Modal logic

The modal logic is the one branch of logic that deals with the possible and necessary inferences about the Modalbegriffe. Thus, for within the modal logic not only statements such as " It is raining " or " All circles are round" analyze, but statements like "Maybe it's raining " and " Necessarily all circles in the round".

• 7.1 Syntactic characterization
• 7.2 Semantic characterization
• 7.3 Deontic modal logic and normative

History

The earliest approaches to modal logic can be found in Aristotle, in the first analysis. There also the modal logic variants are discussed for each categorical syllogism. In the Middle Ages examines, inter alia, Duns Scotus modallogische terms. Gottfried Wilhelm Leibniz coined the term " possible world ", which has become important for the development of modal logic model theory. In the 20th century, to distinguish between two fundamentally different approaches to specify modal statements and their logical relationships, an object- linguistic and axiomatic and a metalanguage -operative.

The first axiomatic approach yielded Clarence Irving Lewis in 1912 in his critique of "material implication " ( Whitehead and Russell), which by no means the traditional " if - then " corresponded. Together with CH Langford he set up in 1932 five logical systems (S1 to S5 ) with different modal axioms that more or less " plausible " appeared. Only in 1963 Saul Kripke was able to develop a semantics for the multitude of proposed hitherto modal logic systems. Used on this axiomatic basis since the 1970s Bas van Fraassen (Toronto), and Mary L. Dalla Chiara (Florence) arrangements in the context of quantum logic.

Largely consequences for the further development of modal logic were the attempts of Husserl 's student Oskar Becker in 1930 to interpret the modal statements of Lewis phenomenologically. Following a suggestion Becker Kurt Gödel showed in 1932 a close connection between the system S4 and the intuitionistic logic.

Fundamentally new was the metalinguistic approach to modalities of Rudolf Carnap 1934. During a sharp criticism of Wittgenstein's conception of the fundamental limitations of linguistic skills, he claimed that the deduction of Lewis extension of classical logic by adding an operator "possible" is not wrong, but unnecessary, " because the meta-language, which is also necessary to describe the axiomatic modal logic, already allow to express both the sequence relationship and the arrangements with exact wording. "

This concept Carnap, modalities only when speaking about a language, to use meta-linguistic, was taken up again until 1952 by Becker and used two years later by Paul Lorenzen in support of its operational modal logic. The Lorenzen (later also of Schwemmer ) developed constructive logic based on a formalized dialogue semantics was first introduced in 1961 by Peter Mittelstaedt in the quantum logic and 1979 further developed by his student Franz Josef Burghardt for " modal Quantenmetalogik ".

The underlying intuition

The terms "possible" and "necessary" provides the language in addition to "true" and "false" an additional way to characterize Statements: Some statements are false but 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. We can imagine for example, that there would be people with green skin, the statement " Some people have green skin " is possible. However, we can not imagine that there would be square circles, the statement " There are square circles" is not possible, that is impossible. In addition, there are statements that are true in every imaginable situation; such statements we refer to as necessary. Necessary statements such as " circles are round" and " bachelors are unmarried ".

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

When one refers to a statement in this sense as possible, you will not comment on whether the statement could also be wrong. For this reason, all the 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. From this option term, the concept of contingency is different: Quota is a statement if and only if it is false in at least one possible world and true in at least one possible world if 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 you, a sub-statement replaced in a statement that contains modallogische expressions by another with the same truth value, the truth value of the overall presentation is not necessarily preserved. For example, consider the statement " It is possible that Socrates is not a philosopher ." This statement is true ( we can imagine that Socrates could never interested in philosophy) and includes as part of the statement false statement " Socrates is not a philosopher ." We now replace this part of the statement by also false statement " There are square circles", we get "It is possible that there are square circles". This is, however, contrary to our initial statement wrong ( because we can, as I said, no square circles imagine ). Thus we have shown that the modal logic is not truth-functional.

Notation

In modal logic, the term " possible " (more precisely, the set operator " it is possible that ...") represented by an upside to the top of the square, which also means "Diamond" (English for rhombus ), and the term " necessary " (more precisely, " it is necessary that ... ") by a small square, which is also " called box ".

Modallogische conclusions

Modal and negation

Connect the modal operators with the negation, ie the " not " ( in formal representation: ), so it makes a difference whether the negation refers to the whole, composed of modal operator and statement expression or just to the the modal operator trailing expression. "It is not possible that Socrates is a philosopher " () thus means something other than " It is possible that Socrates is not a philosopher " ( ), the first statement is false, the second true. It is further noted that statements can be made with the possibility operators in statements with the necessity operator translate and vice versa. "It is possible that Socrates is not a philosopher " is synonymous with "It is not necessary that Socrates is a philosopher ," " It is not possible (it's impossible) that Socrates an elephant " with "It is necessary, that Socrates is not an elephant. " In formal notation:

• Equivalent to
• Equivalent to

"It is possible that Socrates 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's not possible that Socrates not human is ".

• Equivalent to
• Equivalent to

Because of these last two equivalences, the possibility operators can define or reversed by the necessity operator.

Disjunction and conjunction

The disjunction ( OR operation, symbolically ) of two possible statements is equivalent to the possibility of their disjunction. From "It is possible that Socrates is a philosopher or is it possible that he is a carpenter " follows " It is possible that Socrates is a philosopher or a carpenter is " and vice versa.

• Equivalent to

The same applies to the Notwendigkeitssoperator and the conjunction ( AND operation, symbolically ): "It is necessary that all circles are round, and it is necessary that all triangles are square" is equivalent to "It is necessary that all the circles round and all the triangles are square. "

• Equivalent to

The situation is different in the conjunction of possibility and necessity of the disjunction of statements. While implies the possibility of a conjunction of two statements the conjunction of the possibility of the statements, but this is not true vice versa. If it is possible that Socrates both philosopher and carpenter, then it must be possible that he is a philosopher, and also possible that he is a carpenter. In contrast to this example, it is well possible that the number of planets is just as possible that it is odd, but it is not possible that they both straight and is also odd.

• From below, but not vice versa

Similarly, one can from the disjunction of the need for two statements infer the need for the disjunction of individual statements, but not vice versa. Is it necessary that there are infinitely many prime numbers or necessary that Socrates is a philosopher, then it must be necessary that there are infinitely many prime numbers, or that Socrates is a philosopher. However, on the other hand, for example, necessary that Frank weighs more than 75 kg or heavier than 75 kg, but it is neither necessary that he weighs more than 75 kg, more necessary that it is heavier than 75 kg. Therefore:

• Follows, but not vice versa

Quantifiers, Barcan - formulas

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

• Should be equivalent.
• Equivalent to

Here, one direction of the equivalence is always a problem and is accepted:

• It follows from. Is there an object that may have the property, it may need to be something that has the property.
• It follows from. Have necessarily all items the property, so every object necessarily has the property.

These statements apply in most quantified modal logics.

More problematic are the in the Ruth Barcan Marcus named after return directions of the two Aquivalenzbehauptungen ( Barcan formulas ):

• It follows from
• It follows from

The two Barcan formulas are at the usual substitutability of ... by ... and ... by ... equivalent to one another. The debate revolves around the interpretation of the formulas. Is there, for example, someone who can grow a beard ( the so possibly bearded ), so it is possible that there is someone who wears a beard. The Barcan formula corresponding reversal ( If there may be someone who is bearded, so there is someone who may be bearded. ) Leads to the following problem: The front part of the if-then sentence asserts only that there be an individual can, that would be bearded, the rear part assumes that there is an individual that may be bearded. Thus, this subset has a Existenzpräsupposition: Suppose that the quantifier refers to a lot of people who are currently in a particular room, so presupposes the rear part that just someone is in the room, the one to could grow beard, the front part not. So is. e.g. excluded that the room is randomly empty. This becomes more problematic when the quantifier on, to everything there is related '; the Barcan formula would then claim that every possible object ( possibilium ) to which a property can be assigned, now exists and possibly having the property. Take, for example, to, the childless philosopher Ludwig Wittgenstein might have had a son, followed by the formula that now would be a person who is possibly Wittgenstein's son. The contentiousness of the Barcan formula for necessity and universal quantifier can be made clear by using the same example: All ( actually existing ) people are necessarily no sons Wittgenstein, that does not mean that necessarily all (possible) people are not sons of Wittgenstein that Wittgenstein words no sons could have had. Ruth Barcan Marcus himself has set up the formulas, but excluded from these same reasons from normal modal logic system.

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

• Follows, but not vice versa

From the possibility of a universal claim follows the Allquantifikation a way statement, but not vice versa.

The reasons for this are similar to those that were noted above in the combination of conjunction and possibility (see also de re and de dicto ). If it is possible, that all men wear a beard, all men must be able to have a beard. Although anyone can possibly win at backgammon, this does not mean that it is possible for everyone to win (in this game can only be one winner there namely always give ).

• Follows, but not vice versa

The existential quantification of a need statement implies analogous to the necessity of the existence statement, but not vice versa. Example, is there a thing that is necessarily God, it is necessary that there is a God. In backgammon, there is necessary a winner (the game can not justify draw out ), but it does not follow that one of the players wins necessary.

Other interpretations of modal operators

The operators Diamond and Box can be verbalized in other ways than by " necessary" and "possible". In the " deontic " interpretation the operators by the ethical concepts of " allowed" and " necessary " are interpreted, one no longer speaks of modal logic in the narrow sense, but of deontic logic. The modal logic in the narrow sense is then sometimes referred to as " alethic modal logic ". In temporal logic, the operators are, however, interpreted over time. Summarizing the operators, however, as concepts of faith, that is, the subjective -true for holding on, you get to epistemic logic.

It is characteristic of all these interpretations, that the above conclusions remain useful and intuitive. This is only an example, namely, the equivalent of, and will be shown.

• " It is permitted that p " is equivalent to " It is not necessary that not -p"
• " P holds sometime in the future" is equivalent to " It is not the case that not -p is always true in the future"
• " I think it is possible that p" is equivalent to " I do not think it certain that not - p."

Different systems of modal logic

Syntactic characterization

A formal system of modal logic arises from the fact that adding a propositional logic or predicate logic formulas and modallogische additional axioms or rules of inference. Depending on which of the logic one starts, one speaks of modallogischer statements or predicate logic. The language of modal logic contains all statements or predicate logic formulas, and in addition, all formulas of the form and for any modal logic formulas. This box can be defined by Diamond and vice versa according to the equivalences already known:

• Equivalent to
• Equivalent to

In terms of the modal logic derivation term is clear, first, that there are several such terms that can be arranged in different modallogische "systems". This is partly due to the above, various interpretations of the operators box and diamond together.

The vast majority of modal systems rely on the system K (K stands for Kripke ) on. K arises from the fact that it is the axiom schema K and the final rule, the Nezessisierung ( also referred to as " Godel rule " after the logician Kurt Godel ) allowed:

• Axiom schema K:.
• Nezessisierungsregel If true ( i.e., if p is found ), as is also true ( may be derived ).

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

If you add to the system K the axiom schema T to, we obtain the system T.

• Axiom schema T: or

Under the modal interpretation of this scheme is intuitively valid, because it says that true statements also are always possible. Under the deontic interpretation is obtained, that everything that is true is also allowed, and this is intuitively not a valid conclusion, because there are also violations of the rules and so true, but did not allow statements. For deontic applications are therefore weakens the axiom scheme T to the axiom schema D. If you add D to K added, one obtains the system D (D for " deontic " )

• Axiom schema D:

D states under the deontic interpretation that what is offered is also permitted, and is therefore under this interpretation, a reasonable inference dar.

If we extend T to the axiom scheme B, we obtain the system B. ( B stands for Brouwer. )

• Axiom Scheme B:

The S4 system arises from the fact that it extends the system T by the axiom schema 4. ( The term S4 is historic and dates back to the logician CI Lewis. Lewis has developed five modal systems, of which only two, S4 and S5, are in use today. )

• Axiom schema 4: or

The systems S4 and B are both stronger than T and thus also as D. "Stronger " here means that all formulas that are provable in T (or D) in S4 and B are provable, but not vice versa. S4 and B are independent, that is, that in both systems are provable formulas that are not provable in the other.

Add to that the axiom system T Scheme 5 added, one obtains the system S5.

• Axiom Scheme 5:

S5 is both stronger and S4 as well as B. It should be noted that the axiom Scheme 4 from a temporal interpretation is valid, but not 5: If there is at a time in the future, a time in the future, is considered to be the p then there is a time in the future, is considered to be the P (4). But it is not true that if there is a point in the future, applies to p, is such a time there for all time points in the future ( 5). S4, S5, but not so suitable for a temporal interpretation.

In S4 and S5 chains can be reduced by modal operators into a single operator. In S4, but this is only allowed if the chain consists of the same operators. The formula is for example there with equivalent. In S5 can be any chains, including dissimilar reduce. Instead you can simply write there. In all other modal systems mentioned no reduction is possible.

The last -mentioned property of the system makes it possible for many Modallogiker S5 to the most suitable for the actual modal, the strict sense, that is to analyze the terms " can " and "necessary". The reason is that we do. Repeated application of these expressions to a statement, as opposed to a simple application that can assign no real sense intuitively To example, it is difficult to say what "it is necessary that it is possible that it is raining " hot as opposed to simply "It is possible that it rains ." From this perspective, it is an advantage of S5 that returns it repeated applications of the operations to simple, a more intuitive sense with every modal logic formula in this manner are connected.

Semantic characterization

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

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

• Propositional variables are true in the world w, if the interpretation function w them in the "true" value assigns.
• 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 an accessible world w of v and p v is true; otherwise is false in w
• Is true in w if v is valid for all w of accessible worlds that p is true in v; otherwise is false in w

Here you can still additional clauses for any other connectives or quantifiers add. A formula is valid if it is true in all Kripke models. The above-discussed various Modalkalküle can now be mapped on different conditions on the accessibility relation between worlds. The system K is formed when no condition is linked to the accessibility relation. All and only the valid in such arbitrary accessibility relation formulas are thus provable in K. To get the system T, one has to raise the demand for accessibility relation that each world is to be accessible from itself, ie, the relation must be reflexive. Substituting the accessibility relation so tight, it follows that the valid formulas are provable in the system exactly T. For the system D, there must be at least one accessible for each world, such relations are called serial ( or left total). For B symmetry is also required in addition to reflexivity, h d w is accessible from v, so v must also be accessible from w from. In S4, the accessibility relation is reflexive and transitive, h d w is accessible from v and v of the u, w as also by from u. For S5, finally, the accessibility relation must also reflexive, be symmetric, and transitive, that is, there is an equivalence relation.

Deontic modal logic and normative

The logician and philosopher Paul Lorenzen has the modal logic to the deontic and normative modal logic extended to establish the technical and political sciences thereby (constructive philosophy of science ).

The Modalworte "may" and " must " be formally reconstructed as usual. The appropriate above-mentioned characters are only slightly modified. The various forms of modal logic equipped with such terms on the technical and political abstracts of course hypotheses:

• Capacity to act: The girl can jump off the diving board
• Ethical- political NOT: Tilman may get a slice of pizza
• Bio - medical Become: From a cherry pit can create a tree
• Course of hypotheses ( laws of nature ): The house can coincide
• Technical Skills: The car can be built with catalyst

Accordingly, be referred to the " can " make arrangements " must " modalities. All Modalworte (eg need) are initially relaxed in the modal logic Lorenzen, which means that the information contained in the modal logic statements speak only relative to a supposed knowledge. - The different types of modalities also play together. Approximately in the sentence: " accessibility (human assets ) implies possibility (technical can - hypothesis) ."

Are Modalaussagen formally logically true, the alleged underlying knowledge can be cut away. In this way, therefore, can be formed regardless of whether the knowledge underlying true modallogische truths. This follows from the average rate. For Lorenzen is a modal logic Pointe easy to substantiate.