Philosopher Susan Haack

Susan Haack

Susan Haack is an English professor of philosophy and law at the University of Miami in the United States. She has written on logic, the philosophy of language, epistemology, and metaphysics. Her pragmatism follows that of Charles Sanders Peirce.-Career:Haack is a graduate of the University of...

 uses the term "deviant logic" to describe certain non-classical

Non-classical logic

Non-classical logics is the name given to formal systems which differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is done, including by way of extensions, deviations, and variations...

 systems of logic. In these logics,

• the set of well-formed formula
  Well-formed formula
  In mathematical logic, a well-formed formula, shortly wff, often simply formula, is a word which is part of a formal language...
  
  s generated equals the set of well-formed formulas generated by classical logic.
• the set of theorem
  Theorem
  In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
  
  s generated is different from the set of theorems generated by classical logic.

The set of theorems of a deviant logic can differ in any possible way from classical logic's set of theorems: as a proper subset

Subset

In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

, superset, or fully exclusive set. A notable example of this is the trivalent logic developed by Polish

Poles

thumb|right|180px|The state flag of [[Poland]] as used by Polish government and diplomatic authoritiesThe Polish people, or Poles , are a nation indigenous to Poland. They are united by the Polish language, which belongs to the historical Lechitic subgroup of West Slavic languages of Central Europe...

 logician and mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

 Jan Łukasiewicz. Under this system, any theorem necessarily dependent on classical logic's principle of bivalence

Principle of bivalence

In logic, the semantic principle of bivalence states that every declarative sentence expressing a proposition has exactly one truth value, either true or false...

 would fail to be valid. The term first appears in Chapter 6 of W.V.O. Quine's Philosophy of Logic, New Jersey: Prentice Hall (1970), which is cited by Haack on p.15 of her book.

Quasi-deviant and extended logics

Haack also described what she calls a quasi-deviant logic. These logics are different from pure deviant logics in that:

• the set of well-formed formulas generated is a proper superset of the set of well-formed formulas generated by classical logic.
• the set of theorems generated is a proper superset of the set of theorems generated by classical logic, both in that the quasi-deviant logic generates novel theorems using well-formed formulas held in common with classical logic, as well as novel theorems using novel well-formed formulas.

Finally, Haack defined a class of merely extended logics. In these,

• the set of well-formed formulas generated is a proper superset of the set of well-formed formulas generated by classical logic.
• the set of theorems generated is a proper superset of the set of theorems generated by classical logic, but only in that the novel theorems generated by the extended logic are only a result of novel well-formed formulas.

Some systems of modal logic

Modal logic

Modal logic is a type of formal logic 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...

 meet this definition. In such systems, any novel theorem would not parse in classical logic due to modal operators. While deviant and quasi-deviant logics are typically proposed as rivals to classical logic, the impetus behind extended logics is normally only to provide a supplement to it.

Two decades later

Achille Varzi

Achille Varzi (philosopher)

Achille C. Varzi is a professor of philosophy at Columbia University. He specializes in metaphysics and logic.Along with Roberto Casati, he authored "Holes and Other Superficialities", an exploration into the ontology of holes, as well as "Parts and Places: The Structures of Spatial...

 in his review of the 1996 edition of Haack's book writes that the survey did not stand well the test of time, particularly with the "extraordinary proliferation of nonclassical logics in the past two decades—paraconsistent logics, linear logics, substructural logics, nonmonotonic logics, innumerable other logics for AI and computer science." He also finds that Haack's account of vagueness

Vagueness

The term vagueness denotes a property of concepts . A concept is vague:* if the concept's extension is unclear;* if there are objects which one cannot say with certainty whether belong to a group of objects which are identified with this concept or which exhibit characteristics that have this...

"is now seriously defective." He concedes however that "as a defense of a philosophical position, Deviant Logic retains its significance."