The predicates of a theory are the formulas which contain variable free.
One can also regard them as the concepts of a theory.
Fundamental predicates, or concepts first
The logicians thus call the formulas of the sentences, the proposals, the statements, the assertions, the theses, and in certain cases, of the assertions, the laws, the assumptions, the axioms, the theorems, of the equations,…The simplest sentences for the calculation of the predicates are the atomic formulas. One obtains them by assembling a name of fundamental predicate with one or more names of object. For example with three names of object, Socrate, Pierre, Marie, and two names of unary predicate, is a man, is a woman, one can make six atomic formulas: Socrat is a man, Marie is a woman,… “is a woman” is a unary predicate because it applies to an individual, it is a quality. “is in love with” is a binary predicate, because it applies to two individuals. It is a binary relation. There are also ternary relations. X is produced by there and Z is a ternary relation between X, there and Z.
A rule very generally accepted in formal logic consists with always putting a fundamental predicate in first follow-up of its arguments, i.e. the objects to which it is allotted. That gives human Socrate to say that Socrate is human and equal X to say there that X is equal to Y. For a binary Relation R one however often prefers the notation xRy in Rxy.
The equations or the equalities are simple sentences built with a binary predicate, =, very private individual as for his significance, but which does not raise grammatical difficulties.
Names of object and operators
The names of object can be simple or made up. The simple names are for the logicians and the mathematicians of the symbols. They are in general represented by a letter, I, NR, or by a graphic sign, 0,1,…The words made up of several letters, Socrate,…, can be regarded as simple words as long as their letters are not regarded as words. This technique is very useful to develop a formal Système in a semi-natural language. Many treaties of logic are difficult to read, even for a specialist, because it is necessary to familiarize with the formalism sometimes very particular their authors. Into this page, one will not introduce particular logical symbols. All the basic concepts are translated by natural expressions. One limits oneself however to a simplified grammar, that which will be exposed. It is very easy to translate the rules or the axioms which we will state in a formalism more rigorously determined, with symbols. Bonds return in other pages of the encyclopedia where this formalism is exposed. The two approaches are complementary. The rigorous formal methods are appropriate better for the mathematical theories. The semi-natural language shows more clearly why the Calcul of the predicates can be regarded as a universal grammar.
The words (really) made up are obtained by assembling several simple words. They are called also formal expressions.
The names of operators are the principal tools to form made up names starting from simple names. The operators are also called functions, with the mathematical direction. We will further see the logical operators are used to form made up sentences starting from simple sentences.
Starting from the name Pierre, one can form much other names with only one operator, the father of: the father of Pierre, the father of the father of Pierre, the father of the father of the father of Pierre, and so on. “The father of” is a unary operator, or in a place, or of arite 1, because it applies to only one name of object.
From 1 and of the binary operator +, one can form many names: 1+1, (1+1) +1, 1+ (1+1), 1+ ((1+1) +1),… the result of a binary operator O applied to two individuals X and is noted there O (X, there), oxy (in the notation of Łukasiewicz), or xoy.
The names of operators can be introduced, in a semi-natural language. For example, “the child of Pierre and Marie” can be regarded as obtained by the application of the binary operator “the child of… and” with the two beings Pierre and Marie. He can be written E (P, M) or ePM or PEM.
One often makes use of the brackets to clearly indicate the way in which the names made up are built starting from simple names. The notation of Łukasiewicz, makes it possible to occur some, but the formulas become much less readable.
As long as 1 and + has their ordinary significance 1+ (1+1) is equal to (1+1) +1 but it is not true for all the binary operators: the child of (the child of Pierre and Marie) and Socrate are not equal to the child of Pierre and the child of Marie and Socrate, if there exists. In way symbolic system E (E (P, M), S) is different from E (P, E (M, S)).
One could make use of the first notation for the addition and say that + (1, + (1,1)) is equal to + (+ (1,1), 1). But precisely because these two numbers are equal, it is to well better write x+y than + (X, there). That makes it possible to remove all the brackets, 1+1+1, because in this case there is no ambiguity. But one cannot make to the same thing with the operator “the child of” because (PEM) are is different from EP (My). The notation xoy in general makes it possible to save brackets and it is often more readable but it goes only for the binary operators. For a ternary operator or more, it is necessary to return to the first notation, O (X, there, Z), or oxyz, or with more complicated formulations.
The simple names and the names of operators are enough to make all the made up names which one needs in sciences.
With the fundamental predicates, they make it possible to make all the simple sentences.
Boolean algebra of the complex sentences
The logical operators allow to make complex sentences starting from simple sentences.The Boolean operators are the negation not, the conjunction and, not-exclusive disjunction or, the implication implies, and some others.
The negation is the unary operator “not” who allows to make the sentence not P starting from a P. sentence.
This use adopted by the logicians gives “not Socrate is human” to mean that Socrate is not human. This way of denying is practical when one wants to multiply the negations inside a complex sentence. It does not raise other difficulties only to shock the practices.
The other Boolean operators are binary. The conjunction and the Disjonction make it possible to make (P and Q) like (P or Q) starting from P and of Q.
With the conjunction one can gather several assertions relative to the same situation. With disjunction, one can gather several assertions relative to several possible situations. The Boolean operators thus allow to build all that is dicible starting from a finished number of simple sentences.
The concept of consequence can be expressed in the Boolean algebra with the operator of Implication. P implies Q is more often formulated by if P then Q.
The Boolean algebra is thus named because Boole showed that one can write certain logical laws in the form of equations:
- not P = P
- not ( P or Q ) = not P and not Q
- not (if P then Q ) = P and not Q
- and much of others.
Quantifiers in the Boolean algebra of the predicates
For the logical reasoning, the preceding algebra of predicate is insufficient to treat generalizations. Thus, above, the variables P or Q are not specified. If the predicate is always true whatever their value, one can do without this precision, by treating them like constant symbolic systems, but in logic of the predicates, these variables above are known as variable free .On the other hand, if a predicate is true for certain of their value and not for other, it can be necessary to specify it in a more precise way, using a quantifier . In the language running, one often does to specify the quantifier for all in front of the name of a free variable in order to bind it to the quantifier and to form a complete predicate, this is why this quantifier by defect is frequently omitted and the predicate not quantifé considered as equivalent, thus one does not specify:
- for all 1, for all 2,1 + 2 = 2 + 1
But it can be useful to refine the direction (and the value) of 1 and 2:
- (1) for all 1, for all 2, (1 belongs to ) and (2 belongs to NR ) implies 1 + 2 = 2 + 1
- (2) for all X , for all there , ( X belong to ) and ( belongs there to NR ) implies X + there = there + X
- (3) X + there = there + X
It will be noted that it is possible often to shorten the predicates by increasing the syntax of the quantifiers. Thus the predicate (2) is more often met in the form:
- (4) for all X belongs to NR , for all belongs there to NR , X + there = there + X
Finally it should be noted that the logic of the predicates allows many simplications of the predicates comprising of the associated variables, because it is possible to gather the quantifiers out of the under-predicates which they qualify if the variables dependant in these under-predicates are indépandentes between them, i.e. not homonyms (if it is not the case, it should be re-elected in the predicates that they qualify before moving them). Moreover the most usual quantifiers ( for all , there exists ) are commutative and associative, contrary to the majority of the other relations.
The introduction of quantifiers however does not modify the Boolean algebra: it introduces only new objects among the sources units studied: dependant variables belonging to another species that referred objects (although their value is taken in the same units, which makes it possible to note generalizations), and of new binary relations binding these units to a predicate to create another predicate (as already the other relations Boolean binding of the predicates do it between them to create another predicate.
Other algebras of predicates
There also exists of other algebras relating to predicates, using states which are combinations of several normally incompatible predicates according to the Boolean algebra. The Boolean algebra is most common in logical reasoning.However probabilistic models of reasoning result in handling “simultaneously” predicates being able to be incompatible but affected each one of a nonnull probability. These models of reasoning in the usual sense do not induce a contradiction (since the condition of simultaneity is not necessary to the existence of nonnull probabilities for two completely incompatible predicates between them, even complementary).
In this case, the algebra of predicates extends the operators of Boole by giving functions making it possible to calculate the probability of each couple of predicate according to the combination. The simplest algebra used in probabilistic reasoning consists in simultaneously using the basic Boolean predicates in their affecting a probability equalizes to 1 or 0 such as:
- p (not P ) = 1 - p ( P )
In more complex algebras of predicates, where the base of the reasoning is the whole of the axioms defining the studied whole of the predicates, the combinations of predicates are not necessarily linear, and the operators ducalcul Boolean have forms more complex than a simple linear combination of probabilities in a unit segment of the real numbers.
Among those, algebras make it possible to handle combinations of these predicates (so called " states superposés"), axioms being however all affected of a unit probability, according to there too of the functions of combination of nonlinear probabilities. It is the case in physics for calculations relating to the states of quantum particles affected of probabilities between several incompatible states, several distinct particles being able to be dependant by intricate states between them.
In all the cases, the fundamental operators of Boole are modified to become functions of a couple of superimposed states, each of the two members of the couple origin being affected of its probability, and the function determining the probability of their superposition.
Each superimposed state can be represented for example with the notation bra-ket of Dirac (which makes it possible to indicate the states without having to specify a linear base for the whole of the possible states, since such a base is possibly infinite or unknown). By using this notation, certain superimposed states of the whole of the possible states can be, according to certain conditions defined by the experimental constraints, of the linear combinations of basic superimposed states, and allow calculation according to simplified formulas.
If the superimposed states are independent in the probabilistic plan, then the probability of their combination is the product of the probabilities assigned to each state. If P is a bearing predicate finished on a whole of intricate states, one can also see it as the combination of several nonintricate states of the whole of the states of the unit (the exact formulation of the nonintricate predicates can be unknown and thus not be able to be expressed simply by a finished predicate). One then notes that with the notation braket of Dirac:
- p ( |P> R |Q>) = F (p ( |P>), p ( |Q>))
- P and Q is two finished predicates of the same whole of probabilistic predicates, carrying each one on an unknown unit of intricate states,
- |P> and |Q> represents the vectors (of an unknown, but equal nature between P and Q ) of the probabilities assigned to each basic predicate composing the finished predicates P and Q ,
- p ( |P>) and p ( |Q>) represents the affected probability with each predicate finished in the whole of the probabilistic predicates studied,
- |P> R |Q> represents the vector of the probabilities assigned to each basic predicate composing the superposition (according to R ) of the finished predicates P and Q ,
- p ( |P> |Q>) represents the probability resulting from the superposition of the finished predicates P and Q , and
- F is the function of probability theory, representing the mode of superposition of the finished predicates, which depends on R .
(Note: a “finished predicate” indicates here a predicate which is expressed in a finished way, i.e. using a finished number of quantifiers, a finished number of free or quantified variables, a finished number of relations between variables and constants, and a number finished logical operators of composition of the proposals; in theory a complete predicate quantifies all its variables and does not leave any free variable in the quantified proposal defining the complete predicate; this condition is not necessary for the under-predicates sometimes present in the proposal for a made up predicate; the free variables can of the clean vectors independent in the calculation gap of the probabilities, and for this one must to initially assign them by defect to each one a quantifier for tout' to supplement the predicate, but that does not modify the finished character of the predicate).
For example if |P> and |Q> represent the vectors of probability of presence (the opposite predicate is the absence) of two particles isolated in the same physical system (defined by an unspecified number of basic predicates depending on the state of the particle or the whole system), the probability of presence of the two particles in the same system is
- p ( |P> and |Q>) according to the fomule above (and the definite function F for the superposition R = and are not necessarily linear)
If there exists in the system studied a whole of superpositions R of predicates probabilistic for which F is an algebra of body, one can then study this system with an algebra by determining three values of R possible representing the union (or), the intersection (and), the negation (not), which respects the Boolean algebra of the predicates at least for probability 1 (the predicate is always true) or 0 (the predicate is always false), and the expression of F is then simple:
- p ( | not P>) = 1 - p ( |P>)
- p ( |P> and |Q>) = p ( |P>) · p ( |Q>)
- p ( |P> or |Q>) = 1/2 · p ( |P>) + p ( |Q>)
- etc
When the predicates probabibilists result from the superposition of predicates of a simple base in which they are linearly independent, one does without the notation of Dirac most of the time (because a vectorial base of calculation is known), by reformulating the probabilistic predicates according to the basic predicates in linear combination, and affecting this combination of the same probability as the initial probabilistic predicate, by a simple product of the vector broken up by the scalar of this probability. In this case, one does not treat any more that combiniasons linear between linearly independent predicates and the formulas of Boolean superposition above are résuisent with:
- p ( not P ) = 1 - p ( P )
- p ( P and Q ) = p ( P ) · p ( Q )
- p ( P or Q ) = 1/2 · p ('' P '') + p ('' Q '')
- p ( not ( if P then Q )) = 1 - p ( P and not Q ) = 1 - p ( P ) · - p ('' Q '')
- etc
One will note here that the functions F of superposition above, bearing on the probabilities are algebraic, more precisely polynomial, and not necessarily linear in the base of linearly independent predicates ( P , Q ) for the general case. But one can express it as a linear function if one expresses it in a larger vectorial base expressed in the space of the probabilistic predicates (| P >, | Q >, | P > and | Q >). The degree of these polynomials of composed probability, if it can be given and is finished, depends only on the number of possible superpositions considered, not of the number of predicates linearly independent defining the state of the studied system.
In the modelings used in physical science, this degree of the polynomials of probability is potentially infinite (the finished case can exist only locally for certain experimental conditions in which one fixed constraints, like the number of particles supposed to interact to a significant degree on the studied system and in which one can neglect all the other interactions of the universe), and one cannot do without the notation of Dirac completely, to formulate correctly, in a probabilistic vector space of unknown size, the combinations (partners with the relations of interactions) of superimposed states, defined each one by a potentially infinite number of probabilistic predicates, in the vectorial base which represent the entire physical universe.
Assertions of existence and laws general
There is in many ways to formulate laws and statements of existence. The best and by far from the mathematical point of view consists in making use of names of variable, these X and these there which torture sometimes the spirits of the pupils. For the logicians, the mathematicians and all the scientists, the names of variable are however splendid tools. All the difficulties of syllogistic of Aristote become clear as the water of rock as soon as one included/understood what one can do with names of variable.
The practice of the names of variable is very old, at least as former as the geometry. Aristote was even used for about it to state the laws of the syllogistic one. But it did not understand that the syllogistic rules themselves can be deduced starting from a reasoned use of the names of variable. This one was initially developed by algebrists, because the variables are very useful in the equations. Frege and the other inventors of the great logic of the classes and the relations, those which carried out the great idea of Leibniz, the mathesis universalis, the universal rational computation, that one calls today the Calcul of the first order predicates, or calculation of the concepts, included/understood all the profit which one could draw from the use of the variables. The use of the names of variable, it is the power of the general information, it is one of the clearest demonstrations of the size of the reason.
From the point of view of grammar, the variables do not raise with the first access any particular difficulty. The names of variable are words like the others. It is enough to specify what they are variables, object, class, relation, operator,… One can use in a sentence the names of variable like all the other names. X is human.
The laws and the assertions of existence are made with two types of unary operators built on the names of variable. (for any X) and (there exists X such as) are two unary operators associated in the name of variable X. The first is the universal operator in X, or operator of generalization. The second is the existential operator in X, or operator of existentiation.
The assertion that all the human ones are mortals is translated in the calculation of the predicates by, for any X, if X is human then X is mortal. It is the result of the application of the operator (for any X) on the complex sentence (if X is human then X is mortal). The assertion that there is at least an human being is translated in the same way by, it exists X such as X human east.
The occurrences of a name of variable in a sentence are all the places where this name appears. An occurrence can be free or dependant. When an existential or universal operator in X is applied to a complex sentence, all the occurrences of X become dependant, or quantified, by this operator. All the occurrences which are not thus bound are free.
In the first order theories, only the variables of object can be dependant. The field of the objects, or field of existence, or universe, or ontology, of the theory, is also its field of quantification.
The distinction between free variable and dependant variable is one of most important for the grammar of the calculation of the predicates. It is essential to understand that the concepts are predicates.
The Boolean operators and the universal and existential operators are enough to build starting from the atomic formulas all the statements which one needs in sciences.
The concept of prime number can be defined by the following predicate, inside the theory of the numbers, supposed here whole positive.
X is a number and for all numbers there and Z, so there and Z to 1 then X are not equal is not equal to there time Z.
In the formula above, the variables there and Z are dependant. X is the only free variable. This formula is true if and only if X is a prime number. One can thus regard it as a name of the concept of prime number.
The concepts are or many concepts first, being a number, being equal to, being a unit, being in,… or many derived concepts, not to be equal to, to be a prime number,… In all the cases, they are predicates. The calculation of the predicates is the calculation of the concepts.
The most fundamental concepts are in general either qualitative, i.e. unary, or relational binary. The logic of Aristote is almost exclusively devoted to the qualitative concepts.
To make science, concepts should be made. For first order logic, the basic materials are the fundamental concepts first, or predicates, the basic objects, the operators of object, the variables of object, and a small number of logical operators, the operators Boolean and the operators of generalization and existentiation. The rules of construction are simply those which have just been presented. All the grammatically correct formulas of first order logic name concepts as soon as they contain free variables. Logic does not impose others limiting on the thought only those of the grammatical correction. In this direction, one can see the calculation of the predicates like a way of intellectual release, because it makes it possible to think all that one wants, and to do it without falling into the nonsense.
The concepts are predicates
The predicates of a theory are initially its fundamental predicates, those which one can identify with his concepts first. From them, basic objects, variables and operators, one can build many derived, or defined predicates, inside the theory. It is enough to write formulas which contain free variables. A formula which contains only one free variable names a unary predicate. It is a qualitative concept. A formula which contains two and only two free variables names a binary predicate, or binary relation, or binary relational concept. Whatever their number of free variables, provided that it is at least equal to one, the formulas name concepts.
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