Logic: the study of correct reasoning, systems, and applications

Classical Logic

One speaks of classical logic or of a classical logical system exactly when the following semantic conditions are fulfilled:

1. Every statement has exactly one of two truth values, usually called true and false. This principle is called the principle of bivalence or bivalence principle.
2. The truth value of a compound statement is uniquely determined by the truth values of its partial statements and the way in which they are composed. This principle is called the principle of extensionality or compositionality.

The term classical logic is to be understood more in the sense of established, fundamental logic, because the non-classical logics build on it, than as a historical reference. It was rather the case that already Aristotle, so to speak the classical representative of logic, dealt very well with multi-valued logic, i.e. non-classical logic.

The most important branches of formal classical logic are classical propositional logic, first-level predicate logic, and higher-level logic, as developed at the end of the 19th and beginning of the 20th century by Gottlob Frege, Charles Sanders Peirce, Bertrand Russell, and Alfred North Whitehead. In propositional logic, statements are examined to see whether they are in turn composed of statements that are connected to one another by means of junctors (e.g. "and", "or"). If a statement does not consist of sub-statements connected by junctors, then it is atomic from the point of view of propositional logic, i.e. it cannot be further decomposed.

In predicate logic, the inner structure of propositions can also be represented, which cannot be further decomposed in propositional logic. The inner structure of propositions (The apple is red. ) is represented by predicates (also called propositional functions) (is red) on the one hand and by their arguments on the other hand (The apple); for example, the predicate expresses a property (red) that applies to its argument, or a relation that exists between its arguments (x is greater than y). The notion of a propositional function is derived from the mathematical notion of a function. A logical propositional function, just like a mathematical function, has a value, but it is not a numerical value, it is a truth value.

The difference between first-level predicate logic and higher-level predicate logic is what is quantified about by means of the quantifiers ("all", "at least one"): In first-level predicate logic, quantification is only about individuals (e.g. "All pigs are pink"); in higher-level predicate logic, quantification is also about predicates themselves (e.g. "There is a predicate that applies to Socrates").

Formally, predicate logic requires a distinction between different categories of expression such as terms, functors, predictors and quantifiers. This is overcome in step logic, a form of typed lambda calculus. Thus, for example, mathematical induction becomes an ordinary, derivable formula.

Syllogistics, which was dominant until the 19th century and goes back to Aristotle, can be understood as a precursor of predicate logic. A basic concept of syllogistics is the term "terms"; it is not further decomposed there. In predicate logic, terms are expressed as single-digit predicates; multi-digit predicates can additionally be used to analyze the internal structure of terms, and thus to show the validity of arguments that cannot be grasped syllogistically. A frequently cited intuitively catchy example is the argument "All horses are animals; therefore all horse heads are animal heads", which can only be derived in higher logics such as predicate logic.

It is technically possible to extend and modify Aristotle's formal syllogistic in such a way that calculi of equal power to predicate logic emerge. In the 20th century, such undertakings have been undertaken sporadically by philosophers and are philosophically motivated, for example, by the desire to be able to regard terms as elementary components of statements even in purely formal terms and not to have to decompose them predicate-logically. More on such calculi and the philosophical background can be found in the article on conceptual logic.

Calculus types and logical procedures

Modern formal logic is devoted to the task of developing exact criteria for the validity of inferences and the logical validity of statements (semantically valid statements are called tautologies, syntactically valid statements theorems). Various procedures have been developed for this purpose.

In particular in the field of propositional logic (but not only) semantic procedures are in use, i.e. such procedures which are based on the fact that a truth value is attributed to statements. These include, on the one hand:

• Truth Tables

While truth-tables make a complete listing of all truth-value-combinations (and insofar are only usable in the propositional-logical area), the other (also predicate-logically usable) procedures proceed according to the scheme of a reductio ad absurdum: If a tautology is to be proved, one starts from its negation and tries to derive a contradiction. Several variants are common here:

• Resolution,
• Tree calculus or Beth tableaux (after Evert Willem Beth)

Logical calculi that do not involve semantic evaluations include:

• Axiomatic logic calculi
• Systems of natural closure
• Sequence calculus
• Dialogic logics

Non-classical logics

One speaks of non-classical logic or a non-classical logical system if at least one of the two classical principles mentioned above (bivalence and/or extensionality) is abandoned. If the principle of bivalence is abandoned, multivalued logic arises. If the principle of extensionality is abandoned, intensional logic arises. Intensional are for example modal logic and intuitionistic logic. If both principles are abandoned, multivalued intensional logic arises. (See also: Category:Nonclassical logic)

Philosophical logics

Philosophical logic is a fuzzy collective term for various formal logics that modify or extend classical propositional and predicate logic in different ways, usually by enriching their language with additional operators for certain domains of speech. Philosophical logics are usually not of direct interest to mathematics, but are used, for example, in linguistics or computer science. They often deal with questions that go back far into the history of philosophy and have partly been discussed since Aristotle, for example the handling of modalities (possibility and necessity).

The following areas, among others, are attributed to philosophical logic:

• Modal logic introduces modal propositional operators such as "it is possible that..." or "it is necessary that..." and explores the validity conditions of modal arguments;
• epistemic logic or doxastic logic examines and formalizes statements of faith, belief and knowledge, as well as arguments formed from them;
• Deontic logic or norm logic examines and formalizes precepts, prohibitions, and concessions ("it is permitted that...") as well as arguments formed from them;
• Temporal logic of actions, quantum logic, and other temporal logics examine and formalize statements and arguments in which reference is made to points in time or periods of time;
• Intensional logics concern not only the extension (denotation; meaning in the sense of denoted elements), but their intension (sense/meaning; meaning in the sense of denoted properties) of concepts or sentences.
• Interrogative logic examines interrogative sentences and whether logical relations can be established between interrogative sentences;
• Conditional sentence logic examines "if-then" conditions beyond material implication;
• Paraconsistent logics are characterized by the fact that in them it is not possible to derive any statement from two contradictory statements. This also includes the
• Relevance logic, which instead of material implication uses an implication that is true only if its antecedent is relevant to its consequent (see also the following chapter).

Intuitionism, relevance logic and connective logic

The most discussed deviations from classical logic are those logics which renounce certain axioms of classical logic. The non-classical logics in the narrower sense are "weaker" than classical logic, i.e. in these logics fewer statements are valid than in classical logic, but all statements valid there are also classically valid.

These include the intuitionistic logic developed by L. E. J. Brouwer, which uses the "duplex-negatio" axiom (from the double negation of a statement p follows p)

(DN) ¬

does not contain, whereby the sentence "tertium non datur" (for every statement p holds: p or not-p),

(TND) ¬

is no longer derivable, the minimal calculus of Ingebrigt Johansson, with which the proposition "ex falso quodlibet" (from a contradiction follows any statement),

(EFQ) ¬

can no longer be derived, as well as the subsequent relevance logics, in which only such statements of the scheme are valid, in which is causally relevant for (see Implication#Object-Language Implications). In Dialogic Logic and in Sequential Calculi, both classical and non-classical logics are convertible into each other by appropriate additional rules.

On the other hand, we should mention logics that contain principles that are not classically valid. The sentence ¬ seems at first to express an intuitively plausible logical principle: For if p holds, then p, it seems, can no longer be false. Yet this theorem is not a valid theorem in classical logic. Insofar as classical logic is maximally-consistent, i.e. insofar as any genuine amplification of a classical calculus would lead to a contradiction, this proposition could not be added as another axiom either. Hence, connexional logic, which wants to do justice to the pre-formal intuition that the proposition expresses by distinguishing it as a theorem, must reject other classical-logical theorems. Thus, while in intuitionistic, minimal, and relevant logic the provable formulas are each a genuine subset of the classically provable formulas, the relation of connexional and classical logic, on the other hand, is such that in both formulas are also provable that do not hold in the other logic.

Multivalued logic and fuzzy logic

In contrast to this are the multivalued logics, in which the principle of bivalence and often also the Aristotelian theorem of the excluded third do not apply, among them the trivalent and the infinite-valued logic of Jan Łukasiewicz ("Warsaw School"). The infinite-valued fuzzy logic finds numerous applications in control engineering, while, for example, the finite-valued logic of Gotthard Günther ("Günther logic") has been applied to problems of self-fulfilling predictions in sociology.

Non-monotonic logics

A logical system is called monotonic if every valid argument remains valid even if one adds additional premises: What has been proved once always remains valid in a monotonic logic, that is, even if one has new information at a later time. Very many logical systems have this monotonicity property, including all classical logics such as propositional and predicate logic.

However, in everyday reasoning, as well as in scientific reasoning, provisional conclusions are often drawn which are not valid in a strictly logical sense and which may have to be revised at a later date. For example, from the statements "Tux is a bird." and "Most birds can fly." we might tentatively conclude that Tux can fly. However, if we now get the additional information "Tux is a penguin." then we have to correct this conclusion, because penguins are not birds that can fly. Non-monotonic logics were developed to map this kind of reasoning: they dispense with the monotonicity property, which means that a valid argument can be invalidated by adding more premises.

Of course, this is only possible if a different consequence operation is used than in a classical logic. A common approach is to use so-called defaults. A default inference is valid if there is no contradiction to it from a classical logical inference.

The conclusion from the given example would then look like this: "Tux is a bird." remains the prerequisite. We now combine this with a so-called justification: "Birds can normally fly." From this justification we conclude that Tux can fly, as long as there is nothing to the contrary. So the corollary is "Tux can fly." If we now get the information "Tux is a penguin." and "Penguins cannot fly.", we get a contradiction. Via the default inference, we have arrived at the consequence that Tux can fly. However, using a classical logical inference, we could prove that Tux cannot fly. In this case, the default is revised and the consequence of the classical-logical conclusion is further used. This procedure - roughly described here - is also called Reiter's default logic. (See also non-monotonic inductive Bayes logic).