The Connectives

Index

Index #-boolean valuation (# = ∧, ∨, ¬, etc.), 65–71 #-classical (# = ∧, ∨, ¬, etc.), see classicality #b (truth-function associated with # = ∧, ¬, etc., on boolean valuations), 387, 403, 620 & (special conjunction-like notions), 346, 662, 669, 702, 906  (modal operator), 276 , see Church negation Λ, 1182–1185 Ω Porte’s constant for the Ł-modal system, 484 temporary use as a 1-ary connective , 1080  closure relation in the semantics of orthologic, 919 gcr (for a contrast with consequence relations), 392, 844 propositional entailment in Gärdenfors semantics, 643 structural completion of a consequence relation, 163, 179 supervenience determined consequence relation, 1142 !A! (set of valuations, or set of points in a model, verifying A), 138, 279, 643, 811, 895  (sequent separator), 103, 188 “” notation, 355  (IL nand), 1241  component switching operation on formulas, 716, 736 tuple splicing operation, 40 ◦ composition of functions, 9 fusion (or multiplicative conjunction ), 147, 345, 347 hybrid connective (conjunction and disjunction), 469 relative product, 501, 732 various other connectives, 256, 479, 1148 (deductive disjunction), 50, 420, 555, 892  notation, 63 ... ∨ (Church disjunction), 235¨ ∨ (pseudo-disjunction, q.v.), 235, 555, 1320 δ Łukasiewicz’s variable functor notation , 1157 Blok–Pigozzi δ(t), ε(t) notation, 258 superscripted to denote the dual of a truth function, 405 ˙ ∪ (closure of union), 10 κ, 1112–1114 connective to form contraries, 850 converse-forming connective in BCIA logic, 1112 λ, see lambda notation, see lambda ∧b, ∨b, etc. (#b for boolean #), 403 ↔, 83 biconditional connective (see biconditional , equivalence), 48 ↔m (multiplicative biconditional), 1130  -based algebraic semantics, 246– 250 arbitrary partial order (with converse ), 2 ordering of truth-values, 621 partial ordering of valuations, 138, 436, 750, 914, 1173 1440 INDEX 1441 ∨* (theorem disjunction or upper bound for theorems), 573|=|=M with M a matrix, 203 “|=” used for semantic consequence relations, 57 truth relation for equational logic, 32 truth relation for first order logic, 36 truth relation in Kripke semantics for IL, 307 truth relation in Kripke semantics for modal logic, 279 truth relation in Urquhart semantics for relevant logic, 337 μ μBCI (monothetic BCI ), 1108 multiplicity function, 373 temporary use in connection with probability, 657 (linear implication), 345, 349, 669¬c, ¬i (cohabiting intuitionistic and classical negation), 585¬ complementation, 21 negation connective (see negation ), xvi, 48¬d (dual intuitionistic negation), 92, 1222 ω, xv ⊕ (temporary notation for a variant disjunction), 835 ⊗ combined matrix evaluations, 216 direct products of algebras or matrices, 28, 212 Girard’s notion for multiplicative conjunction, 345 π (binary connective for IL), 1068, 1076  (pre-order), 2 L, 14, 15 R, 14 σ connective to form subcontraries, 850 variable over sequents, 120 ( (weak disjunction), 887  and , xiv, 275 , 276 ⊃ additive implication in linear logic, 349 enthymematic implication defined in relevant logic, 1098 material implication as a new primitive in relevant logic, 1260 material implication defined in relevant logic, 327 Meredith’s simulation of classical implication in terms of intuitionistic implication, 335, 1079, 1272 Sasaki hook, 302, 1192 • (ternary connective for IL), 1068 →, 83 Gentzen’s use of, as a sequent separator, 103 implication connective (see conditionals , implication), xvi, 48 relative pseudocomplement in Heyting algebras, 22 →-intuitionistic (consequence relations ), see intuitionistic logic →d (dual intuitionistic implication), 546 L (disjunctive combination on the left), 12 R (disjunction combination on the right), 12 L (conjunctive combination on the left), 12 R (conjunctive combination on the right), 12  consequence relation, 55 generalized consequence relation, 73 · (special consequence relation de- fined in 5.34), 744 DD (Double Disjunction), 64, 600 IL, CL, etc., see intuitionistic logic, classical logic, etc. P P (probability-preserving consequence relation), 652 P V (pair-validity consequence), 747 1442 INDEX SCI, see SCI KK (minimal bimodal inferential consequence relation), 744 K, S4, etc. (inferential consequence relations associated with K, S4, etc.), 294 Suszko (Suszko consequence relation), 204, 208 ∨ disjunction connective (see disjunction ), xvi, 48 lattice join, 7 ∨Kr, ∨Be (cohabiting Kripke and Beth disjunction), 897 ∨c, ∨q (cohabiting classical and quantum disjunction), 587 ∧ conjunction connective (see conjunction ), xvi, 48 lattice meet, 7 ∧∨, ∨∧, etc. (product connectives), 464–468 ℘, 2  (exclusive disjunction), 398, 780 (∧ Left), (¬ Right), (→E) etc., see rules, (∧ Left) etc. .2 (modal principle), 277, 298 .3 (modal principle), 277 → (would counterfactual), 1008 →(might counterfactual), 1008 “!” notation (modal logic), see also exponentials, 278§ (demi-negation), 536, 576 1, 2 (two constants for pair validity ), 760 4 (modal principle), 277, 284 5 (modal principle), 277, 298 A (alternator), 890 a (anticipation connective), 625 AAL, see abstract algebraic logic Abbott, J. C., 237, 238, 241, 274, 275 Abelian groups, see groups, Abelian logic (see also BCIA...