category: logic


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section of the curriculum: Logic.

Subject: Figures categorical syllogism.

Instructor: Nicholas Bartunov Petrovich__________________



figures categorical syllogism

1. Preface

2. Categorically statements

3. Figures categorical syllogism

4. Basic rules of figures.

5. Modes shapes

6. Literature


In more than two thousand years of history of logic nowrepresents one of the most intense periods of its development is very fastgrow and the amount of new information, and the number of new results. InFurthermore, if more recently the logic area of interest was only relativelynarrow circle of specialists, it has now become an important discipline andnecessary for many, and in the field of modern education - for everyone.

doctrine of the syllogism is historically the first completefragment of the logical theory of reasoning. It systematically described
Aristotle in the "Analytics" and the name of syllogistic exists tothis time, having its own value.

categorically statements

Propositional logic reduces the complicated to the simple utterance

It examines the complex expression as a function of simple butsimple at the same time is not dismembered.

remarks, having a structure expressed by the formula ?S is P?called the affirmative, but with the structure of ?S is not P? --negative. This division of quality.

In addition, categorical statements divided by the number ofunit (This S is (or is not) P), general (All S is (or is not) P)and private (Some S is (or is not) P). The words "all" and "some"is called quantifier words.

In the study of reasoning (syllogisms) do not distinguish betweensingular and general statements, because in some common types of signapproved (or denied) on each elementconsidered set of objects. The only difference is that the set ofreferred to in a single utterance consists of one element, butgeneral - from more than one.

Thus, the classification of categorical statements on the qualityand the number contains four types:

n obscheutverditelnye (A) n universal negative (E) n chastnoutverditelnye (I) n chastnootritsatelnye (O)

letters A, E, O, I for symbolic symbols are taken from the Latinwords affirmo - say - for two affirmative sentences and words ofnego - to deny - for negative.

figures categorical syllogism

Let's consider (for example) the structure of the syllogism.

Everyone (M) - Death (R) < / p>

Socrates (S) - man (M)


Socrates (S) - mortal (P)

syllogism comprises three categorical statements (two parcels andone conclusion, which is to be written under the standard recording feature). Subjectconclusion is indicated (usually) by the letter S, and the predicate - P, but in the syllogism
S is called the minor term, and P - high, both are called extremeterms. The term, repeated twice in the premises, called the mean
(Latin - terminus medius) and is denoted by the letter M.

Dispatch also have their own names: the one that contains the term
P, called the major premise, and the term containing S - minor premise.

Thus, the categorical syllogism - it is a deductive inference,in the conclusion that the relationship between extreme terms (S and P)established on the basis of their (recorded in the premises) relationship tomiddle term (M).

In general, the structure of the syllogism can be represented as:

R (X, Y) ^ Q (Y, Z) -> L (XZ), < / p>

where R, Q, L can have values A, E, I, O;

X, Y means a MP or PM,

Y, Z - MS

X, Z - SP

conjunction of the premises in the syllogism can be regarded as antetsendent,and the conclusion - as the consequent.

Taking these considerations, the structure of the example shouldwritten as:

A (MP) ^ I (SM) -> I (SP).

If we consider only the relative positions of the three terms,obtain the following general structure of our output, referred to as the first figuresyllogism:





1-I figure

(1-I figure)

It is clear that in addition to this figure there are three, because the term M canstand in each parcel as to place the subject and predicate in place:



------ ------ ------


2-I figure 3 figure 4 figure

Thus, the figures of the syllogism, it is such varietywhich differ from each other provision of the middle term.

If we take into account the quantitative and qualitative characteristicsoutside the premises and the conclusion of a syllogism, we get the variety,called modes. Modus written three letters (from A, E, I, O) insuch a sequence - the major premise, minor premise, conclusion.

The above example illustrates the mode of AII.

all possible modes of syllogism (four shapes 256). Takingmost general scheme of the syllogism - R (X, Y) ^ Q (Y, Z) -> L (X, Z), then there are 4ways to choose R, 4 ways to Q and 4 ways to choose L; except that 2 wayschoose the order of X, Y, and 2 ways to sequence Y, Z. Soway, there are 4 * 4 * 4 * 2 * 2 = 256 different modes (of 64 in eachfigure). But not all of them are correct. The question of the correctnessany syllogism can be resolved by the construction of Euler diagrams for eachparcels and then combine them.

Modus some syllogism invalid if and only ifany graph corresponding to its parcels, which does not coincide with anydiagram, corresponding to its conclusion.

For example, consider the mode:

E (MP) ^ A (SM) -> E (SP), ie

No, V is the essence of P

All S are M

------------------------- ------- none S are not P

His premise matches any of the two diagrams shown in Fig

Figure 1

Figure 2

Figure 3

is obvious that each of these diagrams can correspond to the conclusion
"No S are not P?. Therefore, this syllogism is correct, and, hence, forassumptions is true, we need to get a true conclusion.

Figure relations between terms in the major premise A (MP) canbe such as is shown in Figure 2, a diagram of the minor premise
E (SM) is shown in Figure 3.

This fully shows that the set S, completely excluded fromset M, can be completely excluded from the set P, which corresponds toAnd the conclusion (SP). These provisions were recorded as S S1 and S2. Apparentlyunambiguous result can not be obtained. This is evidence thatconclusion does not follow logically from the premises (statements of E (SP) and A (SP) does notcan be simultaneously true).

analyzing this example, we assume that the term that takesplace of the subject, is distributed in the general statements (A, E), and the termoccupying the place of the predicate, distributed in the negative utterances (E,
O). Strict adherence to this definition is the basis of the so-callednarrow theory of the syllogism.

But the term that takes the place of the predicate in affirmative statements
(A, I) can be distributed. This fact underlies the socalled the extended theory of the syllogism.

Basic rules of figures

1. Medium term should be distributed in at least one of the parcels.

If the term M will be distributed in at least one of the parcels that uniquely bind the extreme terms in prison is not possible.

2. The term can be in custody only when it is distributed in the premise (usually extreme terms).

3. The number of negative assumptions must be equal to the number of negative opinions.

This rule means that:

1) If one of the parcels is negative, then the conclusion must be negative.

2) Of the two negative premises correct can not be done.

3) Of the two affirmative premisses can not get a negative conclusion

These three rules are necessary and sufficient to excludeall the wrong syllogisms.

sometimes formulated a rule: "In the syllogism should be three and onlythree terms.. " An indication of this requirement is aimed at avoidingerror, which is called the quadrupling of terms (it is based onconsciously or unconsciously using the phenomenon of homonyms).

in the number of additional rules include:

1. At least one of the parcels must be a common statement (from two private statements can not be the correct conclusion).

2. If one of the private parcels, then the conclusion must be private.

Special rules figures

Based on the general rules (in the narrow theory of the syllogism), and consideringposition of the middle term, we can derive the following special rules for the figures.

first figure.

1) major premise must be universal (A, E);

2) minor premise - affirmative (A, I);

The second figure.

1) major premise must be universal (A, E);

2) One of the parcels is negative (E, G);

< br>The third figure.

1) The minor premise must be affirmative (A, I);

2) Conclusion - private (I, O);

Fourth figure.

1) If the major premise - affirmative (A, I), then the smaller should be the total (A, E)

2) If one of the parcels is negative (E, O), the major premise must be total (A, E);

Many believe the logic of a fourth figure on the artificialBased on that reasoning on this figure is not typical in the practice ofevidence. But, first, the arguments of the fourth figure stilloften implemented in practice, and secondly, to complete the theorysyllogism it should be considered.

Based on the rules of the figures and, of course, given the general rulessyllogism, we can derive all the correct modes of each figure. They willexactly six in each figure, the total number of regular modes in such a way,

all possible combinations of parcels will be 16, for each of the fourtypes of sentences (A, E, O, I) can connect with themselves or each other or witheach of the other three:

| AA | EA | IA | OA |
| AE | EE | IE | OE |
| AI | EI | II | OI |
| AO | EO | IO | OO |

Rules of the first figure to exclude, first, all combinationsparcels of the third and fourth columns, because they contradict the firstrule. Secondly, the combination of AE and AO from the first column countersecond rule. The combination of ITS and EG in the second column should also bedeleted because they are contrary to the general rule of inadmissibility of the twonegative assumptions. Remaining combinations of AA, EA,

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