Modal logic

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The modal logic is the one branch of logic that deals with the consequences of the modal notions possible and necessary deals. In this way, not only statements such as “It's raining” or “All circles are round” can be analyzed within the modal logic , but also statements such as “It may be raining” and “All circles are necessarily round”.

Development of modal logic in the 20th century


The earliest beginnings of a modal logic can be found in Aristotle in the first analysis . There, the modal logical variants of each categorical syllogism are also discussed. In the Middle Ages u. a. Duns Scotus modal logic. Gottfried Wilhelm Leibniz coined the expression “possible world”, which has become significant for the development of modal-logical model theory . Since the 20th century, a distinction has to be made between two fundamentally different approaches to the specification of modal statements and their logical relationships, an object-language-axiomatic and a metalinguistic-operational one.

Clarence Irving Lewis provided the first axiomatic approach in 1912 in his criticism of "material implication" (by Whitehead and Russell ), which in no way corresponded to the conventional "if-then". In 1932, together with CH Langford , he set up five logical systems (S1 to S5) with different modal axioms that appeared more or less "plausible". Only in 1963 was Saul Kripke a semantic development for the large number of previously proposed modal logic systems. On this axiomatic basis, Bas van Fraassen ( Toronto ) and Maria L. Dalla Chiara ( Florence ) used modalities within the framework of quantum logic since the 1970s .

The attempts of the Husserl student Oskar Becker in 1930 to interpret the modal statements of Lewis phenomenologically remained largely without consequences for the further development of modal logic . Following a suggestion by Becker, Kurt Gödel showed a close connection between the S4 system and intuitionist logic in 1932 .

The metalinguistic concept for modalities by Rudolf Carnap in 1934 was fundamentally new . In the context of a sharp criticism of Wittgenstein's view of the fundamental limitation of linguistic expressiveness, he claimed that Lewis's extension of classical logic by adding an operator "possible" was not wrong, but superfluous, "since the metalanguage, which is also necessary for describing the axiomatic modal logic, allows both the consequential relationship and the modalities to be expressed with exact formulation."

Carnap's concept of using modalities only when speaking about a language, i.e. metalinguistic, was not taken up again until 1952 by Becker and two years later by Paul Lorenzen to justify his operative modal logic. The constructive logic developed by Lorenzen (later also by Schwemmer ) on the basis of formalized dialog semantics was first introduced into quantum logic by Peter Mittelstaedt in 1961 and further developed in 1979 by his student Franz Josef Burghardt into "modal quantum metalogics".

The underlying intuition

With the terms “possible” and “necessary”, the language offers, in addition to “ true ” and “false”, an additional way of characterizing statements : some false statements are possible, some true statements are also necessary. If we want to determine whether a statement is possible, we can try to imagine a situation in which the statement is true. For example, we can imagine that there could be people with green skin. The statement "Some people have green skin" is therefore possible. However, we cannot imagine that there could be square circles. The statement "There are angular circles" is therefore not possible, i. H. impossible. There are also statements that are true in every imaginable situation. We call such statements necessary. Necessary statements are e.g. B. "Circles are round" and "Bachelors are unmarried".

In modal logic, one speaks of " possible worlds " instead of possible or imaginable situations . The situation in which we actually live it is one of the possible worlds "real world" (Engl. Actual world , so sometimes called current world). A statement is possible if it is true in one possible world; it is necessary if it is true in all possible worlds.

If one describes a statement as possible in this sense, one does not take a position on whether the statement could also be wrong. For this reason, all necessary facts are also possible: if a statement is true in all possible worlds, then it is trivially true in at least one possible world. The concept of contingency differs from this concept of possibility : Contingent is a statement precisely when it is true in at least one possible world and false in at least one possible world, i.e. when it is possible but not necessary.

Truth functionality of modal logic

In contrast to classical propositional logic, modal logic is not truth-functional . This means that if one replaces a partial statement in a statement that contains modal logical expressions by another with the same truth value , the truth value of the overall statement is not necessarily retained. As an example, consider the statement "It is possible that Socrates is not a philosopher". This statement is true (we can imagine that Socrates would never have been interested in philosophy) and contains the false statement “Socrates is not a philosopher” as a partial statement. If we now replace this partial statement with the equally incorrect statement “There are angular circles”, we get “It is possible that there are angular circles”. However, in contrast to our initial statement, this is wrong (because, as I said, we cannot imagine square circles). This shows that the modal logic is not truth-functional.


Notation Way of speaking
It is possible that p
It is necessary that p
p is contingent

In modal logic, the expression “possible” (more precisely: the sentence operator “it is possible that ...”) is represented by a diamond on its tip , which is also called “diamond”, and the expression “necessary "(More precisely:" it is necessary that ... ") with a small square, which is also called a" box ".

Modal logic

Modal operators and negation

If the modal operators combine with the negation , ie the “not” (in formal representation:) , then it makes a difference whether the negation relates to the entire expression composed of the modal operator and statement or only to the expression following the modal operator. “It is not possible that Socrates is a philosopher” ( ) means something other than “It is possible that Socrates is not a philosopher” ( ), the first statement is false, the second true. It should also be noted that statements with the possibility operator can be translated into statements with the necessity operator and vice versa. “It is possible that Socrates is not a philosopher” is synonymous with “It is not necessary that Socrates be a philosopher”, “It is not possible (it is impossible) that Socrates is an elephant” with “It is necessary that Socrates is not an elephant ”. In formal notation:

  • is equivalent to
  • is equivalent to

“It is possible that Socrates is a philosopher” is also synonymous with “It is not necessary that Socrates is not a philosopher” and “It is necessary that Socrates is a man” with “It is not possible that Socrates not be a man is ".

  • is equivalent to
  • is equivalent to

Based on these last two equivalences, the possibility operator can be defined by the necessity operator and vice versa.

Disjunction and conjunction

The disjunction (or connection, symbolically:) of two possible statements is synonymous with the possibility of their disjunction. From “It is possible that Socrates is a philosopher or it is possible that he is a carpenter” follows “It is possible that Socrates is a philosopher or a carpenter” and vice versa.

  • is equivalent to

The same applies to the necessity operator and the conjunction (AND connection, symbolic:) : “It is necessary that all circles are round, and it is necessary that all triangles are angular” is equivalent to “It is necessary that all circles round and all triangles are square ”.

  • is equivalent to

It looks different with the conjunction of statements of possibility and the disjunction of statements of necessity. The possibility of a conjunction of two statements implies the conjunction of the possibility of the statements, but the reverse is not true. If it is possible for Socrates to be both a philosopher and a carpenter, then it must be possible for him to be a philosopher and also possible for him to be a carpenter. In contrast, it is z. B. both possible that the number of planets is even and that it is odd, but it is not possible that it is both even and odd.

  • from , follows , but not the other way around

Similarly, one can deduce from the disjunction of the necessity of two statements the necessity of the disjunction of the individual statements, but not the other way around. If it is necessary that there are infinitely many prime numbers or it is necessary that Socrates be a philosopher, then it must be necessary that there are infinitely many prime numbers or that Socrates is a philosopher. On the other hand, however, it is necessary, for example, that Frank weighs no more than 75 kg or is heavier than 75 kg, but it is neither necessary that he weighs no more than 75 kg, nor is it necessary that he is heavier than 75 kg. Therefore:

  • follows from , but not the other way around

Quantifiers, Barcan formulas

When using quantifiers, it is controversial in philosophical logic whether it should be allowed to exclude modal operators from the scope of quantifiers or vice versa. The following metalinguistic rules (or corresponding axiom schemes) are therefore in dispute. Here stands for an individual variable and for a predicate name in the object language:

  • is equivalent to .
  • is equivalent to

One direction of equivalence is unproblematic and accepted:

  • From follows . If there is an object that possibly has the property , there must possibly be something that has the property .
  • From follows . If all objects necessarily have the property , then every object necessarily has the property .

These statements hold in most of the quantified modal logics.

More problematic, however, are the reversals of the two equivalence claims ( Barcan formulas ) named after Ruth Barcan Marcus :

  • From follows
  • From follows

The two Barcan formulas are equivalent to each other with the usual substitutability of ... by ... and ... by .... The debate revolves around the interpretation of the formulas. For example, if there is someone who can grow a beard (who is possibly bearded), it is possible that there is someone who has a beard. The inverse of the Barcan formula (if there is possibly someone who is bearded, there is someone who is possibly bearded) leads to the following problem: The first part of the if-then sentence only asserts that there is an individual may that would be bearded, the back part presupposes that there is an individual who may be bearded. This sub-clause thus has an existence presupposition : Let us assume that the quantifier relates to a set of people who are currently in a certain room, the rear part assumes that someone is currently in the room who is could grow a beard, but not the front part. So z. B. excluded that the room is randomly empty. This becomes all the more problematic when the quantifier is supposed to refer to 'everything there is'; the Barcan formula would then claim that every possible object ( possibilium ) to which a property can be assigned now exists and possibly has the property . Take e.g. If, for example, the childless philosopher Ludwig Wittgenstein could have had a son, it followed from the formula that there would now be a person who might be Wittgenstein's son. The controversy of the Barcan formula for necessity and universal quantifier can be made clear using the same example: All (actually existing) people are necessarily not sons of Wittgenstein, but that does not mean that all (possible) people are necessarily not sons of Wittgenstein, that is, Wittgenstein could not have had sons. Ruth Barcan Marcus himself set up the formulas, but excluded them from normal modal logic systems for precisely these reasons.

Instead of the Barcan formulas, however, the following schemes are accepted as valid:

  • It follows from , but not the other way around

The universal quantification of a possibility statement follows from the possibility of an universal statement, but not the other way around.

The reasons for this are similar to those stated above for the combination of conjunction and possibility ( see also de re and de dicto ). If it is possible for all men to have a beard, then all men must be able to have a beard. While everyone can potentially win in backgammon , it does not mean that it is possible for everyone to win (there can only be one winner in this game).

  • It follows from , but not the other way around

The existential quantification of a necessity statement analogously implies the necessity of the existential statement, but not vice versa. For example, if there is a thing that is necessarily God, it is necessary that there be a God. In backgammon there is necessarily a winner (the game cannot end in a draw), but it does not follow that one of the players necessarily wins.

Other interpretations of the modal operators

The operators Diamond and Box can also be verbalized in other ways than “necessary” and “possible”. In the “deontic” interpretation, the operators are interpreted using the ethical terms “required” and “permitted”; one then no longer speaks of modal logic in the narrower sense, but of deontic logic . The modal logic in the narrower sense is then sometimes referred to as "alethic modal logic". In the temporal logic, however , the operators are interpreted in terms of time . If one understands the operators as concepts of belief, that is, of the subjective belief that is true, one arrives at epistemic logic .

formula Modal interpretation Deontic interpretation Temporal interpretation Epistemic Interpretation
p It is possible that p It is permissible that p p applies sometime in the future (past) I think it is possible that p
p It is necessary that p It is imperative that p p is always valid in the future (past) I consider it certain that p

It is characteristic of all these interpretations that the above conclusions remain meaningful and intuitive. This is only to be shown here by means of an example, namely the equivalence of and .

  • "It is permissible that p" is equivalent to "It is not required that not-p"
  • "P holds sometime in the future" is equivalent to "It is not the case that non-p always holds in the future"
  • “I think it is possible that p” is equivalent to “I don't think it is certain that not-p”.

Different systems of modal logic

Syntactic characterization

A formal system of modal logic arises by adding modal logic formulas and additional axioms or inference rules to propositional logic or predicate logic . Depending on which logic one starts from, one speaks of modal logic propositional or predicate logic. The language of modal logic contains all propositional or predicate logic formulas as well as all formulas of the shape and for all modal logic formulas . Box can be defined by Diamond and vice versa according to the already known equivalences:

  • is equivalent to
  • is equivalent to

With regard to the modal-logical derivation term , it should first be noted that there are various such terms with which different modal-logical "systems" can be formed. This is partly due to the different interpretations of the operators Box and Diamond mentioned above.

The vast majority of modal systems are based on System K (K stands for Kripke ). K arises by setting the axiom scheme K and allowing the final rule of neezessization (also known as the "Gödel rule", after the logician Kurt Gödel ):

  • Axiom Scheme K: .
  • Necessization rule: If applies: (i.e. if p is derivable ), then also applies ( is derivable).

In System K, all of the above discussed conclusions are already valid, with the exception of the controversial Barcan formulas, one of which may have to be added as a separate axiom (the other then also results).

If the axiom scheme T is added to system K, then system T is obtained .

  • Axiom scheme T: or also

Under the modal interpretation, this schema is intuitively valid, because it says that true statements are always possible. Under the deontic interpretation one gets that everything that is true is also allowed, and this is intuitively not a valid conclusion, because there are also rule violations and thus true, but not allowed statements. For deontic applications, therefore, the axiom scheme T is weakened to axiom scheme D. If you add D to K, you get the system D (D for "deontic")

  • Axiom scheme D:

D means under the deontic interpretation that everything that is required is also allowed, and therefore represents a meaningful conclusion under this interpretation.

If T is extended by axiom scheme B, system B is obtained . (B stands for Brouwer here .)

  • Axiom scheme B:

The system S4 arises by adding the axiom scheme 4 to the system T. (The designation S4 is historical and goes back to the logician CI Lewis . Lewis developed five modal systems, of which only two, S4 and S5, are in use today.)

  • Axiom scheme 4: or also

The systems S4 and B are both stronger than T and thus also as D. "Stronger" here means that all formulas that can be proven in T (or D) can also be proven in S4 and B, but not vice versa. S4 and B are independent of each other, i.e. This means that formulas can be proven in both systems that cannot be proven in the other.

If axiom scheme 5 is added to system T, system S5 is obtained .

  • Axiom scheme 5:

S5 is stronger than both S4 and B. Note that axiom scheme 4 is valid under a temporal interpretation, but not 5: if at some point in the future there is a point in the future at which p holds, then there is a point in time in the future when p holds (4). But it is not true that if there is a point in time in the future at which p applies, there is such a point in time for all points in the future (5). S4, but not S5, is therefore suitable for a temporal interpretation.

In S4 and S5, chains of modal operators can be reduced to a single operator. In S4, however, this is only allowed if the chain consists of the same operators. For example, there the formula is equivalent to . In S5 you can reduce any number of chains, including disparate ones. Instead , you can just write there . No reduction is possible in any of the other modal systems mentioned.

The last mentioned property of the S5 system makes it the most suitable for modal logics in the strict sense of the word, ie for analyzing the expressions “possible” and “necessary”. The reason is that, unlike a simple application, we cannot intuitively assign any real meaning to repeated application of these expressions to a statement. For example, it is difficult to say what "it is necessary that it be possible for it to rain" should mean as opposed to simply "It is possible for it to rain". From this perspective, it is an advantage of S5 that it reduces repeated applications of the operators to simple ones, in this way an intuitive sense can be connected to every modal logic formula.

Semantic characterization

The formal semantics of modal logic are often referred to as "Kripke semantics" after the logician Saul Kripke . The Kripke semantics is the formalization of the intuitive concept of the possible world. A Kripke model consists of a set of such worlds, an accessibility relation (also: reachability relation ) between them and an interpretation function which assigns one of the values ​​“true” or “false” to each statement variable in each of the worlds.

The truth of a formula in a possible world w is then defined as follows:

  • Proposition variables are true in the world w if the interpretation function assigns the value “true” to them in w.
  • is true in w if p is false in w, otherwise false
  • is true in w if p and q are both true in w, otherwise false
  • is true in w if there is a world v accessible from w and p is true in v; otherwise is wrong in w
  • is true in w if for all worlds v accessible from w it holds that p is true in v; otherwise is wrong in w
Modal systems
Surname Axioms Accessibility relation
K any
T K + reflexive
D. K + serial:
B. T + reflexive and symmetrical
S4 T + reflexive and transitive
S5 T + reflexive, transitive and symmetrical

Here you can add additional clauses for any additional connectors or quantifiers. A formula is valid if it is true in all Kripke models. The various modalities discussed above can now be mapped to the accessibility relation between the worlds using various conditions. The system K arises when no condition is attached to the accessibility relation. All and only the formulas that are valid for such an arbitrary accessibility relation can therefore be proven in K. In order to preserve the system T, one must make the requirement of the accessibility relation that every world should be accessible by itself, the relation must therefore be reflexive . If one sets the accessibility relation in this way, it results that the valid formulas are exactly those that can be proven in system T. For system D there must be at least one accessible world for every world, such relations are called serial (or link total ). In addition to reflexivity, symmetry is also required for B ; H. if w is accessible from v, then v must also be accessible from w. In S4 the accessibility relation is reflexive and transitive , i.e. H. if w is accessible from v and v from u, so does w from u. Finally, for S5, the accessibility relation must be reflexive, symmetrical and transitive at the same time, i.e. H. it is an equivalence relation .

Deontic and normative modal logic

The logician and philosopher Paul Lorenzen expanded modal logic to include deontic and normative modal logic in order to justify the technical and political sciences (constructive philosophy of science).

The modal words “can” and “must” are formally reconstructed as usual. The corresponding characters listed above are only slightly modified. The various forms of modal logic have technical and political abbreviations of course hypotheses with such terms:

  • Action: the girl can jump off the springboard
  • Ethical-political permission: Tilman is allowed to get a piece of pizza
  • Biological-medical becoming: A tree can grow from a cherry stone
  • Course hypotheses ( laws of nature ): The house can collapse
  • Technical skills: The car can be built with a catalytic converter

Correspondingly, “must” modalities can be formed for the “can” modalities. All modal words (for example, necessity ) are initially informal in Lorenzen's modal logic , which means that the statements made in modal logic are only valid relative to an alleged knowledge. The different types of modalities also play together. For example in the sentence: “Accessibility (human ability ) implies possibility (technical can hypothesis)”.

If modal statements are formally logically true, the supposed underlying knowledge can be cut away . In this way, modal logic truths can be formed regardless of whether the underlying knowledge is correct. This follows from the cutting set . For Lorenzen, a punch line is simply to substantiate the modal logic.


Individual evidence

  1. ^ Friedemann Buddensiek: The modal logic of Aristoteles in the Analytica Priora A. Hildesheim 1994.
  2. Instead, Lewis used a "strict implication" based on "possibility"; CI Lewis: Implucation and the algebra of logic. In: Mind 21 (1912), pp. 522–531. Ders .: A Survey of Symbolic Logic. Cambridge 1918.
  3. ^ CI Lewis, CH Langford: Symbolic Logic. New York 1932.
  4. ^ SA Kripke: Semantical Analysis of Logic I. Normal propositional Calculi. In: Journal for Mathematical Logic and Fundamentals of Mathematics. 9, pp. 67-96 (1963).
  5. ^ BC van Fraassen; Meaning Relations and Modalities. In: Nous. 3 (1969), pp. 155-167. ML Dalla Chiara: Quantum Logic and Physical Modalities. In: Journal of Philosophical Logic. 6: 391-404 (1977).
  6. O. Becker: On the logic of the modalities. In: Yearbook for Philosophy and Phenomenological Research. 11 (1930), pp. 497-548.
  7. K. Gödel: An intuitionistic interpretation of the propositional calculus. In: Results of a Mathematical Colloquium. 4 (1932), pp. 39-40.
  8. ^ L. Wittgenstein, Logical-philosophical treatise (Tractatus logico-philosophicus). In: Annalen der Naturphilosophie. 14 (1921), especially sentence 4.21-4.24 there.
  9. ^ R. Carnap: Logical Syntax of Language. Vienna 1934, esp. Pp. 196-199. Quote from FJ Burghardt: Modalities in the language of quantum mechanics. Ideas from O. Becker and R. Carnap in today's basic research in physics (PDF; 1.0 MB) , Cologne 1981, p. 8. After his emigration to the USA, Carnap himself no longer pursued this concept, but joined the im Anglo-American space exclusively uses recognized axiomatic modal logic, e.g. B. 1947 in his book Meaning and Necessity .
  10. P. Lorenzen: On the justification of the modal logic. In: Archive for mathematical logic and basic research. 2 (1954), pp. 15-28 (= Archive for Philosophy 5 (1954), pp. 95-108).
  11. P. Mittelstaedt: Quantum Logic. In: Advances in Physics. 9, pp. 106-147 (1961). FJ Burghardt: Modal quantum metalogic with a dialogical justification . Cologne 1979; P. Mittelstaedt: Language and Reality in Modern Physics. Mannheim 1986. Chap. VI: possibility and probability.
  12. see the book Normative Logic and Ethics published in 1969 . that Lorenzens John Locke lectures at Oxford summarizes.

Web links

This version was added to the list of articles worth reading on May 12, 2007 .