A Cube of Opposition for Predicate Logic

Abstract

The traditional square of opposition is generalized and extended to a cube of opposition covering and conveniently visualizing inter-sentential oppositions in relational syllogistic logic with the usual syllogistic logic sentences obtained as special cases. The cube comes about by considering Frege–Russell’s quantifier predicate logic with one relation comprising categorical syllogistic sentence forms. The relationships to Buridan’s octagon, to Aristotelian modal logic, and to Klein’s 4-group are discussed.

Graphic Abstract

The photo shows a prototype sculpture for the cube.

Appendix: Proof of Contraries

Let us take a closer look at the five pair of contraries identified above in the relational cube of opposition. Two sentences together with accompanying existential import sentences are contraries iff the collection of sentences is unsatisfiable.

Consider in turn the five contraries and their requirements for existential import. We proceed by rewriting the considered sentences to logical clauses. A clause is a disjunction of zero, one more predicate-logical atomic formulas having only universally quantified variables. The case of zero formulas yields the unsatisfiable empty clause. Existentially quantified variables are eliminated by Skolemization. Unsatisfiability is ascertained by way of existence of a resolution proof leading tothe empty clause, cf. [22, 23]. Thus, such a successful formal proof entails that the clauses are unsatisfiable and hence contraries in such cases. An exhaustively failing attempt to construct a resolution proof entails that the clauses are satisfiable, since the resolution proof system is refutation-logically complete.

• 1. Theorem: The sentence forms $\forall \forall$ and $\forall \forall \lnot$ are contraries given existential import on both C and D.

  Proof: The $\forall \forall$ sentence $\forall x (Cx \rightarrow \forall (y Dy \rightarrow Rxy))$ and the $\forall \forall \lnot$ sentence $\forall x Cx \rightarrow \forall (y Dy \rightarrow \lnot Rxy)$ have the clausal forms

  $$\begin{aligned}&\lnot Cx \vee \lnot Dy \vee Rxy\\&\lnot Cx \vee \lnot Dy \vee \lnot Rxy \end{aligned}$$

  where x and y are implicitly universally quantified variables as usual in clausal form.

  The only achievable resolvent clause is $\lnot Cx \vee \lnot Dy$, where Rxy and $\lnot Rxy$ are cancelled out. This clause yields the empty clause with addition of existential import, only, that is $\exists x Cx$ together with $\exists x Dx$ (or, alternatively, $\exists x(Cx \wedge Dx)$), yielding the clauses $Ck_{1}$ and $Dk_{2}$, where $k_{1}$ and $k_{2}$ are Skolem constants. With these two clauses added the empty clause is readily achieved in two steps. Thus the opposition requires existential import on C as well as D.

• 2. Theorem: The sentence forms $\forall \exists$ and $\forall \forall \lnot$ are contraries given existential import on C.

  Proof: The sentence $\forall \exists$, that is $\forall x (Cx \rightarrow \exists (y Dy \wedge Rxy))$, is rewritten to $\lnot Cx \vee (Df(x) \wedge Rxf(x))$, where f is a one-argument Skolem function, in turn by a law of distribution giving the two clauses

  $$\begin{aligned}&\lnot Cx \vee Df(x) \\&\lnot Cx \vee Rxf(x) \end{aligned}$$

  The sentence $\forall \forall \lnot$, that is, $\forall x (Cx \rightarrow \forall (y Dy \rightarrow Rxy))$, have the clausal form

  $$\begin{aligned} \lnot Cx \vee \lnot Dy \vee \lnot Rxy \end{aligned}$$

  The pair of clauses $\lnot Cx \vee Df(x)$ and $\lnot Cx \vee \lnot Dy \vee \lnot Rxy$ has the resolvent clause $\lnot Cx \vee \lnot Rxf(x)$, giving further with the clause $\lnot Cx \vee Rxf(x)$ the resolvent clause $\lnot Cx$. By way of the existential import clause Ck, where k is a Skolem constant, one obtains the empty clause. Accordingly, in this case existential import on C is sufficient to achieve contrarity.

• 3. Theorem: The sentence forms $\forall \exists \lnot$ and $\forall \forall$ are contraries given existential import on C. Proof: This case is similar to the former case of $\forall \exists$ and $\forall \forall \lnot$. Again, hence, existential import on C is necessary and sufficient for contrarity.

• 4. Theorem: The sentence forms $\exists \forall \lnot$ and $\forall \forall$ are contraries given existential import on D. Proof: The sentence $\forall \forall$ yields the clause

  $$\begin{aligned} \lnot Cx \vee \lnot Dy \vee Rxy \end{aligned}$$

  The sentence form $\exists \forall \lnot$, that is, $\exists x (Cx \wedge \forall y (Dy \rightarrow \lnot Rxy))$ is rewritten to $Ck_{1} \wedge (\lnot Dy \vee \lnot Rk_{1}y)$, where $k_{1}$ is a Skolem constant, in turn giving rise to the two clauses

  $$\begin{aligned}&Ck_{1} \\&\lnot Dy \vee \lnot Rk_{1}y \end{aligned}$$

  The empty clause is achievable if only by assuming existential import on D, obtained by positing $\exists x Dx$, having the clause form $Dk_{2}$.

• 5. Theorem: The sentence forms $\exists \forall$ and $\forall \forall \lnot$ are contraries given existential import on D. Proof: The sentence $\forall \forall \lnot$ form yields the clause

  $$\begin{aligned} \lnot Cx \vee \lnot Dy \vee \lnot Rxy \end{aligned}$$

  The sentence form $\exists \forall$, that is $\exists x (Cx \wedge \forall y (Dy \rightarrow Rxy))$ is rewritten to $Ck_{1} \wedge (\lnot Dy \vee Rxf(y))$, where $k_{1}$ is a Skolem constant, in turn giving the two clauses

  $$\begin{aligned}&Ck_{1} \\&\lnot Dy \vee Rk_{1}y \end{aligned}$$

  Again, here, the empty clause is achievable only by assuming existential import on D, obtained by positing $\exists x Dx$, giving a clause $Dk_{2}$.

In summary, existential import is required in the cases where both quantifiers in a quantifier position are universal quantification $\forall$.