The expression P1 (x, write my lab report online review R (a)) cannot be considered a formula, because R (a) is not a term.
they are also called "identically true", "logically true", "tautologies", "universally universal".
The third and fourth statements are also not just false, but "always false", ie those whose error does not depend on the logical meaning of simple statements, their components. "Always false" statements (formulas) are also called logical contradictions.
The vast majority of complex statements are those whose true meaning cannot be determined without taking into account the truth or falsity of their components. Such statements are called executable (feasible, indefinite).
In logic, special methods have been developed to determine the type of a complex statement (formula), ie to establish whether it is "always true" (the law of logic), "always false" (logical contradiction) or executable.
Consider one of these methods – the method of truth tables.
Tables of truth of logical connections, which we have already read, can be used to determine the truth value of complex statements. These tables are built according to the scheme.
In the first row of the table enter first simple statements (propositional variables), then those components of the statement that contain one logical connection, followed by those that contain two connections, etc. The line ends with a statement that is analyzed. Each component of the statement in the first row of the table is assigned a cell, each of which begins the corresponding column.
For example, the statement "(AVB) AB" fits into the table:
A B B AvB (AvB) vB
Since the composition of the studied statement includes only two propositional variables (A, B), the rows in the table will be four (if there were three propositional variables, the number of rows would double).
Filling in the table, we will enter in the first and second columns all admissible sets of logical values of proportional variables "A" and "B". The value of "B" is set according to the values of "B" according to the truth table of the relationship "negation".
The value of "AvB" is set according to the values of "A" and "B" according to the truth table is not strict! disjunctions. The logical value of the studied statement "(AVB) ABJ> is set according to the values of" AvB "and" B "according to the conjunction truth table.
A B B AvB (AVB) AB
and and X and X
and X and and and
X and X and X
X X and X X
Because in the last column of the table there were different logical values (ie both "true" and ba "), then this statement is executable.
A B A AlA (AlA) h> B
and and X X and
and X X X iX and and X and
X X and X and
A B A AvB AlA (AvB) l (AlA)
and and X and X X
and X X and X X
X and and and X X
X X and X X X
The method of truth tables is effective in determining the type of complex statements that contain two or three propositional variables. If there are more propositional variables in the statement, then resort to the method of analytical tables and other methods.
Even when a statement contains three propositional variables, truth tables are already cumbersome:
A B with A ALVLA (ALVLA) h> C
and and and X X
and And X X X
and X and X X
and X X X X
X and and and X
X and X and X
X X and and X
X X X and X and
Having clarified the essence and meaning of the logic of statements (propositional logic), it is easy to guess about its non-universal nature. This is manifested in the fact that there are such considerations, the correctness of which can not be justified by the number of statements, ie, abstracting from the internal structure of simple statements.
Thus, the correctness of the reasoning "All metals are electrical conductors, therefore, some electrical conductors are metals" depends not only on the logical connections between the statements, but also on their internal structure. This and other facts indicate the need for such a logical theory, which would take into account the subject-predicate structure of simple statements and would introduce new logical constants: "V" – the quantifier of generality and "C" – the quantifier of existence … The expression "Vx" is read: "for any x …" and the expression "Zx" – "there is such an x …".
Such a theory is created. It is called predicate logic, or quantification theory. In addition, this theory is an extension of the logic of expressions, so all the laws of the latter are simultaneously the laws of predicate logic (but not vice versa!). The subject of this logic is also only descriptive statements, which have two logical meanings: "truth" and "false".
The language of predicate logic is an artificial language adapted to analyze the logical structure of simple expressions. It includes a list of appropriate sign means (alphabet) and the definition of correctly constructed expressions. Such expressions are terms and formulas.
The sign means of the language of predicate logic are divided into technical and non-technical, and the latter, in turn, into logical and illogical. Illogical terms include primarily names and predictors.
Name is a term that denotes any object.
Predictor – a term denoting a property of an object or relationship.
Predicators that express the properties of objects are called single, and predictors that express the relationship between objects – non-single (double, triple, etc.). The subject value of predictors is considered to be sets, the elements of which are either individual objects or their sequences (for example, pairs of objects).
The logical terms that are part of simple statements are the quantifier of generality and the quantifier of existence.
Alphabet of predicate logic
I. Non-technical signs. Non-technical include illogical and logical signs: subject (individual) constants, subject (individual) variables, predicate symbols, signs of logical conjunctions and signs of quantifiers.
1. Subject (individual) constants: a, b, c, a;, br sg .. These signs are used to denote proper names of natural language ("Chernihiv" "Hegel" "Teteriv").
2. Subject (individual) variables: x, y, z, x ‘, yr zr If subject constants are associated with specific proper names, then subject variables replace any name of the corresponding subject area ("city" "man" "river" ).
Predicator constants: Pn, Q "R" Sn, Pnr Qnr Rnv Snr .. These signs indicate the predictors of natural language. The upper index indicates their capacity, and the lower – the serial number. Thus, a single predicate can be written as P \ double – as P2, etc. (an example of a single predicate can be the expression "be electrically conductive" double – "be cheaper than" and triple – "located between").
4. Signs of logical conjunctions (these signs are known to us from the logic of statements): "-" "l" "v" "v" "->" "
5. Signs of quantifiers: V – sign of the quantifier of generality and C – sign of the quantifier of existence.
II. Technical signs.
(- left bracket;
) – right bracket;
, – coma.
These technical signs in the logic of predicates are a kind of punctuation.
The alphabet of the more universal artificial languages of predicate logic is supplemented by some other signs.
Definition of correctly constructed expressions
In the language of predicate logic, there are two types of correctly constructed expressions (noun) – these are terms and formulas.
Definition of term:
Any subject constant is a term. Any subject variable is a term.
In other words, the symbols a, b, c … (as subject constants) and the symbols x, y, z … (as subject variables) are terms, which cannot be said about the symbols P, Q, R , and so on.
Definition of the formula:
If t, r2, …, t are terms and P ". Is a n-place predictor then Pn. (Rr t2, …. tj is a formula. 1 If A is a formula, then A (non- A) If A and B are formulas, then formulas are such statements as "ABB" "AVB" "AVB" "A-> B" "AB". If A is a formula, ah – a subject variable, then "ZxA" and "VxA" are also formulas 2.
No other expression is a formula.
The formulas given in the first paragraph are called simple, or atomic, and all the others are called complex or molecular. Thus, the expression P1 is a sign of a single predicate, ah is a subject variable, which is a term. The expression P1 (x, R (a)) cannot be considered a formula, because R (a) is not a term.
To translate into the language of the logic of the predicates of the expression of natural language, it is necessary:
replace all quantifier words with quantifiers of generality or existence, respectively (V, 3); replace all words that are proper names with subject (individual) constants (a, b, c …); replace all words that are common names with subject (individual) variables (x, y, 2 …); replace all words that denote the properties of objects with single predicate, and words that denote relations with double or multi-place predictors.
You can then write the formula as a whole. Consider some examples of translation of natural language expressions into the language of predicate logic.
1. The superscript "n" (n> 1) indicates what the predictor is: single, double, triple. And the lower "and" indicates the arbitrariness of the predictor.
2. Symbols tr t2 tn; 27 "; A, B do not belong to the signs of the language of the logic of predicates," but to the signs of metalanguage, which speaks of the expressions of the logic of predicates. 1. "All squares are rhombuses." Denoting the quantifier word "all" by the sign "V", "squares" – "x" and "rhombuses" – "P" we obtain the formula "UxR (x)".
This statement can be represented by means of predicate language and otherwise "Ul: (P (x) -Q (x))" where "P" and "
This expression reads: "For any x it is true that when x is a square, it is a rhombus." 2. "Some rhombuses are squares" – 3X (Q (X) AP (X)), which means "There are x for which it is true that x is a rhombus and a square."
3. "Some rhombuses are not squares" – 3X (Q (X) AP (X)), which means: "There are x for which it is true that x is a rhombus and x is not a square."
4. "No square is a triangle" – Vx (P (x) -> -> R (x)), which means: "For any x it is true that when x is a square, it is not a triangle." Related and free variables
Assigning to a predicate a quantifier of generality or a quantifier of existence is called a quantizer binding operation.
Quantification can be performed simultaneously with respect to several propositional functions, as well as with the simultaneous use of several quantifiers. Therefore, it is necessary to take into account the scope of each quantifier, the part of the quantified function to which the action of a particular quantifier.
Thus, in the formula \ / x (P (x) ~> 3y (Q (x) vR (y))) the scope of the quantifier of generality is the whole part of the formula located to the right of this quantifier (ie P (x) ~> 3y Q (x) vR (y)), and the sphere of action of the quantifier of existence is only Q (x) vR (y). A variable that is located immediately after the quantifier and enters its sphere of action is called a bound variable, and a variable that does not fall within the scope of the quantifier – free.
Consider the difference between free and bound variables in the following example:
Ux (P (x) -> R (u)) A3y (Q (x, y) vR (x, z)).