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Modal Logic
 MODAL LOGIC

"In mathematics 'necessary' and 'everything' go together" (Wittgenstein. Philosophical Investigations).

"The fundamental category for the interpretation of reality is not necessity, but possibility. Everything Everything that exists is a possibility that has been realized" (Leibniz).

"If so it was, so it could be; if so it were, so it could be; but since it is not, it is not. That is logic" (Claude Lévi-Strauss).


Modal Logic

Modal logic is the logic of necessary truths and possible truths. It is a logic that employs the concepts of "necessary" and "possible" in its expressions and reasoning. The notation is: For the possible there are 4 values:

N°ValueNotation
1It is possible that x◇x
2It is possible that non-x◇¬x
3It is impossible that x¬◇x
4It is impossible for non-x¬◇¬x

In principle, the opposite of the possible would admit two different answers: the impossible and the necessary. Both options are correct to some extent: Combining the notions of possible and necessary with the values of true and false, we obtain the following two equivalences:
  1. If something is necessarily true, it is not possible for it to be false; and vice versa:

    □x ≡ ¬◇¬x
    ("it is necessary that x" is equivalent to "it is not possible that non-x")

  2. If something can be true, it need not be false; and vice versa:

    ◇x ≡ ¬□¬x
    ("it is possible that x" is equivalent to "it is not necessary not-x")
Considering these equivalences, we have the following table:

N°ValueNotationEquivalenceNotation
1It is possible that x◇xIt is not necessary that non-x¬□¬x
2It is possible that non-x◇¬xIt is not necessary that x¬□x
3It is impossible that x¬◇xIt is necessary that non-x□¬x
4It is impossible that non-x¬◇¬xIt is necessary that x□x

These equivalences are analogous to those between the universal quantifiers ∀ (all) and the existential ∃ (some):
  1. If every element has property P, then no element exists that does not have property P;; and vice versa:
    ∀xPx ≡ ¬∃x¬Px

  2. If there exists an element that has property P;, then not all elements do not have property P;; and vice versa:
    ∃xPx ≡ ¬∀x¬Px
We can then establish the following correspondence between modality and quantification: Since one modal operator can be defined as a function of the other, it is sufficient to take one of them as a primitive. The one of necessity (□) is usually chosen.

In addition to the above two equivalences, we have the rule that the necessity of something implies its possibility: □x → ◇x


The Evolution of Modal Logic

There have been three major eras in which logicians have been concerned with modality: classical antiquity, medieval times, and the twentieth century. The latter has seen the greatest development, the most prominent figures being Rudolf Carnap, Clarence Irving Lewis and Saul Kripke. Modal logic has led to important general results and has opened up new lines of research in semantics, which are being applied in various fields such as demonstration theory, computer science and artificial intelligence.


MENTAL and Modal Logic

MENTAL offers a simple, expressive and intuitive formalization of modal logic, yet extremely generic and powerful.


Beyond modal logic

MENTAL does not require special modality operators to describe and operate with necessity and possibility. These concepts transcend logic, as do quantifiers and predicates. They belong to a higher semantic level than formal logic, which is of a superficial type. Modality covertly hides a reference to human knowledge and, therefore, it is not about logic, but about epistemology. In MENTAL this aspect becomes evident because:
  1. Necessity is linked to a generic expression (parameterized or not), which can be of any type and not only of a logical type. The requirement is a particular case of the generic. Therefore, the "need" operator is not needed as a primitive. The "Generalization" primitive is sufficient. In the case where a condition is used, the antecedent specifies the condition to be detected. The consequent specifies the action to be performed.

  2. The possibility is what is not specified as necessary.
With MENTAL it is emphasized that generic expressions are of the highest level of abstraction, since it allows expressing the universal quantifier and the modal operator of necessity, as well as functions, rules, classes, etc. It also allows to express general or particular knowledge of a domain, which is mandatory, to implement hypotheses or beliefs and to experiment with them, etc. This knowledge is not necessarily linked to logic.


A modal logic without specific axioms

Modal logic is based on axioms. Unlike classical logic, there are different axiomatic systems, with different axioms. The issue of which axioms are the most suitable is a matter of much debate and controversy. MENTAL, on the other hand, does not need specific axioms of modal logic. The primitives and the generic axioms that relate the primitives are enough.


Notación

The modal logic notation □x (it is necessary that x) indicates that x cannot vary. Quine defined "p is necessary" as "p is equal to itself". For Krike, identity is an essential property of each object and individual, an internal relation that each object maintains with itself. In modal logic, identity can be expressed as □(x=x) (it is necessary that x is equal to itself).

In MENTAL, □x is expressed by the generic expression ⟨x⟩. But the expression x must specify a relation, which is the one to be maintained. And □(x=x) is expressed by the generic expression. In MENTAL, identity is a particular case of necessity.

It is possible to change the MENTAL notation to the standard notation, if desired:
Examples
  1. ⟨( x/man → x/mortal )⟩
    (if x is male, x is necessarily mortal)

  2. ⟨( c = a+b )⟩
    (it is necessary that c is always the sum of a and b)
    ((a = 7) (b = 5))
    c // ev. 12
    ((a = 8) (b = 6))
    c // ev. 14

  3. ⟨( a>12 → (a = 12) )⟩
    (it is necessary that a is not greater than 12)
    (a = 75)
    a // ev. 12

  4. ⟨( a = 5 )⟩
    (it is necessary that a is always 5)
    (a = 75)
    a // ev. 5

  5. ⟨( a∉A → (A = {a A↓} )⟩
    (it is necessary that a belongs to the set A)
    (A = {b c})
    A // ev. {a b c}

Addenda

Conceptual pragmatism

C.I. Lewis is also the founder of conceptual pragmatism, which is based on the following main ideas:
Bibliography