147
similarities too. As sickness comes along naturally with the release of bodily fluids, hence the witch doctor battles sickness by bleeding, because it is similar. The witch doctor has seen that they use injection needles in hospitals and he decides that these needles have great magical power. Thus he uses needles too and injects his patients with his own coctails, because this is similar. Similarity can be dangerous. Let us first find out about certainty before we continue with fuzziness.
5.4 Taxonomy of inference 5.4.1 Proof and being decided
We have a whole list to look at: proven, refuted, non-sequitur, decided, undecided, untenable, provable, decidable, and so on. We can devour these in small chunks using the medieval square of contrariness. Let S0 be a specific (constant) set of statements. We may even call it a system but you might also think of a book. Let us consider a statement p that (i) can already be contained in the system, (ii) contradicts it, or (iii) might be added without contradiction. The following gives a review of the possibilities. Let us first recall what we already saw in Chapter 2. † This table uses the concepts of contradiction and contrariness. Instead of one statement we now use a list of statements S0 . Note that there are no quantifiers so that we (must) use constants. These concepts apply to a single conclusion p. AffirmoNego@Ergo@S0 , pDD Proven@pD HS0 ¢ pL
õ à
‡
Contrary
Ÿ pN
õ
Refuted@pD HS0 ¢ Ÿ pL ‡
Not á
JS0
õ â ä
Subcontrary
NonSequitur@ŸpD
õ
JS0
pN
NonSequitur@pD
` è We should recall the threesome IP fi P fi PM. When S0 does not prove or refute p then è è there is the third possibility P = HS0 ¢ pL = JS0 pN fl JS0 Ÿ pN. Let us call this “undecided”. † A system proves p, refutes p, or leaves it undecided. Contrary@3, Ergo@S, pDD è JHS ¢ pL Í HS ¢ Ÿ pL Í HS ¢ pLN