Modal logic. Modal logic is a type of formal logic primarily developed in the 1960s that extends classical propositional and predicate logic to include operators expressing modality. Modals—words that express modalities—qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is usually happy, in which case the term "usually" is functioning as a modal. A formal modal logic represents modalities using modal operators.
For example, "It might rain today" and "It is possible that rain will fall today" both contain the notion of possibility. In a modal logic this is represented as an operator, Possibly, attached to the sentence "It will rain today". The basic unary (1-place) modal operators are usually written □ for Necessarily and ◇ for Possibly. Ignoratio elenchi. Ignoratio elenchi, also known as irrelevant conclusion, is the informal fallacy of presenting an argument that may or may not be logically valid, but fails nonetheless to address the issue in question.
Ignoratio elenchi falls into the broad class of relevance fallacies. It is one of the fallacies identified by Aristotle in his Organon. In a broader sense he asserted that all fallacies are a form of ignoratio elenchi. Ignoratio Elenchi, according to Aristotle, is a fallacy which arises from “ignorance of the nature of refutation.” The phrase ignoratio elenchi is Latin meaning "an ignoring of a refutation". Here elenchi is the genitive singular of the Latin noun elenchus, which is from the Greek ἔλεγχος elenchos, meaning an argument of disproof or refutation. The translation in English of the Latin expression has varied somewhat.
An example might be a situation where A and B are debating whether the law permits A to do something. See also References Lack of belief in gods. Principle of explosion. The principle of explosion, (Latin: ex falso quodlibet, "from a falsehood, anything follows", or ex contradictione sequitur quodlibet, "from a contradiction, anything follows") or the principle of Pseudo-Scotus, is the law of classical logic, intuitionistic logic and similar logical systems, according to which any statement can be proven from a contradiction. That is, once a contradiction has been asserted, any proposition (or its negation) can be inferred from it.
As a demonstration of the principle, consider two contradictory statements - “All lemons are yellow” and "Not all lemons are yellow", and suppose (for the sake of argument) that both are simultaneously true. If that is the case, anything can be proven, e.g. "Santa Claus exists", by using the following argument: Symbolic representation The principle of explosion can be expressed in the following way (where " " symbolizes the relation of logical consequence): Gödel's incompleteness theorems.
Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic.
The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible, giving a negative answer to Hilbert's second problem. The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic).
For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. Background Fuzzy logic. Fuzzy logic is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact.
Compared to traditional binary sets (where variables may take on true or false values) fuzzy logic variables may have a truth value that ranges in degree between 0 and 1. Fuzzy logic has been extended to handle the concept of partial truth, where the truth value may range between completely true and completely false. Furthermore, when linguistic variables are used, these degrees may be managed by specific functions. Irrationality can be described in terms of what is known as the fuzzjective.  The term "fuzzy logic" was introduced with the 1965 proposal of fuzzy set theory by Lotfi A. Overview
Three-valued logic. In logic, a three-valued logic (also trivalent, ternary, trinary logic, or trilean, sometimes abbreviated 3VL) is any of several many-valued logic systems in which there are three truth values indicating true, false and some indeterminate third value.
This is contrasted with the more commonly known bivalent logics (such as classical sentential or Boolean logic) which provide only for true and false. Conceptual form and basic ideas were initially created by Jan Łukasiewicz and C. I. Lewis. These were then re-formulated by Grigore Moisil in an axiomatic algebraic form, and also extended to n-valued logics in 1945.