Existential Import, Aristotelian Logic, and its Generalizations

Abstract

The paper uses the theory of generalized quantifiers to discuss existential import and its implications for Aristotelian logic, namely the square of opposition, conversions and the assertoric syllogistic, as well as for more recent generalizations to intermediate quantifiers like “most”. While this is a systematic discussion of the semantic background one should assume in order to obtain the inferences and oppositions Aristotle proposed, it also sheds some light on the interpretation of his writings. Moreover by applying tools from modern formal semantics to the investigation of classical Aristotelian logic and its extensions, we combine different approaches to the logic of quantification. We will present variants of quantifiers that are associated to the four corners of the square of opposition with and without existential import and discuss their role for the logical square, conversions and the syllogistic. It will turn out that there is no way to ascribe existential import that validates all inferences and relations which one is willing to hold in Aristotelian logic. Two options, however, provide reasonable results. Existential import should either be ascribed only to affirmative statements or only be ascribed to universal quantification. The former option is preferable for a mere reconstruction of the classical Aristotelian logic while the latter option is more attractive if Aristotelian logic is generalized to intermediate quantifiers.

Appendices

Appendix A. Conversions

I use Venn diagrams to illustrate conversions. As usual, empty sets are shaded and non-empty sets have a cross. ‘-X’ indicates that this subset of X is not empty if at least one element of X exists.

Appendix B. Syllogisms

1.1 B.1. First Figure

1.1.1 B.1.1. Barbara and Barbari

The syllogisms Barbara ($MaP, SaM\models SaP$) and its subaltern Barbari are invalid if the conclusion has existential import, but the minor premise has none. The same holds for Barbari. The existential import of the major premise has no consequences. The premises are illustrated in figure 1 with all-(S, M) and figure 2 with all+(S, M) as minor premise, respectively.

1.1.2 B.1.2. Celarent and Celaront

The syllogism Celarent ($MeP, SaM \models SeP$) is valid iff SaM has import or SeP has no import. The same holds for Celaront: It is invalid iff SoP is false for an empty subject term while SaM is vacuously true in this case. Again, the validity of Celarent and Celaront is not affected by the existential import of the major premise. Figure 3 illustrates the syllogisms with all-e(S, M) and figure 4 gives the versions for all+e(S, M).

1.1.3 B.1.3. Darii

Darii ($MaP, SiM \models SiP$) is valid if the conclusion has no import or if the minor premise has import; it is invalid otherwise. Hence, any coherent application of some+e and some-e yields a valid syllogism. Figure 5 gives Darii with some-e(S, M) and figure 6 illustrates Darii with some+e(S, M).

1.1.4 B.1.4. Ferio

The last syllogism of the first figure is Ferio ($MeP, SiM \models SoP$). If the conclusion is understood as notall+(S, P), then the minor premise must have existential import as well. The two Venn diagrams illustrate Ferio with some-e(S, M) in figure 7 and some+e(S, M) in figure 8.

1.2 B.2. Second Figure

1.2.1 B.2.1. Cesare and Cesaro

Cesare ($PeM, SaM\models SeP$) as well as its subaltern Cesaro are invalid if and only if the conclusion has import and the premise SaM is understood as all-e(S, M) as in figure 9. Figure 10 shows the stronger versions with all+e(S, M), where the conclusion might have existential import.

1.2.2 B.2.2. Camestres and Camestrop

Camestres ($PaM, SeM\models SeP$) and its subalternate Camestrop are invalid if SeP or SoP have import and SaM has none. The premises of the syllogism is illustrated with no-e(S, M) in figure 11 and with no+e(S, M) in figure 12.

1.2.3 B.2.3. Festino

Festino ($PeM, SiM\models SoP$) is valid whenever the conclusion requires no existential import where the minor premise has none. Hence, some-e(S, M) only entails notall-e(S, P) as in figure 13. Festino with some+e(S, M) is given in figure 14.

1.2.4 B.2.4. Baroco

Baroco ($PaM, SoM\models SoP$) holds whenever the conclusion is not more demanding with respect to existential import than the minor premise. It is illustrated in figure 15 with notall-e(S, M) and in figure 16 with notall+e(S, M).

1.3 B.3. Third Figure

1.3.1 B.3.1. Darapti

Darapti ($MaP, MaS \models SiP$) requires one premise with existential import. One must apply all+e to at least one premise in order to get a valid inference. Even a conclusion without existential import, i.e. some-e(S, P) requires that at least one premise has existential import. all-e$(M,P), \textsc {all-e}(M,S)\not \models \textsc {some-e}(S,P)$ is made clear in figure 17. The valid variants with all+e(M, P) or all+e(M, S) are illustrated in figure 18.

1.3.2 B.3.2. Disamis

Disamis ($MiP, MaS \models SiP$) also requires one premise with existential import for a valid inference. Figure 19 shows that Disamis is not valid for some-e(M, P) and all-e(M, S). Disamis with some+e(M, P) or all+e(M, S) is given in figure 20. As becomes apparent, the conclusion from these premises might be understood as some-e(S, P) or some+e(S, P).

1.3.3 B.3.3. Datisi

Datisi ($MaP, MiS \models SiP$) also requires one premise with existential import: all-e(M, P) and some-e(M, S), illustrated in figure 21, have no implications for S. If one of the premises has import, no matter which, then SiP can have existential import as well. The Venn diagram for Datisi with all+e(M, P) or some+e(M, S) is given in figure 22.

1.3.4 B.3.4. Felapton

Felapton ($MeP, MaS \models SoP$) requires one premise with existential import. The standard FOL interpretation is therefore invalid. In terms of the defined quantifiers: no-e(M, P), all-e$(M,S) \not \models \textsc {notall+e}(S,P)$ and no-e(M, P), all-e$(M,S) \not \models$notall-e(S, P) as becomes clear in figure 23. Assignments with no+e(M, P) or all+e(M, S) result in a valid Felapton as illustrated in figure 24.

1.3.5 B.3.5. Bocardo

Like all the other syllogisms of the third figure, Bocardo ($MoP, MaS \models SoP$) requires at least one premise with existential import. notall+e(M, P) or all+e(M, S) yield a valid Bocardo that is illustrated in figure 26. The invalid variant with notall-e(M, P) and all-e(M, S) is given in figure 25.

1.3.6 B.3.6. Ferison

Ferison ($MeP, MiS \models SoP$) can be seen as a logically stronger version of Felapton since it requires only a particular minor premise and leads to the same conclusion. Ferison is invalid if none of the premises has existential import: In figure 27 the premises are interpreted as no-e(M, P) and some-e(M, S). The Venn diagram for the valid variants with no+e(M, P) or some+e(M, S) is represented in figure 28.

1.4 B.4. Fourth Figure

1.4.1 Bramantip

Bramantip ($PaM,MaS \models SiP$) requires that the major premise is interpreted as all+e(P, M) as in figure 30. Given this requirement is fulfilled the conclusion may have existential import. Bramantip is invalid for all-e(P, M). The Venn diagram for all-e(P, M) and all-e(M, S) is seen in figure 29. The import of MaS is, however, not relevant for SiP: Even if $S\cap M$ is non-empty, its element may be outside of the intersection with P.

1.4.2 B.4.2. Camenes and Camenop

Camenes ($PaM, MeS \models SeP$) and its subalternate Camenop ($PaM, MeS \models SoP$) are valid if the conclusion is not required to have existential import. The premises do not need to have existential import. In figure 31, it becomes clear that all-e(P, M), no-e$(M,S) \models$no-e(S, P). The conclusion may not have existential import even if both premises have import. In the Venn diagram in figure 32 it is shown that all+e$(P,M), \textsc {no+e}(M,S) \not \models \textsc {no+e}(S,P)$.

1.4.3 B.4.3. Dimaris

Dimaris ($PiM, MaS \models SiP$) is valid for some+(P, M) as in figure 34 but is invalid for some-(P, M) as becomes apparant in figure 33. The existential import of MaS is irrelevant to the validity of Dimaris.

1.4.4 B.4.4. Fesapo

Feasapo ($PeM, MaS \models SoP$) is valid if the minor premise has existential import as demonstated in figure 36. Figure 35 shows that the syllogism is invalid for all-e(M, S).

1.4.5 B.4.5. Fresison

Fresison ($PeM, MaS \models SoP$) differs from Fesapo with respect to the minor premise which is only particular but allows for the same conclusion. Fresison is valid for all+e(M, S), as shown in the Venn diagram in figure 38. Figure 37 demonstrates that Fresison is invalid for all-e(M, S).