Existential graph
Existential Graphs ( the German translations of " existential graphs" and " existence graphs" are not very common) are a logical system of the American logician and philosopher Charles Sanders Peirce. They include both its own graphical notation (notation ) for logical statements as well as a logical calculus, that is (essentially) a formal system of rules of inference with which existing statements can be reshaped so that the construction of new statements from follow the former.
- 3.1 Notation of Beta Graphs
- 3.2 rules of inference of the Beta graphs
- 3.3 Further Examples
- 5.1 Primary Sources
- 5.2 secondary literature 5.2.1 monographs
- 5.2.2 Article
Introduction
Peirce felt the algebraic notation ( ie formula notation ) of the logic, especially that of his lifetime still very new, much co-developed by himself predicate logic as unsatisfactory, because the symbols zukomme philosophical significance by mere convention. In contrast, he aspired to a notation in which the characters literally carry their meaning in itself - in the terminology of his theory of signs: a system of iconic characters that are similar or identical to the designated objects and relations.
Such was the development of an iconic, graphic and - as he intended - so intuitive and easy to learn logical system, a project that Peirce busy life. After at least an aborted approach - the " entitative Graphs " - finally emerged from 1896 to the closed system of Existential Graphs. Although considered by its creator as a clearly superior and more intuitive system than they were, then as calculus without major influence on the history of logic; returned the one hand that Peirce on this subject but little publicized and published texts were not written very understandable; and on the other hand, the fact that the linear formula notation in the hands of professionals is the less expensive manageable tool. Thus, the Existential Graphs were little noticed or considered unwieldy notation.
For a better understanding led from 1963 works by Don D. Roberts and J. Jay Zeman, in which Peirce's graphical systems have been systematically investigated and presented. A practical role but now only a modern application, introduced by John F. Sowa 1976 term graphs that are used in computer science for knowledge representation. As a research subject, the Existential Graphs occur in the context of a growing interest in graphical logic again propagated in appearance, which is also reflected in attempts to replace the specified Peirce's inference rules through intuitive.
The overall system of Existential Graphs consists of three consecutive sub- systems, the alpha graphs, the graph beta and gamma graph. The Alpha graphs are a purely logical system statements. In building it emerge as the true extension beta graphs, a predicate logic of first-stage system. The still not fully explored, and Peirce's unfinished gamma graph to be understood as a development of alpha and beta graphs. With appropriate interpretation of the gamma graph predicate logic and modal logic cover higher level. Even in 1903 Peirce began with a new approach, the " Existential Graphs tinctured, " with whom he wanted to replace the existing systems of the alpha, beta and gamma graphs and combine their expressiveness and performance in a single new system. As the gamma graph remained the " Existential Graphs tinctured " unfinished.
When calculi the alpha and beta graphs are both correct ( ie all deductible as an alpha or beta graph expressions are statements or predicate logic semantically valid ) and complete ( ie all statements or predicate logic semantically valid expressions can be derived as an alpha or beta graphs).
The choice of the term " Existential Graphs " Peirce on the grounds that the simplest meaningful and well-formed beta graph meets an existence statement. Peirce used this term for the first time the end of 1897; earlier, he speaks of " positive logical graphs " or simply from his system of logic diagrams.
The present article deals with the alpha and beta graph as the consummate and most researched part of Peirce's system. Any additional information the offer referred to in the bibliography works.
Alpha graphs
Notation of Alpha graphs
Atomic propositions, ie those statements that are not themselves composed of other statements that are - as usual in propositional logic - expressed by letters; For example, the atomic statement " It is raining " with the letter "P" are expressed. The conjunction of several - atomic or non- atomic - statements is expressed by their juxtaposition writing. To say that two statements P and Q are true, therefore, to write " PQ ".
In addition, the system comprises the conjunction negation. It is expressed by the negative to expression - whether simple or compound - is surrounded by a closed line, so to speak, " eingeringelt " is. Special requirements to the shape of the trace are not made , but it is common to use a circle or an oval. The closed line, which denies a statement, Peirce calls the Cut (literally cut). Figurative background of the cuts is that on the sheet of paper on which is written - the sheet of assertion, assumption sheet - which are written as true statements adopted. False statements must be excluded, separated from the realm of true statements that are "cut off", and it is this function takes the cut.
To express a conditional, i.e., to tell that statement P is a sufficient condition for a statement Q, a spelling is selected as the " P scrolls Q ", "P curls Q a 'in English is called the statement Q, ie, the conditional sentence is within one 's own cuts with his condition, the statement P, in a second, outer cut ( see figure, point c1 and c2). This notation is atomic introduced in the logical system of Existential Graphs, but in the knowledge that it is the cut to the negation and the juxtaposition of writing is the conjunction, easily be reconciled with the truth conditions of these two links in coverage: The Conditional, P → Q is equivalent to the negation ¬ (P ∧ ¬ Q), and this is precisely the statement of the "P scrolls Q" which defines precisely the case of the true P and Q from the wrong sheet of assertion, " cut out ".
The disjunction is expressed by the two disjoint - set, each in individual cuts - side by side written and with an additional external cut. It is easily seen that this spelling in modern notation is the statement ¬ (¬ P ∧ ¬ Q), a statement that is P ∨ Q equivalent. Figurative background is again that the disjunction of the case from the sheet of assertion exclude that both P and Q are false.
With the two links of the Alpha graphs of negation ( the cut ) and the AND operation ( writing several statements on the acceptance sheet ) can be - as for the conditional and disjunction has been shown by way of example - represent all the other links of the two-valued propositional logic (see functional completeness of connectives ). Alpha graphs are thus a full notation for the propositional logic.
If statements are to be processed in the notation of Alpha graphs of electronic computing equipment, or simply play with word processing systems or earlier typewriters, often remedy is to thus express the Cuts by parantheses. Instead of drawing a closed polyline around the set P, to write in this case ( P). The conditional, "P scrolls Q", is in this notation to (P ( Q) ). For typographical reasons, the procedure in this article so.
Final rules of Alpha graphs
In order to formulate the rules, it is first necessary to level the concept of a statement ( in the literature: " proposition level" ) to define. The level of a - elementary or composite - statement is defined as the number of cuts, of which this statement is enclosed directly or indirectly. For example, in the expression ( P ( Q) ) of the plane P and that of (Q ) 1, because both P and (Q) are only part of the outer cuts. The level of Q, however, is 2, because Q is surrounded not only directly from a cut, but this in turn is part of the outer cuts.
Having said this, the rules of inference can be stated as follows:
Beta graphs
The beta graphs are the predicate logic system of Existential Graphs. Expand the system of Alpha graphs to the language means the identity line ( "line of identity" ) and generalize the existing rules of inference.
The atomic expressions are in the Beta graphs no longer statement letters (P, Q, R, ... ) or statements ( " It's raining, " " Peirce died in poverty "), but predicates in the sense of predicate logic (for details see there), possibly abbreviated to predicate letters ( F, G, H, ...). A predicate in the sense of predicate logic is a sequence of words with clearly defined spaces, which is a declarative sentence, if one uses a proper name in any blank space. For example, the phrase " _ died in poverty " is a predicate, because it the declarative sentence " Peirce died in poverty " arises when one enters the proper name " Peirce " in the blank space. Likewise, the words " _1 _2 is richer than " a predicate, because from it the statement " Socrates is richer than Plato " arises when you move into the vacancies the proper name " Socrates " and " Plato " begins.
Notation of the Beta graphs
The basic language elements is the identity line ( "line of identity" ), a thick line of any shape. The identity line locks onto the empty space of a predicate to show that the predicate is true to at least an individual. To express that the predicate "_ is a man" at least one individual is true - that is to say that there are people ( one at least ) -, to write thus a line of identity in the space of the predicate "_ is a human being ,"
Connects an identity line two or more spaces - whether different predicates or predicate of the same - then it expresses that there is at least one individual who - written in the respective space - each of these predicates makes both true. A simple example is followed by beta graph. In this graph, the line of identity expresses that there is at least one object that both the predicate "_ is American " and the predicate "_ died in poverty " at the same time fulfilled - in other words, that there is at least one American, who in poverty died.
From this beta graphs clearly distinguish one has the following, which is composed according to the rules of the alpha graph:
In this case, there are two mutually written individual Beta graphs. The upper part of graph indicates that at least one individual, the predicate "_ is Americans ' are satisfied, that is, that there are Americans. The lower part of graph says analog out that at least one individual, the predicate "_ died in poverty 'are satisfied, namely that an individual died in poverty at least. Two Beta graphs to write beside each other or one another means following the rules of Alpha graphs to testify the truth of both. The combined graph indicates, therefore, that there is at least an American and that at least one individual died in poverty - but he does not claim that the individuals to which this is true, a predicate, also applies the other predicate.
By a suitable combination of the line of identity with the known propositional means of Alpha graphs can already almost all predicate logic statements formulated.
A simple case is the negation of an existential statement. In the following example, the statement of the first example, that is the statement that there are people in the negative, by being written within a cut. Thus, it is stated that it is not the case that there are people - more beautiful in German: that there are no people.
From this graph differs from the following, in which the identity line seems to stand out from the Cut:
After the reading of the Beta graphs here is the connection of two graphs before: One outer, empty identity line that says simply: "Something exists ," and a line of identity within the Cuts, which says for itself: It is not the case that there is at least one individual is that the predicate "_ is a man" met. The connection between the two lines in the point in which they intersect with the cut, pressed the identity of the subject individuals: "It is something and that something is not human. " So pushes above beta Graph nothing else than the statement that there are things out there that are not human.
Just as well, can be an identity line lying within a cut combine with an outer line of identity, which in turn binds to a predicate. The following graph is an example of this configuration. Taken alone, the cut says: It is not the case that there is an individual who died in poverty; and, taken alone, indicates an outward expression that there is at least one individual who is an American. Since both expressions identity lines touch each other in the cut, the overall expression expresses the identity of both individuals, so says: There is at least one American who did not die in poverty.
A universal statement of the type " All pigs are pink " would be represented by a beta graph of the following type. Literally, these are the negation of a sentence of the type of the preceding example, specifically to the negation of "There is at least one pig that is not pink. " To deny that there are non- pink pigs, now mean to testify, that all pigs are actually pink.
Is a line of identity, such as selected in the following example, with a blank cut, pushes the non- identity of the individuals who satisfy those spaces in which to dock the line of identity. In this sense, for example in this indicates that there are at least a pig and that there is at least one pink individual, both are not identical.
Analogy with the preceding expresses the following beta graph, there are at least two pigs: "There is a pig, and there are ( still ) pig, which is not identical with the former. "
Inference rules of the Beta graphs
In the system of the Beta graphs no genuine predicate logic inference rules are added, but it can be modified the existing rules. Specifically, the well-known inference rules obtained thus following new wording: