Harold T. Hodes

Some theorems on the expressive limitations of modal languages

Fix a set of predicates Pred, each of a particular number of places, and a countably infinite set of variables Var; for a set of names C form the language L(c) by following the usual formation rules, with the primitives ‘l’, ‘7, ‘Y’ and ‘0’. A term of L(c) is a member of Var U C. Use and mention shall be freely confused. Let 16, (3~)

Three-valued logics: An introduction, a comparison of various logical lexica, and some philosophical remarks

, Og, and Eu abbreviate 0 E 1, l(Vv)l#, lol@ and (3~) (v = u) where u is a term of L(C) and v is a variable distinct from a; ‘82, ‘v’ and ‘z are defined as usual. We work entirely within the modal logic S5. So we may take a frame to be a pair F = (IU, A), W a non-empty set, A a function on W so that .4(w) is a set for all w E IV, and U{A(w) 1 w E IV) = 1 is non-empty. An F-valuation for L(C) is a function V with domain C U (IV x Pred), V(c) E 1 for c E C; forwEWandPEPred,Pn-place:ifn>l, V(w,P)gZn;forn=O, V(w, P) E {t, f). A structure for L(C) is a triple J/= (W, A, v), Ya (W, A)valuation for L(C). For w E W, we say that w is from &, rf is an assignment fordiff d: Var +A. Let:

Jumping Through the Transfinite: The Master Code Hierarchy of Turing Degrees

Etude comparative des differents lexiques logiques en logique trivalente et caracterisation algebrique de leur puissance expressive

Individual-actualism and three-valued modal logics, part 2: Natural-deduction formalizations

Where a is a Turing degree and ξ is an ordinal 1 ) L1 , the result of performing ξ jumps on a, a (ξ) , is defined set-theoretically, using Jensens fine-structure results. This operation appears to be the natural extension through (ℵ 1 ) L1 of the ordinary jump operations. We describe this operation in more degree-theoretic terms, examine how much of it could be defined in degree-theoretic terms and compare it to the single jump operation.

On some concepts associated with finite cardinal numbers

5. FORMALIZING POSSIBILISTIC LOGICS BASED ON K A sequent calculus X may be viewed as a class-function assigning each appropriate language L = L, to a simultaneous inductive definition of two sets of sequents in L: Th s(L) = the theorems of z(L); WkTh z(L) = the weak theorems of z(L). The base-clauses of this definition shall be called axioms; the inductive clauses shall be rules. 5 will be sound relative to a given logic X iff for any appropriate L: all members of Th Ii(L) are X-valid; all members of WkTh J&L) are weakly X-valid. & will be complete relative to X iff for any appropriate L: all X-valid sequents of L belong to Th g(L); all weakly X-valid sequents of L belong to Wk Th z(L). Where g(L) is fixed, use these abbreviations: l-, A t 4 : (I-, A, 4) E Th s(L); I-, A t” 4 : (I-, A, 4) E Wk Th X(L). For I E A E fml(L), (I, A) is X(L)-inconsistent iff I, A b I; otherwise (I, A) is X(L)-consistent. Where x(L) is fixed, we’ll just write “consistent” or “inconsistent”. Notation: where @ E fml(L), let: 00 = {m#J:c

Axioms for actuality

EcP);O-‘@ = {&of

On modal logics which enrich first-order S5

Eq; define ~0 and 0-l CD similarly.

The composition of Fregean thoughts

I catalog several concepts associated with finite cardinals, and then invoke them to interpret and evaluate several passages in Rips et al.s target article. Like the literature it discusses, the article seems overly quick to ascribe the possession of certain concepts to children (and of set-theoretic concepts to non-mathematicians).