Logical Squares for Classical Logic Sentences

In this paper, with reference to relationships of the traditional square of opposition, we establish all the relations of the square of opposition between complex sentences built from the 16 binary and four unary propositional connectives of the classical propositional calculus (CPC). We illustrate them by means of many squares of opposition and, corresponding to them—octagons, hexagons or other geometrical objects.


Introduction: Basic Definitions
For any sentences α, β, ϕ , ψ of CPC we assume the following definitions: α, β are contrary iff α/β is a tautology of CPC, where the stroke/is the Sheffer's connective; ϕ , ψ are subcontrary iff ϕ ∨ ψ is a tautology of CPC, where ∨ is the disjunction connective; α entails ϕ iff α → ϕ is a tautology of CPC, where → is the implication connective; α and ψ are contradictory iff (α ∧ ψ) ∨ (∼ α ∧∼ ψ) is a counter-tautology of CPC, i.e. α and ψ never agree in truth-values. We will illustrate the above relationships between sentences α, β, ϕ , ψ in a square of opposition graphically in a non-standard way 1 by means of Fig. 1, where the dotted lines indicate contradictory sentences and the downward arrows the implication. Among the 16 binary classical sentence-forming connectives there are only four for which sentences that are built by means of them are true only in one case (see Table 1). They are the following: both . . . and. . . , . . . unless. . . , not. . . because. . . , neither . . . nor. . . (binegation ).
We define them by means of variables p, q and the classical connectives: the conjunction ∧ (or implication →) and negation ∼ as follows: both p and q = df p ∧ q; p unless q = df ∼(p → q) ≡ p ∧ ∼q; not p because q (or not p though q) = df ∼ (q → p) ≡ ∼p ∧ q; neither p nor q (p q) = df ∼ p ∧ ∼q.
The third connective is also called the dual implication: d(p → q) = df ∼ (q → p) (see [16,17]) and the last one is known as binegation .
The truth-value table for these connectives is the following: The above conjunctive sentences (conjunctions) are pairwise contrary, i.e. the Sheffer's disjunction of two sentences of each of the six pairs of the above 2 These connectives of natural language fulfill in sentences not only a logical, descriptive (communicative) role but also an expressive one (expressing psychical states of a speaker). Defining these connectives by means of well-known logical connectives of CPC, we omit of course their expressive functions in composed propositions.
Vol. 10 (2016) Logical Squares for Classical Logic Sentences 295 conjunctions is true (is a tautology), so the sentences can never both be true, but can both be false.
To each of the 6 pairs of contrary conjunctions from the following: (i) p ∧ q, p ∧ ∼q, ∼p ∧ q, ∼p ∧ ∼q there exists a pair of contradictory implications from the following pairwise subcontrary implications: (ii) p → q, p → ∼q, ∼p → q, ∼p → ∼q so the sentences that can both be true, but cannot both be false, i.e. the disjunction of which is true. The implications (ii), of course, are true in three cases depending on the truth value of sentences p and q. Each pair of (ii), together with the suitable pair of contrary conjunctions of (i), creates one of the 6 squares of opposition for complex sentences of classical logic. The squares are given below (see  To each of the 6 pairs of contrary conjunctions from the following: (i) p ∧ q, p ∧ ∼q, ∼p ∧ q, ∼p ∧ ∼q there exists a pair of contradictory Sheffer's disjunctions (denial alternatives) from the following pairwise subcontrary Sheffer's disjunctions, i.e. the disjunction of which is true: (iii) p/q, p/∼q, ∼p/q, ∼p/∼q. Each pair of (iii) together with the suitable pair of conjunctions of (i), creates one of the six squares of opposition for complex sentences of classical logic (see Squares 1 -6 ). 3 To each of the six pairs of contrary conjunctions from (i) there exists also a pair of contradictory disjunctions from the following pairwise subcontrary disjunctions, i.e. the disjunction of which is true: Each pair of (iv), juxtaposed with the suitable pair of conjunctions from (i), creates also one of the six the squares of opposition for complex classical sentences (see Squares 1 -6 ). 4 Squares 1 -6 and 1 -6 are given below: As it turns out, at least one of them, Square 3 : p q p q p q p q / was known much earlier (see [4,7,[11][12][13]19,21], 5 [8]. So, if we have, for example, p ≈ John is a scientist, q ≈ John is a priest, then their conjunction: John is a scientist and John is a priest is contrary to their binegation: neither John is a scientist nor John is a priest; their disjunction: John is a scientist or John is a priest is subcontrary to the disjunction of their negations; their conjunction is contradictory to the disjunction of their negations and their binegation is contradictory to their disjunction; 298 U. Wybraniec-Skardowska Log. Univers.
from their conjunction f ollows their disjunction and from their binegation f ollows disjunction of their negations.
For Squares 1-6 and 1 -6 we obtain Octagon 1 and Octagon 2 in which disjunctions of (iv) are either replaced by equivalent implications from (ii) or by equivalent Sheffer's disjunctions from (iii), respectively.
Among the 16 connectives of the set F16 of the all binary connectives of classical logic we may find six which form true sentences in two cases and one which forms a true sentence in four cases. We will consider them in the next sections.
We see that sentences in the same line on the right side and on the left side are contradictory.
The connective ⊥ (∨∨, ∨) is well known as the strong or exclusive disjunction connective. The sentence: even if p, q (p | > q) can be read: even if p then q, and the sentence: even if ∼p, ∼q (∼p | > ∼ q) can be read: even if not p then not q.
The truth-value table for the connectives < |, | > , ≡ and ⊥ is the following (see Table 2) 6 : Table 2. The truth-value table for connectives: < |, | >, ≡, ⊥ p q p < | q ∼p < | ∼q p | > q ∼p | > ∼q p ≡ q p ⊥ q In every two successive columns with composed propositions we have two contradictory sentences. For each pair of contradictory sentences in Table 2 we can build four squares of opposition and two hexagons of opposition corresponding to them.
The idea of constructing hexagons built from the squares of oppositions (rectangles of opposition corresponding to them) differs from the main idea of Blanché's hexagon (1966) presented clearly by Béziau [1] and based on putting together two triangles of opposition: the triangle of contrariety and the triangle of subcontrariety.

3.2.
For p < | q and ∼p < | ∼q (∼p < | q) we have Squares 7 and 8 and Hexagon 1 (in which Square 6 is also illustrated) and Squares 7 and 8 and Hexagon 1 (in which Square 1 is illustrated p < |~ q p q / Let us return to the first square of the series of squares. Let us consider an example of using Square 7 with contradictory sentences p < | q and ∼p < | ∼q.
Let p be a sentence: He goes for a walk, and q a sentence: It rains. Then • the sentence p < | q : He goes for a walk even if it rainsis contrary to the sentence ∼ p ∧ q: He does not go for a walk because (and) it rains; • the sentence p < | q : He goes for a walk even if it rains is contradictory to the sentence ∼p < | ∼q: He does not go for a walk even if it does not rain; 302 U. Wybraniec-Skardowska Log. Univers.
• the sentence ∼ p ∧ q: He does not go for a walk because (and) it rains is contradictory to the sentence p ∨ ∼q: He goes for a walk or it does not rain; • the sentence p < | q: He goes for a walk even if it rains implies the sentence p ∨ ∼q: He goes for a walk or it does not rain; • the sentence ∼ p ∧ q: He does not go for a walk because (and)it rains implies the sentence ∼p < | ∼q: He does not go for a walk even if it does not rain; • the sentence p ∨ ∼q: He goes for a walk or it does not rain and the sentence ∼p < | ∼q: He does not go for a walk even if it does not rain are subcontrary.
Below we have Squares 9 and 10: p | > q p q / Let us consider an example of using Square 9 with contradictory sentences p | > q and ∼p | > ∼q. Let us recall that the sentence: p unless q = df ∼ (p → q) ≡ p ∧ ∼ q. And let p be a sentence: The reviewer rejected this paper, and q a sentence: It will be published.
Then • the sentence p | > q : Even if the reviewer rejected this paper, it will be published is contrary to the sentence p ∧ ∼ q (p unless q, ∼ (p → q)): It is not truth that if the reviewer rejected this paper it will be published; Vol. 10 (2016) Logical Squares for Classical Logic Sentences 303 • the sentence p | > q : Even if the reviewer rejected this paper, it will be published, is contradictory to the sentence ∼p | > ∼q: Even if the reviewer did not reject this paper, it will not be published; • the sentence p ∧ ∼ q: The reviewer rejected this paper unless it will be published is contradictory to the sentence p → q : If the reviewer rejected this paper then it will be published; • the sentence p | > q: Even if the reviewer rejected this paper, it will be published implies the sentence p → q: If the reviewer rejected this paper then it will be published; • the sentence p ∧ ∼ q: The reviewer rejected this paper unless it will be published implies the sentence ∼p | > ∼q: Even if the reviewer did not reject this paper, it will not be published; • the sentence p → q: If the reviewer rejected this paper then it will be published, and the sentence ∼p | > ∼q: Even if the reviewer did not reject this paper, it will not be published, are subcontrary. Below we present Squares 9 and 10 :

Square for Unary Connectives
The last squares correspond to the known square of opposition for unary connectives: assertion a, falsum f, verum v and negation ∼, which has the following form: v p p Square for unary connectives f p a p /

Conclusion
In Sects. 2-5 we were able to consider all the squares of opposition relationships for all sentences built from connectives of classical logic (CPC): 16 binary and 4 unary. 9 In Sect. 2 we showed that on the basis of four composed sentences in Table 1 we can built six basic squares of opposition with numbers 1-6 (so called "balanced" squares in literature). In Sect. 3 we showed that on the basis of Table 2 we can built 12 squares of opposition, so call "unbalanced" squares, with numbers 7-12 and 7 -12 (and 6 hexagons with numbers 1-3 and 1 -3 ). Moreover, it is also possible to generate three more "degenerate" squares (see Sect. 3.6).
Generally, we can say that on the basis of Tables 1 and 2, CPC contains 21 squares of opposition.
In Sect. 4 we considered additionally 14 squares of opposition for the binary connectives: verum and falsum. In Sect. 5 we also present one square of opposition for unary connectives of CPC.
As we could see, replacing categorical sentences in the traditional square of opposition with the suitable formulas of the classical propositional calculus (CPC) is a fully justified generalization of the idea of the logical square of opposition. This idea was known earlier ever since Blanché and Sauriol in the literature but based on other methods presented squares of oppositions for CPC.