In addition to the considerations presented in the last chapter, some important extensions of the propositional logic must be mentioned here in any case, in order not to let the reader believe that he or she has already become acquainted with a large part of the logic through propositional logic. The possibility of expression of the propositional logic is still very limited and an extension in this or that direction will soon be desired, if one is occupied longer with it. However, the knowledge I have provided so far about the origin and meaning of the concluding rules is sufficient to understand the further considerations in this book.
With the propositional logic one can find rules of inference that lead from true statements necessarily to true statements. The modus ponens here is the prototype of such a concluding rule. But we know from Aristotelian logic also other conclusions which necessarily lead from true to true, e.g. the two sentences “All men are mortal” and “Socrates is a man” lead to “Socrates is mortal”.
Here in the proposition “All men are mortal” the word “All” appears. One could continue to consider the proposition as a whole and its truth value and thus stick to the propositional logic. But you can also look into the “inside” of the sentence and notice that here the number of elements of a set comes into play. This is not always the case, but with a view to expressing oneself in as much detail as possible, this possibility should also be taken into account in a formal language. This has led to the extension of the propositional logic to the so-called predicate logic, in which a special structure is now provided for the propositions, namely a quantification with “quantifiers” such as “all” or “none”.
The decisive factor for such an extension was that one learned to give a more general meaning to the term “predicate”, which we know from the grammar of a natural language. In logic, a predicate is everything that can be meaningfully attributed to an object, i.e. “predicated”. So, in the sentence “Socrates thinks”, “thinks” is the predicate. Thus, in addition to properties, these can also be verbs. But also multi-digit relations, e.g. R(x,y) can be predicates; here now two objects x and y are assigned to a relation R.
The best translation of such sentences into character strings has proven to be, for example, that a one-digit predicate is seen as a function of a variable and the object to which the predicate is assigned is seen as an argument. The function s(x) then means that to the object “x” the predicate s≔ “is mortal” is predicted, i.e. assigned. One can then easily formulate that the predicate belongs to several or all objects from a given set or to none. This therefore gives the possibility of quantification.
This function s(…) has as function value the truth value of this statement; s(x) is therefore 1 or true, if x is mortal. If we now define the function M(x) with M(…):= “is a human being”, we can now formulate:
For all x applies: If x is a human being, then x is mortal, or
∀x (M(x) → s(x)).
where we have introduced the symbol ∀x for “For all x”. The symbol “∀” is called a quantor. Of course, you have to determine beforehand to which quantity “all” refers.
Useful is also the symbol “∃x” for “There is an x”. In principle, however, the symbol “∀x” alone can be used for all quantifications.
The predicate logic therefore includes the propositional logic. Additional concluding rules in predicate logic are first:
∀x M(x) ∧ (x = c) ⊨ M(c),
i.e. “If a predicate M is assigned to all x from a given set, then also to a single element c of this set”, and second:
M(c) ⊨ ∃x M(x),
i.e. if an object c has the property M, then there exists an object x that has the property s.
After all, it does not work without the mode ponens, e.g. in the form
M(c), ∀x (M(x) → s(x)) ⊨ s(c),
where “c” stands for a specific element x from a predetermined set.
All syllogisms of Aristotelian logic can also be formulated in predicate logic. The syllogism “Barbara” reads here for example:
∀x (G(x) → M(x)), ∀x (M(x) → s(x)) ⊨ ∀x (G(x) → s(x))
where “G” stands for the predicate “is Greek”. Here a concluding rule is applied which can be derived from the modus ponens. But it can also be derived directly from the tautology
(A → B) ∧ (B → C) → (A → C).
So far, quantification has always extended to object x, an element from a given set. One speaks here of a first-order logic. You can also introduce a second-order logic by extending the quantification to the predicates as well.
We want to study this from a famous example: Let us first consider the statement
P(x) → P(S(x)),
i.e. if the predicate is assigned to object x, then also to object S(x). This makes sense, for example, if x is a number and S(x) = x + 1, the successor of x. If, for example, an equation for x is fulfilled, then it should also be fulfilled for x + 1.
This one can be required for all natural numbers, and one gets
∀x (P(x) → P(S(x))).
This is one of the prerequisites for a particularly prominent concluding rule in number theory, the “mathematical induction”. One should not confuse the word “induction” with the logical induction. The danger of confusing it with physical induction is probably less. In order to arrive at a conclusion, there must still be an induction start, e.g. P(0), i.e. the property P is assigned to the number 0. Then, with the above implication we may conclude: ∀x P(x), i.e. the property P applies to all natural numbers x.
If, on the other hand, one wants to demand this statement ∀x (P(x) → P(S(x))) for all predicates P, then one writes:
∀P [∀x (P(x) → P(S(x))].
So, this is a statement in the language of second-order logic; the quantifier operates also on a predicate P.
You can also continue predicting, i.e. assigning a property, by assigning properties, i.e. predicates, to properties again. If F designates a property that can be assigned to a predicate P, you can, for example, first formulate:
F(P) → P(x),
That means: If a predicate P has the property F, then x has this predicate P. If you require this for all predicates P of a certain set, you can, for example, formulate:
∃x ∀P(F(P) → P(x))
i.e. for all predicates P applies: If they have the property F, then there is also an object x, which has this predicate P.
If the property F means “is good”, whatever that is, then the proposition says: “There is an object x that has all good properties.” Of course, this does not say anything about whether this statement is true or can be derived from any axioms. One can only formulate it.
About the differences of the first- and second-order logics with regard to their means of expression there are deeply rooted propositions in mathematical logic to which one can gain access only through intensive study. Here it can only be shown how to extend the language of propositional logic. It should be noted that the concluding rules of first-order logic are “sufficient as building blocks for all mathematical modes of argumentation” (Ebbinghaus & Thomas, 2018, p. 62).
Modal and Deontic Logic
In the propositional and predicate logic, one always restricts oneself to a certain type of proposition, namely to those that can either be true or false. The law of the excluded third applies here, i.e. A ∨ ¬A is a tautology.
But often we do not know for sure whether the statement A about a fact is really true. We now also want to consider that it is only possible that a proposition p is true. (In the following we describe propositions with small letters p, q, …, so that the notation remains clear later.) However, for statements p, which we have derived from accepted assumptions according to logically correct concluding rules, we would then say that the proposition is necessarily true.
Finally, we also know the situation in which we can only say about a fact that it can be possible but does not have to be necessary, that it is therefore “contingent”. If the language of predicate logic is extended by characters for such terms, one speaks of modal logic.
Analogously, another extension of the language of propositional logic can be developed, which deals with duties such as obligations or prohibitions. You then get a deontic logic (δέῖ, δέῖ gr. = you have to). But you can’t make proposition in this area, but you can make actions: The symbol “p” then stands for an action instead of a proposition. And one can also allow actions, i.e. neither forbid nor command them.
Here we see parallels between the terms “necessary, impossible, contingent” and ” obligatory, forbidden, permitted”. Fig. 1 also shows them clearly.
It turns out that the modal terms “necessary”, “impossible” and “contingent” are disjunctive to the other two terms: What is not necessary can be impossible, but also contingent. What is not impossible may be contingent, but even necessary, and what is not contingent is either necessary or impossible. The same applies to deontological terms: What is not obligatory may be forbidden, but also permitted, etc.
With these three terms one must now expand the formal language of propositional logic by introducing new symbols. Actually, you only need a symbol for one term, because it shows that you can express all the others with the help of negations through this one symbol.
This new symbol shall be the sign “N” in modal logic, which can also be interpreted as an operator, which acts on a proposition p: “Np” shall mean: “It is necessary, that p is true”.
In modal logic we introduce the operator “O”, “Op” now means: “It is obligatory that the action p is performed”.
All in all, you get this:
Np: It is necessary that p is true, i.e. it is impossible that p is false,
¬Np: It is not necessary for p to be true, i.e. it is not impossible for p to be false,
N¬p: It is necessary that p is false, i.e. it is impossible that p is true,
¬N¬p: It is not necessary for p to be false, i.e. it is not impossible for p to be true,
¬Np ∧ ¬N¬p: The statement p is contingent true because it is not necessary and not impossible to be true.
One can formulate accordingly:
Op: It is obligatory to perform the action p,
¬Op: It is not obligatory to perform the action p,
O¬p: It is obligatory not to perform the action p, i.e. it is forbidden to perform the action p,
¬O¬p: It is not obligatory not to perform the action p. So, it is allowed to perform the action p.
¬Op ∧ ¬O¬p: It is neither obligatory nor forbidden to perform p.
A distinction must be made here between norms that are to be set with a law and propositions about norms. The norms are brought into effect by setting, the propositions about norms can be true or false.
Fig. 2 shows the different operators in the so-called modal or deontic hexagon. With these terms and their symbols, which are comprehensible for everyone, all specifications can be clearly represented.
Further explanations such as the discussion of possible axioms and a calculus in connection with the calculus of propositional logic go beyond the scope of this book.
However, the list of extensions to the propositional logic has not yet been completed in any way. There are also so-called non-classical logics in which, for example, the law of the excluded third does not apply. One then does not assume that propositions are only either true or false. In modal logic we have assumed this, only considering the possibility that one does not know it exactly. Here now it is to be renounced that there is “ontologically” only this alternative. It will no longer be adequate to map the Boolean association of statements to a binary Boolean association. There is a wealth of approaches to such multi-valued logics, and “intuitionist logic” is a prominent example of this.
The so-called fuzzy logic must also be mentioned here. It still permits nuances of the predicates such as “very” or “rather”, however treats them quantitatively again. It plays an important role in many control engineering applications.
But the most elaborated approach is “dealing with uncertain knowledge”, which is based on mathematical probability theory. Here a proposition is assigned a measure of the probability that it is true. This can also be seen as a measure of the credibility of a statement.
Thus, we can pick up the theme of Aristotle, which he hinted at in the classification of conclusions, when he spoke of a dialectical conclusion. Here already questions arose, which could not be answered at that time: What can be said about a conclusion in dialectical inferences? Can one even develop a calculus for propositions that are merely credible? How could conclusions be drawn as strictly as possible in such a calculus? What would “strict” even mean here? These questions will be addressed later.