rules can be as exact as definitory rules.
Fallacies
The formal study of fallacies was established by Aristotle and is one of the oldest branches of logic. Many of the fallacies that Aristotle identified are still recognized in introductory textbooks on logic and reasoning.
Formal fallacies
Deductive logic is the study of the structure of deductively valid arguments—i.e., those whose structure is such that the truth of the premises guarantees the truth of the conclusion. Because the rules of inference of deductive logic are definitory, there cannot exist a theory of deductive fallacies that is independent of the study of these rules. A theory of deductive fallacies, therefore, is limited to examining common violations of inference rules and the sources of their superficial plausibility.
Fallacies that exemplify invalid inference patterns are traditionally called formal fallacies. Among the best known are denying the antecedent (“If A, then B; not-A; therefore, not-B”) and affirming the consequent (“If A, then B; B; therefore, A”). The invalid nature of these fallacies is illustrated in the following examples:
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If Othello is a bachelor, then he is male; Othello is not a bachelor; therefore, Othello is not male.
If Moby Dick is a fish, then he is an animal; Moby Dick is an animal; therefore, Moby Dick is a fish.
Verbal fallacies
One main source of temptations to commit a fallacy is a misleading or misunderstood linguistic form of a purported inference; mistakes due to this kind of temptation are known as verbal fallacies. Aristotle recognized six verbal fallacies: those due to equivocation, amphiboly, combination or division of words, accent, and form of expression. Whereas equivocation involves the ambiguity of a single word, amphiboly consists of the ambiguity of a complex expression (e.g., “I shot an elephant in my pajamas”). A typical fallacy due to the combination or division of words is an ambiguity of scope. Thus, “He can walk even when he is sitting” can mean either “He can walk while he is sitting” or “While he is sitting, he has (retains) the capacity to walk.” Another manifestation of the same mistake is a confusion between the distributive and the collective senses of an expression, as for example in “Jack and Jim can lift the table.”
Fallacies of accent, according to Aristotle, occur when the accent makes a difference in the force of a word. By a fallacy due to the form of an expression (or the “figure of speech”), Aristotle apparently meant mistakes concerning a linguistic form. An example might be to take “inflammable” to mean “not flammable,” in analogy with “insecure” or “infrequent.”
The most common characteristic of verbal fallacies is a discrepancy between the syntactic and the semantic form of a sentence, or between its structure and its meaning. A general theory of linguistic fallacies must therefore address the question of whether all semantic distinctions can be recognized on the basis of the syntactic form of linguistic expressions.