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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. Predicate LogicWith thepropositional logic one can find rules of inference that lead from truestatements necessarily to true statements. The modus ponens here is theprototype of such a concluding rule.&#160; Butwe know from Aristotelian logic also other conclusions which necessarily leadfrom true to true, e.g. the two sentences "All men are mortal" and"Socrates is a man" lead to "Socrates is mortal".&#160; Here in the proposition"All men are mortal" the word "All" appears.&#160; One could continue to consider theproposition as a whole and its truth value and thus stick to the propositionallogic. But you can also look into the "inside" of the sentence andnotice that here the number of elements of a set comes into play. This is notalways the case, but with a view to expressing oneself in as much detail aspossible, this possibility should also be taken into account in a formallanguage. This has led to the extension of the propositional logic to theso-called predicate logic, in which a special structure is now provided for thepropositions, namely a quantification with "quantifiers" such as"all" or "none". The decisivefactor for such an extension was that one learned to give a more generalmeaning to the term "predicate", which we know from the grammar of anatural language. In logic, a predicate is everything that can be meaningfullyattributed to an object, i.e. "predicated". So, in the sentence"Socrates thinks", "thinks" is the predicate. Thus, inaddition 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 arelation R. The besttranslation of such sentences into character strings has proven to be, forexample, that a one-digit predicate is seen as a function of a variable and theobject to which the predicate is assigned is seen as an argument. The functions(x) then means that to the object "x" the predicate s≔ "is mortal" is predicted, i.e. assigned. One can then easilyformulate that the predicate belongs to several or all objects from a given setor to none. This therefore gives the possibility of quantification. This functions(...) has as function value the truth value of this statement; s(x) istherefore 1 or true, if x is mortal. If we now define the function M(x) withM(...):= "is a human being", we can now formulate:For all xapplies: If x is a human being, then x is mortal, or ∀x (M(x) → s(x)).where we haveintroduced the symbol ∀x for "For all x".&#160; The symbol "∀" is called a quantor. Of course, you have to determine beforehandto which quantity "all" refers. Useful is alsothe symbol "∃x" for "There is anx". In principle, however, the symbol "∀x" alone can be used for all quantifications.The predicatelogic therefore includes the propositional logic. Additional concluding rulesin predicate logic are first: ∀x M(x) ∧ (x = c) ⊨ M(c),i.e. "If apredicate M is assigned to all x from a given set, then also to a singleelement c of this set", and second:M(c) ⊨ ∃x M(x),i.e. if anobject c has the property M, then there exists an object x that has theproperty s.After all, itdoes not work without the mode ponens, e.g. in the formM(c), ∀x (M(x) → s(x)) ⊨ s(c),where"c" stands for a specific element x from a predetermined set. All syllogismsof 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 concludingrule is applied which can be derived from the modus ponens. But it can also bederived directly from the tautology &#160;(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-orderlogic by extending the quantification to the predicates as well. We want to studythis from a famous example: Let us first consider the statementP(x) → P(S(x)),i.e. if thepredicate 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, forexample, an equation for x is fulfilled, then it should also be fulfilled for x+ 1.This one can berequired for all natural numbers, and one gets∀x (P(x) → P(S(x))).This is one ofthe prerequisites for a particularly prominent concluding rule in numbertheory, the "mathematical induction". One should not confuse the word"induction" with the logical induction. The danger of confusing itwith 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 isassigned 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 otherhand, one wants to demand this statement ∀x (P(x) → P(S(x))) forall predicates P, then one writes:∀P [∀x (P(x) → P(S(x))].So, this is astatement in the language of second-order logic; the quantifier operates alsoon a predicate P.You can alsocontinue predicting, i.e. assigning a property, by assigning properties, i.e.predicates, to properties again. If F designates a property that can beassigned to a predicate P, you can, for example, first formulate: F(P) → P(x), That means: If apredicate P has the property F, then x has this predicate P. If you requirethis for all predicates P of a certain set, you can, for example, formulate:∃x ∀P(F(P) → P(x))i.e. for all predicatesP applies: If they have the property F, then there is also an object x, which hasthis predicate P.If the propertyF means "is good", whatever that is, then the proposition says:"There is an object x that has all good properties." Of course, thisdoes not say anything about whether this statement is true or can be derivedfrom any axioms. One can only formulate it.About thedifferences of the first- and second-order logics with regard to their means ofexpression there are deeply rooted propositions in mathematical logic to whichone can gain access only through intensive study. Here it can only be shown howto extend the language of propositional logic. It should be noted that theconcluding rules of first-order logic are "sufficient as building blocksfor all mathematical modes of argumentation" (Ebbinghaus &#38; Thomas,2018, p. 62).&#160; Modal and Deontic LogicIn thepropositional and predicate logic, one always restricts oneself to a certain typeof proposition, namely to those that can either be true or false. The law ofthe excluded third applies here, i.e. A ∨ ¬A is atautology.&#160; 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 alsoknow the situation in which we can only say about a fact that it can bepossible but does not have to be necessary, that it is therefore"contingent". If the language of predicate logic is extended bycharacters 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.  Fig.1 The three basic modal and deontological terms It turns outthat the modal terms "necessary", "impossible" and"contingent" are disjunctive to the other two terms: What is notnecessary can be impossible, but also contingent. What is not impossible may becontingent, but even necessary, and what is not contingent is either necessaryor impossible. The same applies to deontological terms: What is not obligatorymay be forbidden, but also permitted, etc.With these threeterms one must now expand the formal language of propositional logic byintroducing new symbols. Actually, you only need a symbol for one term, becauseit shows that you can express all the others with the help of negations throughthis one symbol. This new symbolshall be the sign "N" in modal logic, which can also be interpretedas an operator, which acts on a proposition p: "Np" shall mean:"It is necessary, that p is true".In modal logicwe introduce the operator "O", "Op" now means: "It isobligatory that the action p is performed".&#160; All in all, youget 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,&#160;  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 canformulate 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 distinctionmust be made here between norms that are to be set with a law and propositionsabout norms. The norms are brought into effect by setting, the propositionsabout 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. Fig.2: The modal and deontic hexagon (after (Joerden, 2010), (Honerkamp,  2015)). The arrows indicate an implication (what is obligatory is also  permitted), " &#62;-&#60;" a contravalence or contradictory opposition  (e.g. what is not forbidden is permitted), "----"an exclusion or  contrary opposition (e.g. being obligatory and being forbidden are  mutually exclusive) and "..."" a disjunction or subcontrary opposition  (e.g. contravalence &#62;-&#60; to "permitted" and to "not obligatory").  Furtherexplanations such as the discussion of possible axioms and a calculus inconnection with the calculus of propositional logic go beyond the scope of thisbook. Outlook However, thelist of extensions to the propositional logic has not yet been completed in anyway. There are also so-called non-classical logics in which, for example, thelaw of the excluded third does not apply. One then does not assume thatpropositions are only either true or false. In modal logic we have assumedthis, only considering the possibility that one does not know it exactly. Herenow it is to be renounced that there is "ontologically" only thisalternative. It will no longer be adequate to map the Boolean association ofstatements to a binary Boolean association. There is a wealth of approaches tosuch multi-valued logics, and "intuitionist logic" is a prominentexample of this. The so-calledfuzzy logic must also be mentioned here. It still permits nuances of thepredicates such as "very" or "rather", however treats themquantitatively again. It plays an important role in many control engineeringapplications. But the mostelaborated approach is "dealing with uncertain knowledge", which isbased on mathematical probability theory. Here a proposition is assigned ameasure of the probability that it is true. This can also be seen as a measureof the credibility of a statement. Thus, we canpick up the theme of Aristotle, which he hinted at in the classification ofconclusions, when he spoke of a dialectical conclusion. Here already questionsarose, which could not be answered at that time: What can be said about aconclusion in dialectical inferences? Can one even develop a calculus for propositionsthat are merely credible? How could conclusions be drawn as strictly aspossible in such a calculus? What would "strict" even mean here?These questions will be addressed later.