Transplicature

In his contribution on partial logic to the Handbook of Philosophical Logic [1], Stephen Blamey introduces a ‘value gap introducing’ connective named ‘transplication’ (/) to the standard 3-valued partial logic, the Strong Kleene logic. Where t stands for ‘true’, f stands for ‘false’ and n stands for ‘neither true nor false’, the truth table for this connective is: / 1 n 0 1 1 n 0 n n n n 0 n n n Blamey suggests the possibility of reading the transplication connective as a type of conditional. Basically, the idea is that conditional sentences of the form ‘if A then B’ are neither true nor false when A is false. They are also neither true nor false when either A or B is neither true nor false. I was interested to see how the transplication connective fares as a conditional by testing it against a list of inferences concerning conditionals. Here are the results: (1) q p/q × (2) ¬p p/q × (3) (p ∧ q)/r (p/r) ∨ (q/r) √ (4) (p/q) ∧ (r/s) (p/s) ∨ (r/q) √ (5) ¬(p/q) p √ (6) p/r (p ∧ q)/r × (7) p/q, q/r p/r √ (8) p/q ¬q/¬p × (9) p/(q ∨ ¬q) × (10) (p ∧ ¬p)/q × Paraconsistent Transplication What would the transplication connective look like when added to the 3-valued LP (Logic of Paradox), which treats the third truth value b as both true and false. Well, to begin with, application of the transplication connective’s behaviour to the truth value b forces a step outside of the 3-valued system into a 4-valued system, with truth values n (again neither true nor false) plus b. This transplication connective thus finds a home in the many-valued logic FDE (First Degree Entailment) system. The truth table for this connective is: / 1 b n 0 1 1 b n 0 b 1 b n 0 n n n n n 0 n n n n 1 Here are the results for the transplication connective based on the logic FDE, which turns out to be the same as that for the transplication connective based on Strong Kleene logic: (1) q p/q × (2) ¬p p/q × (3) (p ∧ q)/r (p/r) ∨ (q/r) √ (4) (p/q) ∧ (r/s) (p/s) ∨ (r/q) √ (5) ¬(p/q) p √ (6) p/r (p ∧ q)/r × (7) p/q, q/r p/r √ (8) p/q ¬q/¬p × (9) p/(q ∨ ¬q) × (10) (p ∧ ¬p)/q × References [1] Blamey, Stephen. ‘Partial Logic’, In D. Gabbay and F. Guenthner, (eds.). Handbook of Philosophical Logic Volume III. Dordrecht, D. Reidel Publishing Company, 1986, pp. 1-70.