Conflicting statements and arguments are the objects of research of logic as a science, and special mechanisms exist that are designed to determine connections among the individual components of various hypotheses and their veracity. One of such tools is the square of opposition that is a common technique for analyzing judgments and interpreting them. As the key components of this scheme, particular and general arguments are applied, and this method allows correlating basic data and drawing the right conclusions.
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Categorical Logic and the Square of Opposition
In contrast with formal logic, categorical one implies dividing judgments and hypotheses into separate segments, or categories. According to Jacquette (2016), one of the examples of such interrelations is the square of opposition, a mechanism that is believed to have been developed by Aristotle. Its purpose is to demonstrate the scheme of the mutual relation of four-type claims. Jacquette (2016) notes the main features of this principle of statement interaction and argues that the edges of the square allow identifying the relationship between simple judgments, including both compatibility and contradiction. The sides and diagonals symbolize those logical connections that arise between the parts of assumptions. As a result, when placing the signs of quality and the number of judgments on the vertices of the square, one can note the principles of hypotheses intersection and their relationship.
The Components of the Square of Opposition
The vertices of the square indicate the type of judgment by the combined classification of A, E, O, and I. Murinová and Novák (2016) decode these components and give their meaning in the context of categorical logic. The upper corner on the left is marked with the letter A – a sign of universally affirmative statements. The upper corner on the right is marked with the letter E – a sign of universally negative statements.
The lower corner on the left is marked with the letter I – a sign of particular affirmatives, and the lower corner on the right is marked with the letter O – a sign of particular negative judgments. When placing the signs of quality and the number of statements on the vertices of the square, one can note that the sides of the square AI and EO represent the relations of submission.
The Example of Using the Square of Opposition
When citing an example of using the square of opposition in practice, one can utilize a neutral judgment and determine how its components relate based on the scheme. The following example from legal practice is the true statement (A): all socio-economic systems have certain forms of ownership. Based on this claim, one can apply the aforementioned scheme and conclude by using specific signs:
- E – No socio-economic system has certain forms of ownership – a false judgment.
- I – Some socio-economic systems have certain forms of ownership – a true statement.
- O – Some socio-economic systems do not have certain forms of ownership – a false statement.
The presented example allows concluding the proposed judgment and interpreting possible claims. Since proposition A is universally affirmative, E, respectively, emphasizes the opposite. Concerning particular judgments of I and O, they are dependent on universal ones and are formed by the same principle. Therefore, the correlation of judgments monitored through the square of opposition is obvious, which indicates the usefulness of this scheme in categorical logic.
The square of opposition as a logical mechanism for evaluating the truth and falsity of judgments is a valuable scheme that allows determining the relationship of dependence and contradiction among individual judgments. The conclusions obtained through the use of categorical logic are the result of analytical work with segmental statements utilized. The example of applying the square of opposition provides an opportunity to determine how claims can depend on and oppose one another.
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Jacquette, D. (2016). Subalternation and existence presuppositions in an unconventionally formalized canonical square of opposition. Logica Universalis, 10(2-3), 191-213. doi:10.1007/s11787-016-0147-y
Murinová, P., & Novák, V. (2016). Syllogisms and 5-square of opposition with intermediate quantifiers in fuzzy natural logic. Logica Universalis, 10(2-3), 339-357. doi:10.1007/s11787-016-0146-z