Home Prosthetics and implantation What does this sign mean in logic. The language of logic

Prosthetics and implantation

What does this sign mean in logic. The language of logic

Symbolism is logical

a system of signs (symbols) used in logic to designate terms, predicates, propositions, logical functions, relations between propositions. Different logical systems can use different notation systems, so below we give only the most common symbols used in the literature on logic:

The initial letters of the Latin alphabet are usually used to denote individual constant expressions, terms;

Capital initial letters of the Latin alphabet are usually used to denote specific statements;

Letters at the end of the Latin alphabet are usually used to denote individual variables;

Uppercase letters at the end of the Latin alphabet are usually used to denote propositional variables or propositional variables; for the same purpose, small letters of the middle of the Latin alphabet are often used: p, q, r, ...;

logical symbolism; u

Signs that serve to indicate negation; read: "not", "it is not true that";

Signs for designating a conjunction - a logical connective and a statement containing such a connective as the main sign; read: "and";

A sign for designating a non-exclusive disjunction - a logical connective and a statement containing such a connective as the main sign; read: "or";

A sign to denote a strict, or exclusive, disjunction; read: "either, or";

Signs for designating an implication - a logical connective and a statement containing such a connective as the main sign; read: "if, then";

Signs to indicate the equivalence of statements; read: "if and only if";

A sign denoting the deducibility of one statement from another, from a set of statements; read: "derivable" (if the statement A is derivable from an empty set of premises, which is written as "A", then the sign " " reads: "provable");

Truth (from English true - truth); - lie (from English false - lie);

General quantifier; read "for everyone", "everyone";

Existence quantifier; read: "exists", "there is at least one";

Signs to indicate the modal operator of necessity; read: "it is necessary that";

Signs to indicate the modal possibility operator; read: "possibly".

Along with those listed in multi-valued, temporary, deontic and other systems of logic, their own specific symbols are used, however, each time it is explained what exactly this or that symbol means and how it is read (see: Logical sign).

Dictionary of logic. - M.: Tumanit, ed. center VLADOS. A.A. Ivin, A.L. Nikiforov. 1997 .

See what "logical symbolism" is in other dictionaries:

- (Logical constants) terms related to the logical form of reasoning (proof, conclusion) and are a means of conveying human thoughts and conclusions, conclusions in any field. L. to. include such words as not, and, or, there are ... Glossary of Logic Terms

GOST R ISO 22742-2006: Automatic identification. Bar coding. Linear barcode and 2D symbols on product packaging- Terminology GOST R ISO 22742 2006: Automatic identification. Bar coding. Linear barcode symbols and two-dimensional symbols on product packaging original document: 3.8 Data Matrix: Two-dimensional matrix symbology with correction ... ...

- (Wittgenstein) Ludwig (1889 1951) Austro English. philosopher, prof. philosophy at Cambridge University in 1939 1947. Philos. V.'s views were formed as under the influence of certain phenomena in the Austrian. culture early. 20th century, and as a result of creative ... ... Philosophical Encyclopedia

- (Greek logike̅́) the science of acceptable ways of reasoning. The word "L." in its modern use is ambiguous, although not as rich in semantic shades as ancient Greek. logos from which it comes. In the spirit of tradition with the concept of L ... Great Soviet Encyclopedia

- (from the Greek semeiot sign) a general theory of sign systems that studies the properties of sign complexes of a very different nature. Such systems include natural languages, written and oral, a variety of artificial languages, starting with formalized ... Philosophical Encyclopedia

This term has other meanings, see Cow (meanings). ? Domestic cow ... Wikipedia

Concept Calculus- “CALCULUS OF CONCEPTS” (“Record in concepts”), the work of the German mathematician and logician Gottlob Frege, which laid the foundation for modern form mathematical (symbolic) logic. The full title of this work included an indication that in ... ... Encyclopedia of Epistemology and Philosophy of Science

Wittgenstein (WITTGENSTEIN) Ludwig- (1889 1951) austrian philosopher. Prof. philosophy at the University of Cambridge in 1939 47 . The philosophical views of V. were formed both under the influence of certain phenomena in the Austrian. culture of the beginning of the 20th century, and as a result of the creative development of new achievements ... ... Modern Western Philosophy. encyclopedic Dictionary

code- 01.01.14 code [code]: A set of rules that match elements of one set with elements of another set. [ISO/IEC 2382-4, 02/04/01] Source ... Dictionary-reference book of terms of normative and technical documentation

- (Comte) founder of positivism, b. January 19, 1798 in Montpellier, where his father was a tax collector. At the Lyceum, he excelled in mathematics. Entering the Polytechnic School, he surprised professors and comrades with his mental development. IN… … Encyclopedic Dictionary F.A. Brockhaus and I.A. Efron

PROPERTIES OF LOGICAL OPERATIONS

1. Notation

1.1. Notation for logical connectives (operations):

a) negation(inversion, logical NOT) is denoted by ¬ (for example, ¬A);

b) conjunction(logical multiplication, logical AND) is denoted by /\
(for example, A /\ B) or & (for example, A & B);

c) disjunction(logical addition, logical OR) is denoted by \/
(for example, A \/ B);

d) following(implication) is denoted by → (for example, A → B);

e) identity denoted by ≡ (for example, A ≡ B). The expression A ≡ B is true if and only if the values of A and B are the same (either they are both true or they are both false);

f) symbol 1 is used to denote truth (true statement); symbol 0 - to denote a lie (false statement).

1.2. Two boolean expressions containing variables are called equivalent (equivalent) if the values of these expressions are the same for any values of the variables. So, the expressions A → B and (¬A) \/ B are equivalent, but A /\ B and A \/ B are not (the meanings of the expressions are different, for example, when A \u003d 1, B \u003d 0).

1.3. Priorities of logical operations: inversion (negation), conjunction (logical multiplication), disjunction (logical addition), implication (following), identity. Thus, ¬A \/ B \/ C \/ D means the same as

((¬A) \/ B)\/ (C \/ D).

It is possible to write A \/ B \/ C instead of (A \/ B) \/ C. The same applies to the conjunction: it is possible to write A / \ B / \ C instead of (A / \ B) / \ C.

2. Properties

The list below is NOT meant to be exhaustive, but is hopefully representative.

2.1. General properties

For a set of n boolean variables exist exactly 2 n different values. Truth table for boolean expression from n variables contains n+1 column and 2 n lines.

2.2 Disjunction

If at least one of the subexpressions to which the disjunction is applied is true on some set of variable values, then the entire disjunction is true for this set of values.
If all expressions from some list are true on some set of variable values, then the disjunction of these expressions is also true.
If all expressions from some list are false on some set of variable values, then the disjunction of these expressions is also false.
The value of a disjunction does not depend on the order of the subexpressions to which it is applied.

2.3. Conjunction

If at least one of the subexpressions to which the conjunction is applied is false on some set of variable values, then the entire conjunction is false for that set of values.
If all expressions from some list are true on some set of variable values, then the conjunction of these expressions is also true.
If all expressions from some list are false on some set of variable values, then the conjunction of these expressions is also false.
The meaning of a conjunction does not depend on the order of subexpressions to which it is applied.

2.4. Simple disjunctions and conjunctions

We call (for convenience) the conjunction simple if the subexpressions to which the conjunction is applied are distinct variables or their negations. Similarly, the disjunction is called simple if the subexpressions to which the disjunction is applied are distinct variables or their negations.

A simple conjunction evaluates to 1 (true) on exactly one set of variable values.
A simple disjunction evaluates to 0 (false) on exactly one set of variable values.

2.5. implication

implication A →B is tantamount to disjunction (¬ A) \/ B. This disjunction can also be written as: A\/B.
implication A →B takes the value 0 (false) only if A=1 And B=0. If A=0, then the implication A →B true for any value b.

Logical symbols link events according to their causal relationships. A logical sign can have one or more inputs, but only one output or output event.

The output event of the logical sign "AND" occurs if all input events occur simultaneously. The output event of the logical sign "OR" occurs if any of the input events occurs.

The causal relationships expressed by the logical signs "AND" and "OR" are deterministic, since the appearance of the output event is completely determined by the input events. There are causal relationships that are not deterministic, but probabilistic.

The hexagon, which is a logical prohibition sign, is used to represent probabilistic causal relationships. An event placed under the logical inhibit sign is called an input event, while an event placed to the side of the logical sign is called a conditional event. A conditional event takes the form of an event when the input event occurs. An output event occurs if both the input and conditional events occur, i.e. an input event triggers an output event with a (usually constant) probability of occurrence of the conditional event.

The preemptive AND logical sign is equivalent to the logical AND sign, with the additional requirement that input events occur in a specific order. An output event appears if the input events occur in a certain sequence (from left to right). The occurrence of input events in a different order does not cause an output event.

The XOR gate describes the situation in which an output event occurs if one of two (but not both) events occurs on the input.

IN general case new logical signs can be introduced to represent special types of causal connections. It should be noted that most of the special logical signs can be replaced by a combination of logical signs "AND" or "OR".

table 2

Logic symbols

No. p / p	Boolean sign symbol	The name of the logical sign	Causal relationship
		Sign "I"	An output event occurs if all input events occur at the same time
		Sign "OR"	An output event occurs if either of the input events occurs.
		Prohibition sign	The presence of an input causes an output to appear when the conditional event occurs.
		Sign "Priority AND"	An output event occurs if all input events occur in the correct order from left to right
		XOR sign	An output event occurs if one (but not both) of the input events occurs

So much has already been done in the field of the logic of meaning (meaning) that there is no need to present spatial arguments in support of the theory on which we all rely here; perhaps it is sufficient to indicate in general terms facts or, if you like, assumptions on which my further considerations are based.

Meaning has both logical and psychological aspects.

In a psychological sense, any object that has meaning can be used as a sign or symbol; that is, for someone it must be a sign or a symbol. In a logical sense, it must be able to convey meaning, be the kind of thing that can be used in this way. In some value relationships, such a logical requirement is trivial and is tacitly accepted; in others it is of the utmost importance and may even lead us, in some amusing way, through labyrinths of nonsense. These two aspects - logical and psychological - are completely confused by the use of the obscure verb "to mean"; for sometimes it is right to say "this means," and sometimes "I mean." Obviously, one word such as "London" does not "mean" a city in exactly the same sense that one uses the word "means" for a given place.

Both aspects, the logical and the psychological, are always present, and their interaction gives rise to a huge variety of meaning connections that have puzzled philosophers and over which they have struggled for the past fifty years. The analysis of "meaning" must have a particularly complicated history. The word is used in many different senses, and much of the discussion has been aimed at clarifying correct use, on the subject of clarifying the meaning of "meaning". Whenever people discover several kinds of genius, they are always looking for the primary form, that archetype which is supposed to be revealed in each case in its own way; for a long time philosophers have hoped to discover the true quality of meaning by collecting all its various manifestations and looking for some common ingredient. They spoke more and more generally about "symbolic situations", believing that through generalization it is possible to achieve an understanding of the essence of all such situations. But a generalization based on obscure special theories can never give us a clear general theory. That kind of generalization which simply replaces the "symbolic situation" with "denotation-or-connotation-or-denotation-or-association-etc" is scientific point sight useless; for the whole purpose of general concepts is to make the distinctions between individual classes clear, and to relate all subspecies to each other in a certain way. But if such general concepts are merely composite photographs of well-known types of meaning, they can only obscure rather than clarify the connections that are derived from the special senses of the word.

Charles Pierce, who was probably the first to seriously study semantics, began to compile a list of all "symbolic situations", hoping that if all the possible meanings of "meaning" were brought together, their differences would be revealed, whereby it would be possible to separate the necessary from the unnecessary. But this disorderly heap (instead of a clear classification) has been divided and subdivided into the most terrifying system of signs, characteristics and traits, without any hope that the original 59,049 types can really be reduced to a simple 6637.

Subsequently, several attempts were made to capture by empirical methods the essential quality of meaning. But the more diversity was discovered, the less hope there was for identifying a common essence. Husserl, who characterized each type of meaning as a distinct concept, ended up with as many theories as there are "meanings"38. But we still have the necessary and the unnecessary, as well as all their derivatives, and it still seems surprising why one "family" name "Meaning" should be attached to all these concepts, although no family resemblance can be determined here.

In fact, no quality of meaning exists; its essence lies in the realm of logic, where one has nothing to do with qualities, but considers only connections and relations. The words "meaning is relation" are unclear, as they suggest that the matter is too simple. Most people think of relationship as something two-sided - "A in relation to B"; but meaning includes several aspects, and different types of meaning consist of different types and degrees of relation. Perhaps it would be better to say: "Meaning is not a quality, but a function of a term." A function is a pattern (model) considered in relation to one single term around which it is centered; this pattern occurs the moment we look at the term in its full relation to other cognate terms. The whole can be quite confusing. For example, a musical chord can be viewed as a function of one note, known as a "capital bass", it can be interpreted by writing that one note and revealing its relation to all the other notes that should beat the first one. In a hundred

In real organ music, the chord Ъц (would be written as SHI \

which means: "A chord with sixth, fourth and third notes above A". This chord is considered as a pattern that surrounds and includes A. It is expressed as a function of la.

Similarly, the meaning of a term is a function; it is based on a model in which this term occupies a key position. Even in the simplest kinds of meaning, there must be at least two other things associated with the term that "means," that is, the object "designated" and the subject that uses the term. In the same way that a chord must have at least two notes other than the "capital bass" in order to determine which chord it is (one of them may simply be "understood" by musicians, but without it the given combination will not define a chord). The same can be said about a term with a meaning; the existence of the subject is often implicitly assumed, but if at least one designated object and some mind for which it is designated are missing, then there is not a complete meaning, but only a partial pattern that can be performed in various ways.

Any term in the general model can be perceived as key term with which the others are connected. For example,

chord can be seen as a function of itself

lower H01Y and can be expressed through such a description.

or it can be interpreted by referring to the note on which it is built from the point of view of harmony, the one that, apparently, is the note D. The musician, analyzing this harmony, would call this chord "the second inversion of the seventh chord on the dominant in the key of G". The "dominant" of this clef is re, not la. He would interpret all this as a function of the note D; this sounds more confusing than the other interpretation, which fixed the notes from A and above, but of course this is not the case at all, because in the latter case one arrives at the same pattern (model).

Similarly, we can look at a pattern of meaning in terms of any term it contains, and accordingly our descriptions of it will be different. We can say that for some individual a certain symbol "means" some object, or that this individual "means" this object by a given symbol. The first description interprets meaning in a logical sense, the second in a psychological one. The former takes symbols as the key, and the latter as the subject39. Thus the two most contradictory kinds of meaning, logical and psychological, are distinct and at the same time related to each other through general principle a view of meaning not as a property, but as a function of terms.

In subsequent analyses, "meaning" will be taken in the objective sense, unless some other sense is emphasized; that is, I will speak of terms (such as words) as "meaning" something, not people as "meaning" this or that. Later we will need to isolate the various subjective functions; but for now let's consider the relations of terms to their objects. That which links terms to their objects is, of course, the subject; this has always been understood.

First of all, there are two separate functions of terms, each of which has full right be called "meaning": for any significant sound, gesture, thing, event (for example, an explosion) can be either a sign or a symbol.

The sign indicates the existence - in the past, present or future - of a thing, event or condition. Wet streets are a sign that it has rained. The sound of raindrops on the roof is a sign that it is raining. The fall of the barometer or the appearance of the ring of the moon - that it will rain soon. The presence of abundant greenery in a non-irrigated area indicates that it often rains here. The smell of smoke signals the presence of fire. The scar is evidence of an accident in the past. Dawn is the herald of the rising sun. Sleek healthy body- a sign of frequent and abundant nutrition.

All examples given here are natural signs. A natural sign is part of a larger event or complex condition, in relation to the observer experiencing it, it means the remainder of that situation, hallmark which he is. It is a symptom of the state of affairs.

The logical connection between a sign and its object is very simple: they are connected in such a way as to form a pair; that is, they are in a one-to-one relationship. Each sign corresponds to one specific object, which is its object, the thing (or event, or condition) it denotes. The entire remainder of this important function of designation includes a third term, the subject, which uses a pair of things; and the relation of the subject to the other two terms is much more interesting than their own mere logical pair.

The subject is essentially related to the other two terms as a pair. They are characterized just by the fact that they are paired. Thus, a white bulge on a man's hand - as a mere sensuous fact - is probably not interesting enough to have its own name, but such a fact, in connection with the relationship to the past, is noted and called a "scar". Note, however, that although the subject relation is paired with other terms, it is also related to each of them individually, making one of them a sign and the other an object. What is the difference between a sign and its object, whereby they are not equivalent? The two terms are simply connected as a pair like two sandals, two scales, two ends of a stick, etc.—the two terms could be used interchangeably without any harm.

The difference lies in the fact that the subject for whom they are a couple must consider one of them more interesting than the other, and the second more accessible than the first. If we are interested in the weather for tomorrow, then the current events, if they are connected with tomorrow's weather, are signs for us. The ring around the moon or the cirrus clouds in the sky are not important in and of themselves; but as currently observable phenomena connected with something important - though not at the moment - they have "meaning". If the sign and the object did not exist for the subject, or the interpreter, then they would be equivalent. Thunder can just as well be a sign that there has been lightning, just as lightning can mean that there will be thunder. By themselves, these phenomena are simply related. This connection is important only where one of these phenomena is perceived, and the other (which is more difficult or impossible to perceive) is of interest, here we actually have the case when the designation belongs to a certain term41.

Now, just as in nature, certain events are related to each other in such a way that a less important event can be taken as a sign of a more important one; we can produce conditional events that are intentionally linked to those important events that should be their values. The whistle means that the train will start soon. A cannon shot is a sign that the sun has just risen. A mourning bandage on the door signifies that someone has died. They are artificial signs because they are not part of the state of which the remainder (or anything in that remainder) they naturally signal. Their logical connection with their objects is, however, the same as that of natural signs—that is, a one-to-one correspondence between sign and object, by virtue of which the interpreter, interested in the object and receiving the sign, can foresee the existence of that term which interests him.

The interpretation of signs is the basis of the animal mind. Probably animals do not distinguish between natural signs and artificial or accidental signs; but in their practical activity they use both kinds of signs. We do the same throughout the day. We answer calls, look at the clock, obey warning signals, follow the directions indicated by the arrows, remove the kettle from the fire when we hear a characteristic whistle, approach crying baby close the windows when we hear thunder. The logical basis of all these interpretations, the simple interconnection of trivial and important events, is in fact very simple and common - so much so that there is no limit to the meaning of any sign. This seems to be even more true of artificial signs than of natural ones. A certain shot can mean: the beginning of a race, sunrise, the danger of aimed fire, the beginning of a parade. As for calls, the world just went crazy because of them. Someone rings the doorbell, someone - on the phone; here the call means that the toast is ready, there - that the line has ended when typing on a typewriter; the beginning of classes at school, the beginning of work, the beginning of the church service, the end of the church service; the tram moves off, the ticket office clicks; time to get out of bed, time to dine; fire in the city - calls are heard everywhere!

Since a sign can mean so many different things, we are quite apt to misinterpret it, especially if it is artificial. Ring tones, of course, can either be incorrectly associated with their objects, or the sound of one ring can be confused with the sound of another. However, natural signs can also be misunderstood. Wet streets are not at all a reliable sign that it has recently rained if a watering truck has passed before. Misinterpreting signs is the simplest form of error. For purposes practical life this is the most important form of error and the easiest to detect; for its usual manifestation is the experience called disappointment.

Where we find the simplest form delusion, we may also hope to find its correlate and the simplest form of knowledge. Of course, we are talking about the interpretation of signs. It is the most elementary and most tangible kind of thought, the kind of knowledge we share with animals; we master it entirely through the experience of a clearly biological origin, with equally obvious criteria of truth and falsity. Its mechanism can be understood as the development conditioned reflex, linking a certain function of the brain ("switch") and the correct or incorrect "number" for that sense organ, which the musculature "calls" and expects to receive some answer in the language of changed sensations. This thinking has all the virtues of simplicity, internal coherence, and rationality that are recommended for scientific presentation. Thus, it is not surprising that the followers of genetic psychology took advantage of the understanding of the sign as the archetype of all cognition, it is not surprising that they perceive signs as the original carriers of meaning and interpret all other terms with semantic properties as subspecies, that is, "substitute signs" that act as representatives of their objects and are forced to conform to the latter, and not to conform to themselves.

But "substitute signs", although they can be placed alongside symbols, are signs of a very specific kind and play a rather limited role in the entire process of mental life. I will return to them later when discussing the relationship between symbols and signs, as they are part of each of these areas. However, first of all, one should continue to record the characteristics of symbols in general and their significant differences from signs.

A term that is used symbolically and non-significantly does not align the action with the presence of the object. If I say: "Napoleon", you will not worship the conqueror of Europe, as if I introduced you to him, and not just named him. If I mention our mutual acquaintance Mr. Smith, you can say something about him behind his back that you certainly would not say in his presence. Thus, a symbol referring to Mr. Smith, his name, can successfully provoke such an action, which is only appropriate in his absence. Raised eyebrows and a glance at the door, taken as a sign that Mr. Smith had entered, would have stopped you in the middle of your story; this action would have been addressed personally to Mr. Smith.

Symbols do not represent objects themselves, but are carriers of a certain concept about objects. To comprehend a thing or situation is not the same as "reacting" to it in an obvious way, or becoming aware of its presence. Speaking of things, we do not have things as such, but ideas about them; symbols, on the other hand, directly "imply" precisely concepts, not objects. Behavior in relation to concepts is what words usually induce; this is a typical thought process.

Of course, the word can be used as a sign, but this is not its primary purpose. The sign character of a word is revealed by a special modification - by the tone of voice, gesture (for example, pointing or staring), or by the very location of the announcement in which this word is used. By its very essence, a word is a symbol associated with a concept1, and not directly with any social object or event. The fundamental difference between signs and symbols is the difference in associations and, consequently, the difference in their application by the third participant in the function of meaning - the subject; signs announce their objects to him, while symbols make him perceive their objects. The fact that the same object (say, a small noise effect that we call a "word") can serve as both a sign and a symbol does not at all obliterate the cardinal difference between the two functions, as might be supposed.

Probably the simplest kind of symbolic meaning is that which belongs to proper names. A personal name gives rise to the concept of something given as a unit in the experience of the subject, something concrete and, therefore, easily reproduced in representation. Since the name, so obviously belonging to the representation, is unambiguously derived from the individual object, it is often assumed that it "means" that the object, as a sign, must "mean" it. This view is strengthened by the fact that the name borne by a living person is always both a symbol by which we think of that person and a vocative name by which we signal to him. Because of the confusion between these two functions, the proper name is often considered to be a bridge from animal semantics, or animals' use of signs, to human language, which uses symbols. Dogs, as we have already said, understand names - not only their own, but also their owners. They, of course, understand names only as vocatives. If you say "James" to a dog whose owner bears this name, the dog will understand this sound as a sign and will look for James with his eyes. But if you say the same thing to a person who knows someone with that name, he will ask: "What did you want to say about James?" This simple question always remains on the other side of the understanding of the dog; the designation is only the meaning of the name that it can have for the dog, the meaning that the owner's name shares with the owner's smile, with his ability to play football and the characteristic sound of the bell on his door. However, for a human being, the name calls to mind the representation of the particular person who is so called, and prepares the mind for further representations in which

"Note that I have named the terms of our mental representations, not concepts. Concepts are abstract forms embodied in representations; their bare representation may be roughly called 'abstract thought', but in ordinary mental life they figure no more as mere factors than the skeletons seen walking down the street. Concepts, like the skeletons mentioned, are always embodied—sometimes too much so. : "What do you want to say about James?"

There is a famous passage in Helen Keller's autobiography in which this wonderful woman describes the first appearance of Language in her mind. Of course, she had already used signs that formed associations before, having learned to anticipate certain phenomena and identify people and places; but on a momentous day all the significations of the signs faded and were eclipsed by the discovery that a certain fact in her limited sensible world had a certain meaning, that a certain action of her fingers constituted a word. This event required a long preparation; the child has learned many finger actions, but so far they have been a meaningless game. Then one day the teacher took her for a walk - and there the great advent of Language took place.

“She brought me a hat,” says her memoir, “and I knew that I had to go outside, where it was warm and the sun was shining. This thought, if a wordless sensation can be called a thought, made me jump up and jump for joy.

We walked up to the shed over the well, attracted by the fragrance of honeysuckle exuding from there. Someone was fetching water, and my teacher put my hand under the stream of water. When cold water filled her palm, the teacher said the word "water" to another person - first slowly, and then quickly. As I stood, all my attention was drawn to the movement of her fingers. Suddenly I felt a vague movement of consciousness, like something forgotten - a kind of flutter of a returning thought; and somehow the secret of language was revealed to me. Now I knew that "w-o-d-a" means something wonderful, something cold that flows through my palm. I realized that the living word awakened my soul, gave it light, hope, joy, made it free! True, there were still obstacles, but these obstacles in time could be swept away.

Feeling a thirst for knowledge, I left the well under a canopy. Everything had its own name, and each name gave birth to a new thought. When we returned to the house, every object I touched seemed to tremble with life overflowing into it. It was because I was looking at everything from a strange new perspective that came to me"42.

This passage is the best written evidence that can be found for pointing out the actual difference between sign and symbol. A sign is something in accordance with which an action is performed, or a certain means for denoting an action; and a symbol is an instrument of thought. Notice how Miss Keller qualifies the mental process immediately preceding her discovery of words: "That thought, if a wordless sensation can be called a thought." Real thinking is possible only in the light of a genuine language, no matter how limited or primitive; in her case, this becomes possible with the discovery that "w-o-d-a" was not necessarily a sign that water was wanted or expected, but was the name of this substance, through which it could be mentioned, remembered and thought.

Since the name (name) - the simplest kind of symbol - is directly related to the representation and is mentioned by the subject in order to realize given representation, this easily leads to the fact that the name is treated as a "conceptual sign", an artificial sign that announces the presence of a certain idea. In a sense, this is entirely justified; yet it strikes out that wrong and unnatural note which usually honestly warns that the attempted interpretation misses the most important feature in its material. IN this case what is missing is the relation of representations to a concrete world, which is so close and so important that it enters into the very structure of "names". And finally, the name points to something (has its own denotation *). "James" may represent a concept, but it names a specific person. In the case of proper names, this relation of the symbol to what it points to is so striking that such pointing has been confused with a direct connection between sign and object. As a matter of fact, "James" does not designate a person, without further ado; this name points to it as a denotation - it is associated with a representation that "fits" a particular person. The relation between symbol and object, usually expressed as "S points to O," is not the simple two-valued relation that S has to O; This - difficult case: for a certain subject, S is associated with a representation that suits O, that is, with a concept that is satisfactory for O.

For the usual sign function, there are three essential terms: subject, sign, and object. For a denotation (denotation), which is the most common type of symbol function, there must be four terms: subject, symbol, representation (concept) and object. The fundamental difference between sign-value and symbol-value can therefore be brought out in a logical way, since it is based on a difference in model, it is, strictly speaking, a different function.

Thus, a denotation (denotation) is a complex relation of a name to an object that bears this name; but what would be the more direct relation of the name (or symbol) to the representation associated with it? We will call it in the traditional way - connotation*. The connotation of a word is the representation that the word conveys. Because the connotation remains with the symbol while the object being referred to is neither present nor sought, we are able to think about the object at all without explicitly reacting to it.

Hence, there are three most familiar meanings of the word "meaning" itself: denotation, denotation, and connotation. All three are equally entirely valid, but are by no means interchangeable.

In any analysis of the use of a sign or the use of a symbol, we must be able to take into account not only the origin of knowledge, but also the most characteristic feature of man - error. How a sign can be misinterpreted has already been shown; but an unfortunate denotation or confusion of connotation is unfortunately just as common, and should also be brought to our attention.

In each case of denotation, which could be called the application of a term to an object, a certain psychological act takes place. For example, the word "water" indicates a certain substance, because people traditionally apply it to this substance. Such an application has fixed its connotation. We may ask, quite reasonably, whether a certain colorless liquid is water or not, but one can hardly ask whether water "really" means that substance which is found in ponds, falls to the earth from clouds, has the chemical structure of H2O. The connotation of this word, although derived from long-term usage, is now more definite than some of the applicability of the word. When we misuse a term, that is, apply it to an object that does not satisfy its connotations, we do not say that the term "indicated" that object; in this case, one feature is missing in the fourfold meaning-relationship, and therefore there is no real pinning, but only a psychological act of application, and even that is a mistake. The word "water" is never involved in referring to the drink that killed little Willie in the well-known pathetic laboratory stanza:

We had little WILLY.

Now he is gone, After all, what he took for H2O It turned out to be H2SO4.

Willy mistook one object for another; he misapplied a term whose connotation he knew fairly well. But since connotations are usually fixed on a word initially by applying it to certain objects whose properties are sufficiently known, we may also be mistaken about connotation when we use this term as a vehicle of thought. We may know that the symbol "James" is attached to our neighbor who lives opposite, and it is quite a mistake to assume that this symbol means a person in general with all his virtues or shortcomings. This time we don't mistake James for someone else, but we are wrong about James.

The peculiarity of proper names is that they have their own connotation for each denotation. Since their connotation is not fixed, they can be applied arbitrarily. There is no connotation in the proper name itself; sometimes it takes on the most general kind of conceptual meaning - it means a genus, or race, or denomination (for example, "Christian", "Welsh", "Jewish people"), but there is no real error in calling a boy "Mary", a girl - " Frank", a German - "Pierre" or a Jew -

"Luther". In a civilized society, the connotation of a proper name is not seen as a meaning attached to the bearer of the name; when a name is used to refer to a specific person, it takes on the connotation required by such a function. In primitive societies this happens less often; names are often changed because their accepted connotations do not suit the bearer. The same person may be called "Lightfoot", then "Hawkeye", then "Whistling Death", etc. In Indian society, the class of people with the name "Hawkeye" is most likely a subclass of "sharp people". But in our society, ladies called "Blanche" are not necessarily albinos or even blondes. A word that functions as a proper name is excluded from the scope of normal application rules.

This is all that concerns the venerable "logic of terms". It looks a little more complex than the logic in medieval books, since we must add to the long recognized functions - connotation and indication (denotation) a third -

a designation that is fundamentally different from the first two; and since, in discussing the semantic functions of terms, we had to make the rare discovery that they really are functions, and not powers or arcane properties or anything else, we had to treat them accordingly. The traditional "logic of terms" is really a metaphysics of meaning; the new philosophy of meaning is above all a logic of terms - signs and symbols, an analysis of related instances in which "meaning" can be found.

But the semantics of individual symbols is only a rudimentary basis for the more interesting aspect of meaning. Until we come to discourse, everything is mere propaedeutics. It is in discursive thinking that truth and falsity are born. And before that, the terms are built into assumptions, they do not assert anything and do not exclude anything; in fact, although they may name things and convey certain ideas about these things, they say nothing. I have been discussing them for so long for the simple reason that most logicians have given them such an unceremonious interpretation that even such an obvious distinction as the distinction between sign-function and symbol-function has passed unnoticed by them; so careless philosophers are guilty of allowing ambitious followers of genetic psychology to argue with them on topics from the conditioned reflex to the wisdom of J. Bernard Shaw - all in one swift generalization.

The logic of discourse has been handled much more adequately, so well that there is practically nothing new to be said here; nevertheless, it should at least be mentioned here, since an understanding of discursive symbolism, the vehicle of judgment-based thinking, is essential to any theory of the human mind; for without it no literal meaning would be possible, and consequently no scientific knowledge.

Anyone who has ever studied a foreign language knows that learning a dictionary alone will not make a person proficient in a new language. Even if he memorized the entire vocabulary, he would not be able to form the simplest sentence correctly without certain grammar principles. He must know that some words are nouns and some are verbs; he must know that there are active and passive forms of verbs, and also know the gender and number; he must know where the verb is in the sentence in order to give the sentence the meaning it implies. Simple separate names of objects (even actions that are "called" by infinitives) do not constitute a sentence. A number of words that we can extract from the dictionary, running our eyes from left to right and down the columns (for example, "possessed - dressed - approval - illumine - mischief") do not say anything. Each word has its own meaning, but arbitrary row- none.

Thus, grammatical structure serves as an additional source of meaning. We cannot call it a symbol, because it is not even a term; but it has a symbolic mission. The grammar links together several symbols, each with at least a fragmentary connotation of its scope, in order to create one complex term whose meaning is a particular constellation of all the connotations involved. What a separate galaxy is depends on syntactic links within a complex symbol or judgment.

The structure of judgment is of greater interest to the logicians of the present generation than any other aspect of symbolism. Since Bertrand Russell1 pointed out that the Aristotelian metaphysics of substance and its properties is an integral part of the Aristotelian logic of the subject and predicate (that the common sense point of view on objects and properties, factor and object of action, subject and action, etc. is an undoubted part of the fact that the logic of common sense is embodied in parts of speech), the links between expressibility and intelligibility, forms of language and forms of experience, judgments and facts emerge more and more clearly. It turned out that a judgment fits a fact not only because it contains the names of the objects and actions that make up this fact, but also because it combines them into a pattern, similar in some way to the one in which the named objects are combined "in fact". A judgment is a picture of a structure - the structure of a state of affairs. Unity of judgment is the same kind of unity that belongs to an image that is one scene, no matter how many things can be distinguished within this image.

What property must an image have in order to represent its object? Does it really have to share the visual appearance of the object? Certainly not to any extent. For example, an object might be black on white, or red on grey, or any color at all on any other background; the image may be bright while the object itself is dim; it can be much larger or much smaller than the object; it is, of course, flat, and although the devices of perspective sometimes give a perfect illusion of three-dimensionality, an image without perspective, such as the "vertical projection" made by an architect, is undoubtedly still an image representing an object.

The reason for this wide acceptance is that the image is essentially a symbol and not a copy of what it represents. An image has certain characteristics that enable it to function as a symbol for its object. For example, in children's drawing(Fig. 1) the rabbit is immediately recognizable, and although in fact it looks completely different, even a person with poor eyesight will not doubt for a moment that he sees a sitting rabbit on the page of the book. All that the image has in common with "reality" is a certain proportion of parts, the position and relative length of the "ears", a point where the "eye" should be, a certain ratio of the size of the "head" and "torso", etc. e. Next to this image is exactly the same drawing, only with different ears and tail (Fig. 2); any child will mistake him for a cat. Although in reality, cats do not look like long-tailed short-eared rabbits. Neither the rabbit nor the cat is flat and white, they are not paper and they do not have black outlines. But all these features of the painted cat are irrelevant, since it is just a symbol and not a pseudo-cat43.

Of course, the more detailed the image is drawn, the more undoubtedly it becomes a reference to a specific moment. A good portrait is "true" in relation to a particular person. However, even good portraits are not copies. In portraiture, as in other arts, there are various styles. We can paint with sublime warm soft colors or cold pastels; we can choose from the clear pinions characteristic of Holbein's drawings to the shimmering hues characteristic of French Impressionism; and there is no need to change the object anyway. The variable is our idea of the object.

3 Susan Langer

The image is a symbol, and the so-called "means" is a kind of symbolism. However, there is, of course, something that connects the image with its original and makes it represent, for example, a Dutch interior, and not a crucifixion. What an image can represent is dictated purely by its logic - the arrangement of its elements. The mutual arrangement of pale and dark, dull and bright colors, or thin and thick lines and variously delineated white spaces, gives certainty to those forms that imply specific moments. They can refer to those and only those objects in which we recognize similar forms. All other aspects of the image, such as what artists call the "distribution of light and shadow", the "technique" and "tonality" of the entire work, serve purposes other than mere reproduction. The only thing an image must have in order to be an image of a particular subject is an arrangement of elements similar to the arrangement of conspicuous visible elements in an object. The image of a rabbit must have long ears; a person should be depicted with arms and legs.

In the case of the so-called "realistic" image, this analogy goes to the smallest detail, to such that many people begin to consider the statue or drawing as a copy of the corresponding object. But notice how we encounter such quirks of style as modern commercial art produces: ladies with bright green faces or aluminum hair, men with perfectly round heads, horses made entirely of cylinders. We still recognize the objects they represent, for we find some element to match the head, and some element to fit the eye, a white mark signifying a starched chest, some line being where there might be hand. With astonishing speed, our vision picks up these traits and allows fantasy to convey the human form.

One step away from the "stylized image" is the diagram. Here, any attempt to imitate parts of the object is already abandoned. These parts are simply expressed by conventional symbols such as dots, arcs, crosses, or something else like that. The only thing that is "depicted" is the ratio of the parts. A diagram is a "picture" of a form alone.

Notice the photograph, the painting, the pencil sketch, the architect's elevation, and the builder's diagram, all showing the front view of the same house. Without much effort, you will recognize this house in any kind of reproduction. Why?

Because each of these very different images expresses the same ratio of parts that has already been fixed in your mind when you formed the idea of \u200b\u200bthe house. Some of these versions show more of these proportions than others; they are more detailed. The same images that do not show specific details, at least do not show anything instead of them, could be perceived as if these details were missed. All the items shown in the simplest image - in the diagram - are contained in a more thoughtful transmission. In addition, they are contained in our idea of the house; thus all representations respond, each in its own way, to our representation, although the latter may contain such details as are not shown at all. In the same way, another person's idea of the same house will be essentially consistent with the images and with our idea, although it may have many particular aspects.

Due to such fundamental features, which are usually present in any correct conception of the house, we can talk together about the "same" house, despite partial differences in sensory experience, opinions and purely personal associations. What all adequate representations must normally contain is the concept of an object. One and the same concept is embodied in a variety of representations. It is the form that arises in all versions of thought or imagination that can designate a certain object in a question, a form that is dressed for each separate mind in its veils of sensations. There are probably no two people who see anything in exactly the same way. Their sense organs are different, their attention and representation and feeling are so different that it is impossible even to assume that they have the same impressions. But if their respective ideas about any object (or event, person, etc.) are embodied in the same concept, they will certainly understand each other.

A concept is anything that actually conveys a symbol. But the concept is immediately symbolized for us, our own imagination dresses it in a personal representation, which we can distinguish from the generally accepted concept only by means of abstraction. Whenever we deal with a concept, we must have some separate idea of it, through which we comprehend the concept. What we actually have "in the mind" is always the universalium in re1. When we express this universalium, we use another symbol in order to discover it, and another res will embody it for the mind that "looks" through our symbol and comprehends the concept in its own way.

The power of understanding symbols, that is, of treating everything that concerns sense data as an irrelevant exclusion of the particular form they embody, is the most characteristic feature of the human mind. This culminates in an unconscious spontaneous process of abstraction that goes on all the time in our mind - the process of recognizing the concept in any configuration that comes across in experience and forming the corresponding representation. This is the real meaning of Aristotle's definition of man as a "rational animal." Abstract vision is the basis of our rationality and is its certain guarantee long before the appearance of any conscious generalization or syllogism44. This is a feature that no other animal has. Animals do not recognize symbols; that's why they don't see the images. We sometimes say that dogs don't react to even the best portraits because they live more in a world of smell than of sight; but the behavior of a dog that watches a real, motionless cat through a window pane refutes such an explanation. Dogs ignore our paintings because they see colored canvases, not images. Reproducing a cat in a painting will not make a dog "think" of it.

Since any particular sense datum can, in a logical sense, be a symbol for any particular thing, any conventional label or token can signify a representation - or, to put it bluntly, a concept - of any particular thing, and thus refers to that thing as such. The movement of the fingers, perceived as a single action, became the name of the substance for the little deaf-blind-mute Helen Keller. In a similar way, a word taken as a sound unit becomes for us a symbol of a certain object existing in this world. And now the power of seeing configurations as symbols comes into play: we produce models of pointing symbols, and they immediately symbolize completely different, albeit similar, configurations of pointed objects. The temporal order of words corresponds to the order of the relationship of objects. If pure word order becomes insufficient, word endings and prefixes "imply" relationships; from them prepositions and other purely correlative symbols are born. In the form of mnemonic dots and crosses, symbols indicating objects can also enter into diagrams or simple pictures, thus produce sounds, as soon as they are words, enter into verbal descriptions or suggestions. A sentence is a symbol of a state of affairs and depicts the nature of this state.

Consequently, in an ordinary image, the terms of the reproduced complex are symbolized by very many visual means, that is, by colored areas, and the relationship of terms is shown by the relationship of these means. Thus drawing, being static, can only represent a momentary state; it may suggest, but it can never really tell a story. We can produce a series of images, but nothing in these images can really guarantee that multiple scenes will be connected into one sequence of events. The five children's drawings of the little Dion sisters in various activities can be taken either as a series of reproductions of the successful actions of one child, or as separate points of view of five little girls in the corresponding field of activity. There is no reliable way to choose between these two interpretations, taken without signatures or other similar indications.

But most of our interests are focused on events and not on objects in static spatial relationships. Causality, activity, time and change are what we most want to understand and consider. And for this purpose, the pictures are hardly suitable. Therefore, we will resort to a more powerful, flexible and adaptable symbolism of the language.

How are relationships expressed in language? For the most part, they are not symbolized by other relationships, as in the pictures, but are named exactly the same as nouns. We name two objects and between them we place the names of the relation; this means that a relation holds two things together. The phrase "Brutus killed Caesar" shows that "murder" is something that is common to the relationship between Brutus and Caesar. Where the relationship is not symmetrical, the word order and grammatical forms (adverbial, mood, tense, etc.) of the words symbolize its direction. The phrase "Brutus killed Caesar" means something different from the phrase "Caesar killed Brutus", and the phrase "Caesar killed Brutus" is not a sentence at all. Word order partially determines the meaning of the structure.

The ability to name relationships, and not just describe them, gives the language a huge scope; one word can thus cover a situation that would take a whole page of pictures to describe. Pay attention to this sentence: "Your chance of winning is one in a thousand." Imagine expressing this relatively simple sentence in pictures! First you would need the symbol for "you winning" and then for "you losing" a thousand times! Of course, a thousand-fold image of anything is completely beyond clear understanding based on a simple visual gestalt. We can distinguish between three, four, five, and maybe a few more visible images, for example:

But a thousand becomes just "a lot". The exact fixation of a thousand requires such an order of concepts in which it occupies a certain place, since each quantitative concept in our numerical system has its own place. But in order to point to such a multitude of concepts and keep their relations to each other correct, we need a symbolism that can express both terms and relations more economically than pictures, gestures, or mnemonic signs.

It was noted earlier that a symbol and an object, having the usual logical form, can be interchanged without any psychological factors , namely: the object is of interest, but it is difficult to fix it, while the symbol is perceived easily, although in itself, perhaps, is completely insignificant. Thus, the small vocal sounds from which we make words are extremely easy to reproduce in all varieties of subtle shades and are easy to perceive and distinguish. Bertrand Russell wrote: “Of course, to a large extent, the fact that we do not use words of a different kind (non-voice) is connected precisely with convenience. There is a language of the deaf and dumb; shrugging the shoulders among the French is also a word; in a word, if it be prescribed by common usage in society. But the convention which has given speech its predominance has a sound basis, since there is no other way of producing so many different bodily movements perceived with such rapidity or with such little muscular effort. Speech would be very tiring. if statesmen had to use the language of the deaf and dumb, and completely exhausting if all words included as much muscular effort as a shrug of the shoulders"46. Speech not only costs little effort, but above all it does not require any means other than the vocal apparatus and auditory organs, which we usually have with us as part of ourselves; thus words are naturally available symbols, and very economical ones at that. Another advantage of words is that they have no other meaning than symbolic (or sign); in themselves they are quite trivial. This is a greater advantage than philosophers of language usually realize. A symbol that is of interest to us distracts our attention in the same way as an object. It cannot convey its meaning freely. For example, if the word "abundance" were replaced by a real, juicy and ripe peach, few people could fully turn their attention only to the simple concept of perfectly sufficient if they encountered such a symbol. The poorer and more indifferent the symbol, the greater its semantic power. Peaches are too good to be words; we are too much interested in peaches as such. But short sounds (words) are ideal transmitters of concepts, because they give us nothing but their meaning. This is the reason for the "transparency" of language, which has already been pointed out by some scholars. The vocabularies themselves are so worthless that we cease to be aware of their physical presence at all and become aware only of their connotations, indications, or other meanings. It seems that our conceptual activity proceeds through them, and does not simply accompany them, like other experiences that we give meaning to. They are not able to impress us as "experiences" until we have mastered them as a foreign language or technical jargon.

But the greatest advantage of word symbols is probably their great readiness to enter into combinations. There is practically no limit to the selection and relative position with which we can give them. It does so in a huge groan because of the economy Lord Russell noticed, the speed with which each word is generated, presented and completed, paving the way for the next. This enables us to grasp a whole group of meanings at the same time and produce a new, complete and complex concept from the individual connotations of rapidly following words.

This is the basis of the power of language, embodying the concepts of not only objects and their combinations, but also situations. The combination of words denoting a situation-concept is a descriptive phrase; if the relation-word in such a phrase is given by a grammatical form called a "verb", then the phrase becomes a sentence. Verbs are symbols with a double function; they express a relation and furthermore imply that the relation is preserved, that is, that the symbol has an indication47. Logically, they combine the value of the function φ and the statement-sign; verb, has the power of "assert f ()".

If a word is given by a conditional denotation, which may be a simple object or a complex phenomenon, then it is simply a name; for example, in a language I invented, the word "muf" can mean a cat, a state of mind, or the government of a country. I can give this name to anything I want. The name may be clumsy or familiar, ugly or pretty, but in itself it is neither true nor false. But if it already has a connotation, it can no longer be given a conditional indication, I cannot use the word "kitten" with its conventional connotation to indicate an elephant. The use of a word with its connotation is equivalent to the statement: "This is so-and-so." To call an elephant the word "kitten" not as a proper name, but as an ordinary noun, is a mistake, because this word does not illustrate the concept denoted. Similarly, a word with a fixed indication cannot be given an arbitrary connotation, because since the given word is a name (ordinary or proper), to give it a definite connotation is to predicate the designated concept, regardless of its name. If the words "clumsy animal" refer to an elephant, they cannot be given the connotation "something fluffy" because the clumsy animal is supposed to be non-furry.

Therefore, the connection between connotation and denotation is the most obvious focus of truth and falsity. Conditional expressions of this connection are sentences asserting that something is so-and-so, or something has such and such properties; in the technical language of the approval of the forms "x e y (fu)" and "fx". The difference between these two forms is simply which aspect of the name we have determined first, its connotation or its denotation; for both kinds of truth or falsity statements have the same basis.

In such a complex symbolic structure as a sentence that connects several elements to each other by means of a verb that expresses a developed model of relations, we have a "logical image", the applicability of which depends on the denotations of many words and the connotations of many symbols of relations (word order, particles, circumstances etc.). If the names have denotations, the sentence is talking about something; then its truth or falsity depends on whether any relations actually contained among the indicated objects illustrate the concepts of relations expressed by the given sentence, that is, whether the indicated model of objects (or properties, events, etc.) is similar to the syntactic model complex character.

There are many subtleties of logic that give rise to special symbolic situations, ambiguities and strange mathematical devices, as well as a legion of those differences that Charles Pierce was able to identify. But the main lines of logical structure in all relations of meaning are those which I have just discussed: the correlation of signs with their meanings through a selective mental process; the relation of symbols to concepts and concepts to objects, which give rise to a shorthand relationship between names and objects, known as denotation; and the assignment of carefully formed symbols to certain analogies in experience, the basis of all interpretation and thought. In fact, there are relationships that we use in weaving the inner web of meaning that is the actual fabric of human life.

Further, no special logical symbols are used. Given, however, that the reader may have to read books in which such symbols are used, we will give as an example the main, most frequently used logical symbols.

For more than two thousand years, traditional logic has used ordinary language to describe thinking. Only in the 19th century the idea was gradually established that for the purposes of logic, a special artificial language is needed, built according to strictly formulated rules. This language is not intended for communication. It should serve only one task - to reveal the logical connections of our thoughts, but this task should be solved with the utmost efficiency.

The principles of constructing an artificial logical language are well developed in modern logic. Its creation had about the same significance in the field of thinking for the technique of logical inference, which in the field of production had the transition from manual labor to mechanized labor.

A language specially created for the purposes of logic is called formalized. The words of the ordinary language are replaced in it by individual letters and various special characters. A formalized language is a "thoroughly symbolic" language in which there is not a single word of ordinary language. In a formalized language, meaningful expressions are replaced by letters, and as logical symbols

(logical constants) symbols with a strictly defined meaning are used.

In the logical literature, various notation systems are used, so two or more variants of symbols are given below.

Signs that serve to indicate negation; read: “not”, “it is not true”;

Signs for designating a logical connective called a conjunction; read: "and";

A sign to denote a logical connective called a non-exclusive disjunction; read: "or";

A sign to denote a strict, or exclusive, disjunction; read: "either, or";

Signs to indicate implication; read: "if, then";

Signs to indicate the equivalence of statements; read: "if and only if";

General quantifier; it reads: “for everyone”, “everyone”;

Existence quantifier; read "exists", "there is at least one";

L, N, - signs to designate the modal operator of necessity; read: "it is necessary that";

M - a sign for designation of the modal operator of possibility; reads: "possibly".

Along with the above, other specific symbols are used in diverse systems of logic, and each time it is explained what exactly this or that symbol means and how it is read.

As punctuation marks in artificial languages of logic, brackets are used, as in the language of mathematics.

Let's take, for example, some meaningful statements and give a side by side their record in the language of logic:

A) "He who thinks clearly, speaks clearly" -; the letter A denotes the statement “A person thinks clearly”, B - the statement “A person speaks clearly”, - a bunch of “if, then”;

B) "He is an educated person and it is not true that he is not familiar with Shakespeare's sonnets" -; A - the statement "He is an educated person", B - "He is not familiar with Shakespeare's sonnets", - a bunch of "and",

C) “If light has a wave nature, then when it is presented as a stream of particles (corpuscles), an error is made” -

; A - “Light has a wave nature”, B - “Light is represented as a stream of particles”, C - “A mistake is made”;

D) “If you were in Paris, then you saw the Louvre or saw the Eiffel Tower” - “You were in Paris”, B - “You saw the Louvre”, C - “You saw the Eiffel Tower”;

4. Logical symbolism

E) “If a substance is heated, it will melt or evaporate, but it can also explode” - (A ^ (B v C v D)); A - “The substance heats up”, B - “The substance melts”, C - “The substance evaporates”, D - “The substance explodes”.

Let us give one more simple example of the transition from an artificial language of logic to an ordinary language. Let variable A represent the statement "Darwin's theory is scientific", B - "Darwin's theory can be confirmed by experimental data", C - "Darwin's theory can be refuted by experimental data". What meaningful statements are expressed by formulas:

A) A ^ (B ^ C);