By Imre Ruzsa (auth.)

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Extra resources for Modal Logic with Descriptions

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Furthermore, either I involves some descriptions, or e pre-degenerates I at w (cf. 3), and the same holds for ''f&g(x/c)''. 3), e(3x(j& g(x/c»)w= 2. By these, (1) holds. g(x/c»w=2. Then, by (Val. g(x/c) at w. In the first case, I &g(x/c) involves some descriptions, and in the second case, e pre" degenerates "3x([ &g(x/c»)" at w, and in both cases, V(j&g(x/c»w=2 (for all V). Hence, e(3x(j &g(x/c»))w=2. g(x/c»w=2. By these, (1) holds true. (c) None of (a), (b) holds. ,e2. Hence, e does not pre-degenerate 3x(j & g(x/c») at w.

In the remaining case j= h, af-I'f-:Jh holds, since f-I'h-:Jh, by PC. If n> 1, and h is obtained from g and g-:Jh via {a,j}-relevant detachment (which means thatg is weakly relevantto {a,j, "'h}), then we finish the deduction as follows. j-:Jg-:Jh } (. j-:Jg III . ass. 3. (f-:J g-:J h) -:J (f-:Jg)-:Jj-:J h 4. j-:Jh (A3) (l-:J4= 3) (2-:J5=4) For line 4, the relevance condition holds obviously. For line 5: By the remark in parentheses above, g is weakly relevant to {a,f, ",h}, and hence,j-:Jg is weakly relevant to {a, "'(j-:Jh)} = {a,f & "-'h}.

References of form "k=j-::::Jn" in a deduction will mean that line n follows from lines k and j via restricted detachment. ;:~~~ } (assumptions) 3. (f-::::Jg)-::::J(g-::::Jh)-::::Jj-::::Jh 4. j-::::Jh (by PC) (1-::::J4=3) (2-::::J5=4) To get line 4 from I and 3,J-::::Jg must be weakly relevant to rv «g-::::J h)-::::Jj-::::J h). Since j and g are Boolean components of the latter sentence, this condition holds trivially. , to j & rv h. Since h is a Boolean component ofj & rv h, the assumption that g is weakly relevant to j & rv h suffices.

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