ERRATA

Possibilities and Paradox: An Introduction to Modal and Many-Valued Logic

• Page 20, exercise 8.
              Delete last two words ("on X").
  
• Page 30, first paragraph and table.
              Replace "Celarent" by "Dimaris".
  
• Page 31, last sentence of section 3.5.
              Replace "When available" with "When the tableaux are finite".
  
• Page 40, ftn. 4.
              Replace "o" by "0".
  
• Page 55, exercise 2.
              Replace "Van Ormen" by "Van Orman".
• Page 56, third to last displayed line.
              Replace "(w'A)" by "(w',A)".
  
• Page 57

  "semantic consequence (or implication) is truth-preservation over all worlds of all LKmodels": this is to be understood as follows: X semantically implies A in language LK if and only if, for every model M of LK and every world w in that model, if M satisfiesX at w then M satisfies A at w.

• Page 73, second paragraph.
             The last occurrence of "w" should be bold and unscripted.
             (This paragraph is meant to introduce the convention of boldface unscripted font to indicate that
             the letter "w", with or without subscript, is introduced as a new world name.)
  
• Page 75, 1ine -7.
              After "sentences in b" add "and those where the formula starts with a box"
  
• Page 78,third to last line.
              The second 'w" should be bold, unscripted.
              In footnote 31, replace "-introduction" by "-elimination".
• Page 83, problem 5 of Exercises 5.6.1.
              Replace the second occurrence of the fishhook symbol by a double fishhook (to symbolize the biconditional).
  
• Page 90.
              After "BECKER’S RULE:" insert "For all sentences A and B,"
• Page 99,before the line " Proof-Theoretic Definitions".

  Insert "Finally, a branch is closed if and only if it contains two nodes in which the same sentence is accompanied by different value-markers, and a tree is closed iff all its branches are closed."

• Page 110, first paragraph of section 7.3, line 5.
              Replace "given tautology" by "given (classical) tautology".
  lines 9/10.

  Replace "yield the same language." by "yield essentially the same language, that is, languages with the same syntax and same semantic consequence relation."

• Page 118, end of page.
              Replace "St. Matthew, Matthew 19:21" by "St. Matthew, Matthew 19:26"
  
• Page 135, second last sentence.
              Replace "hence the latter is an extension of L" by "hence L is an extension of L^S"
  
• Page 137, in proof of Lemma 9.1.
              Replace "L_K3 happens to be an extension of L_CPL" by "L_CPL happens to be an extension of L_K3"
• Page 142: In the proof of Fact 9.2, last line.
              Replace "... which is just v(B)." with "... which is just 1 -v(B)."
  
• Page 151, second last line.
              Replace "The answer is" by "Intuitively, the answer is"
• Page 152, in first paragraph of the proof of Lemma 10.1.
              Replace "and if x bears R to y in N* then y belongs to N* too" by "and if x in N* bears R to y then y belongs to N* too"
  
• Page 158, exercise 6 of Exercises 10.5.1.
              Replace "exercise 3 of Exercises 3.6.1 (page 33)" by "exercise 3 of Exercises 3.3.1 (page 28)".
              Replace "Burge" by "Burgess".
• Page 163, top.
              Replace the top five lines by:
   
  DN) derive E (D ^ A ^ C) F from E (D ^ ~~A ^ C) F.
  NK) derive E (D ^ ~A ^ C) (D ^ ~B ^ C) F from E (D ^ ~(A ^ B) ^ C) F.
  
  In these rules any of the sentences E, D, C and F may be absent. For example, DN) allows the inference of A from ~~A. Less formally: DN) allows replacement of ~~A by A, while NK) allows replacement of a disjunct containing ~(A ^ B) by a disjunction of two disjuncts, one containing ~A and the other ~B.
   
  
• Page 164, penultimate line of second paragraph.
              Replace "Lemma 4 (page 163)" by "Lemma 11.1 (page 163)".
• Page 178, definition 11.9
              Replace "(of LS" by "(of LS)"
• page 180, line 2.
              Delete the phrase "although X* |- A_m". Also delete the last two sentences of that paragraph.
  note 14
              Replace "see exercises, page 167" by "see page 166"
• Page 181, proof of Lemma 11.9.
              Replace "Hence, by Lemma 9" by "Hence, by Lemma 7" and replace "By (T4)and" by "By (T5), (T6)and"
  
• Page188.

  Replace the first two lines by "completeness proof template we have introduced will apply not only to the many-valued logics above but to most of the other logics as well (FDE will be our most salient exception; see section 12.5)."

• Page 189.
              In the second line immediately after Table 12.1, after the words "as follows" insert ", choosing a world w_o to be thought of as the actual world"
• Page 195, Lemma 12.2.
              Replace second occurrence of "j = K^ser" by "j = K^rst"
  
• Page 196, line 6 from below.
              Replace second occurrence of "X₂" by X_n"
  
• Page 197, line 10.
              Replace second occurrence of the single turnstile by a double turnstile
  
• Page 198, line 17.
              Replace "iff v(w',A)=1," by "iff v(w',B) = 1,"
  
• Page 201, line 6.
              Replace "then B is in w " by "then B is in w' "
  
• Page 205, under heading "COMPLETENESSS".
              Replace " prime theories " by " non-trivial prime theories".
  
• Page 206, lines 7 and 8.
              Exchange "rld" and "lrd".
  
• Page 207, line 4.
              Replace "First of all, X is" by "First of all, X* is".
  
• Page 215, second line before the table.
              Replace "corresponding values" by "corresponding values (natural numbers)".
  
• Page 219.
              Replace "PROOF THM 12.4" by "PROOF THM 12.3".

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