Relations between complex judgments, Negation of judgments...

Relationships between complex judgments

We already know that the relationship of truth and falsity between complex judgments can be established only when they have at least one common simple proposition; that this is done with the help of truth tables and that for this it is necessary to formalize complex statements (to present them in the form of formulas).

For example, take two statements.

If no opposing opinions are expressed, then there is nothing to choose the best (Herodotus).

"It is not true that if the opposite opinions are expressed, that is, from which to choose the best."

1. Let us single out in them elementary statements (and their negations):

t - Opposite views are expressed & quot ;;

n - There is something to choose the best; & quot ;;

- •/л - Opposite opinions are not expressed & quot ;;

- * n - There is nothing to choose the best.

If there are no opposing opinions (^ т), then there is nothing to choose the best (-, and) & quot ;;

It is not true that if the opposing opinions are expressed (t), that is, from which to choose the best () .

2. We define logical unions and compose formulas:

We will open the brackets in the second formula according to the laws of negation of judgments (about which below):

In the transformed form (after the parentheses are opened), the formula reads as follows: "Opposite opinions are expressed (t), but the best choice is not what (- • quot;)

3. Let's compose the truth table:

- we will write the initial value of the logical variables type;

- we express the meaning of the subformulas ^ t and -, and;

- enter the values ​​ t, n, -th, - * n into the formulas of judgments:

4. Define the truth values ​​of formulas, taking into account the functions of logical connectives:

- the first formula is an implication: it is true in all cases except one, when a false consequence is deduced from the true basis (the third line of the formula);

- the second formula is a conjunction: it is true only if all the variables are true (second line of the formula).

The final results are as follows:

5. Comparing the truth values ​​of the formulas, we find:

- truth compatibility (in the second line);

- False compatibility (in the third line);

- one-sided logical sequencing (in the second line): if the second formula is true, the first formula is also true; there is no inverse logical sequence, since with the truth of the first formula (in the first, second and fourth rows) the second takes both true and false values.

Conclusion: these characteristics indicate that there are subordination relations between judgments, while the second proposition subordinates the first.

Negation of judgments

Negation of judgments - is a logical operation, as a result of which a new judgment is formed that contradicts the original judgment.

Recall that for contradiction relations incompatibility in truth and incompatibility by falsity are characteristic; that this area of ​​the law of the excluded middle: Of the two contradictory propositions one is true, the other false, and there is no third & quot ;. Accordingly, the denial of true judgment is tantamount to the assertion of a lie, and the denial of a false one is to the affirmation of the truth.

Negation of simple judgments

For simple categorical judgments, the negation operation is a jump along the diagonals of the logical square (Figure 5.3):

Denying Simple Judgments

Fig. 5.3. Denial of simple judgments

The laws of the negation of simple categorical judgments: -Л = 0; - ^ 0 = A; - & pound; = 1; -,/= & pound;.


• The denial of universal affirmative judgment: -A = About Take proposition Nietzsche: Every man

has a price - this is incorrect & quot ;. We express negation in an explicit logical form and perform the operation:

(- A) It is not true that every person (.9) has a price (P). (A) Some people (5) do not have a price (P).

• Negation of a private negative judgment: -.O = A:

(-10) It is not true that some do not have the right to stupidity, (l) "Everyone has the right to be stupid" (G. Heine).

• Negative negative judgment: - & pound; = D.

(- & pound;) It is not true that no one (5) hides his mind (P). (D) Some people hide their mind (D. Swift).

• Negation of a private confirmation: -, 1 = E:

(- 1 /) It is not true that the new (5) is perfect (P). (& pound;) "Nothing new (5) is perfect (P) (Cicero).

• Denial of a single judgment gives an opposite in quality judgment:

It would be incorrect to say that Russia (5) is a country of young culture (P) (N. A. Berdyaev).

Russia (5) is not a country of young culture (P).

Denial of complex judgments is carried out according to certain laws. Consider this in more detail.

• Negative conjunctive judgment is equivalent to a disjunction of negations:

( It is not true that the judgments t and n together are true, at least one of them is false ").

We use the statement of the outstanding domestic philosopher SN Bulgakov: "Not all orientations are practically convenient and accessible to a thinking being". This complex judgment is expressed in the form: 5 - (P (n P), which is equivalent to (5 - Px) n (5 - P), or (m А n) -

Not all orientations are practical (t) and are available to the thinking creature (n).

Some orientations are almost inconvenient (- gn) or are inaccessible to a thinking creature (- and).

Denial of this judgment can be carried out by analogy with the denial of a generally valid statement: "It is not true that all 5 are P - is equivalent to: Some 5 are not P & quot ;. But P = (P (P), respectively: ~ pP = -pP, V -> P. As a result, we get the formula: "Some 5 are -rP, V -nR".

• Negative conjunctive judgment is also equivalent to implicative judgment:

("It is not true that judgments type are together true, even if one of them is true, then the other is false").

Not all orientations are practical (t) and are available to the thinking creature (n).

If it is true that some orientations are practical (t), , then it is false that they are all available to a thinking being (~ y?).

• Denial of a weak disjunction is equivalent to a conjunction of negations:

("It's not true that any of the judgments t or n is true, they are all false" ;).

"It is not true that people know what is best for him (/ or), or wants it (n) (Mr. Hegel).

People do not know what is best for him (- t), and does not want to know (-> gt).

• Denial of strong disjunction will give a more complex picture:

("It's not true that either of, or n is true: they are together true or together are false"),

"A woman," Publius Sire asserted, "loves or hates: she has no third". We will challenge the Roman poet:

It is not true that a woman only loves (from) or hates (/?) .

She can and love, and hate (from l n) or not at all experience these feelings (be indifferent)

(-, 171 L-NI).

• Negative implication:

("It is not true that the truth is from the truth n: despite what is true, n false. ).

As an example, take the saying of Aristotle, the founding father of the science of logic: "It is not true that there is nothing wrong with using one word instead of another if they mean the same thing." Let's express it in an explicit logical form:

It is not true that if words mean the same thing (from),

then do not misuse!) one word instead of another (/?) .

It happens that words mean the same thing (from), but their replacement can be bad (- hp).

• Negative equivalent judgment:

(It is not true that only with truth from true and l: from and and can be true independently of each other ).

It is not true that only wealth can make a person happy.

You can be rich (t), but not happy (-1/2) or not rich (->) and at the same time a happy (n).


A logical analysis of the relationship between judgments can be viewed as a way of implementing a more general principle of intellectual activity - "saving thinking", which in logic and methodology of cognition is regarded as one of the criteria of truth. The essence of this principle is to achieve the maximum of knowledge with the help of a minimum of cognitive means. In the scientific revolution, the concept of "saving thinking" introduced by positivist philosophers, the methodologists of science Ernst Mach (1838-1916) and Richard Avenarius (1843-1896). But as a methodological device known since the Middle Ages, thanks to the English logic William Ockham (1285-1349). The rule formulated by him (the so-called Occam's Razor) reads: "Entities should not be multiplied beyond necessity," or "It is useless to do much by what can be done by means of a smaller".

It is quite obvious that the logical analysis of judgments and relations between them makes it possible in many situations to establish the truth of one or another utterance exclusively by reasoning, without procedures of empirical verification (experimental verification). Thus, significant savings in cognitive effort are achieved. But this is possible only with a thorough knowledge of the science of logic.


Judgments are comparable and incomparable. Comparable judgments have at least a partial general content, for incomparable - no. Logical relationships are possible only between comparable judgments.

Comparability of simple judgments is ensured by the presence of the same terms (including their negations, as well as the change of logical functions).

Comparability of complex judgments is ensured by the presence in their composition of at least one common simple utterance.

The main goal of the logical analysis of the relationship between judgments is to establish the truth values ​​of some statements based on reliable information about the truth or falsity of others.

The main types of relationship between judgments are: truth compatibility, consistency in accordance with the logic and the logical sequence relationship.

Different combinations of these characteristics form the following types of logical relationships: equivalence, opposition, contradiction, subordination and opposite.

In simple categorical judgments, these kinds of relationships are easily established by a logical square. For complex judgments it is necessary to compile tables of truth values.

On the basis of contradiction relations, a logical operation of denying judgments is performed, which ensures an unambiguous evaluation of the truth or falsity of the statement.

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