Can the Laws of Form Represent Syllogisms?

摘要

The laws of form were created to represent propositional logic and Boolean algebra. But Spencer-Brown (and later commentators) also claimed that it could represent Aristotelian syllogistic logic although, as he showed in his book, at least one invalid syllogism appeared to be valid. This paper explores the extent to which the laws of form can correctly deal with all syllogisms. There are in fact 256 possible syllogisms and only fifteen of them are uncontroversially valid (a further nine are valid if certain existence assumptions are made). Using truth tables implemented in a spread sheet, all 256 syllogisms were evaluated and it was discovered that, in fact, 83 invalid syllogisms appear to be valid when simply represented in laws of form notation (in the primary algebra) and ignoring Spencer-Brown's interpretative theorem 2, an issue that will be explored in detail in the paper. This is clearly a significant number. Further investigation show that the problem might be caused by the way that "some/some not" propositions are conventionally represented and a variety of alternative are explored, some related to free logic. One particular interpretation reduces the number of wrongly categorised syllogisms to only seventeen and, surprisingly, fifteen of the seventeen are mirror images of the fifteen valid ones.