### Table 3. Full Propositional Contrastive Logic: Semantics

"... In PAGE 12: ...ote that ` apos; is idempotent, i.e., ( ( )) = ( ). I now modify (in Table3 ) the de nition of satisfaction of contrastive proposition to accommodate nesting, obtaining the previous de nition as a special case when the nested operands are propositional. A simple calculation shows that, for p; q; r propositional, for example, hS; Ai j= (p q) r iff S j= (p^q) ! :r and A j= p^q^r An immediate consequence of the above de nitions is expressed in the next propo- sition, stating the associativity of the binary contrastive operator.... ..."

### Table 3. Full Propositional Contrastive Logic: Semantics

"... In PAGE 56: ...ote that ` apos; is idempotent, i.e., ( ( )) = ( ). I now modify (in Table3 ) the de nition of satisfaction of contrastive proposition to accommodate nesting, obtaining the previous de nition as a special case when the nested operands are propositional. A simple calculation shows that, for p; q; r propositional, for example, hS; Ai j= (p q) r iff S j= (p^q) ! :r and A j= p^q^r An immediate consequence of the above de nitions is expressed in the next propo- sition, stating the associativity of the binary contrastive operator.... ..."

### Table 2. Flat Propositional Contrastive Logic: Semantics

"... In PAGE 48: ... The syntax of CL1 is presented in Table 1. I now turn to the semantic de nition of CL1, presented in Table2 . I use `j= apos; to ambiguously denote satisfaction both of a classical propositional formula over one world, and for the satisfaction of a contrastive formula over two worlds, leaving the distinction to context.... In PAGE 126: ...Strong Cut-Elimination for Constant Domain First-Order S5 name of condition de nition general for every n, n is accessible from reverse for every n, is accessible from n re exivity is accessible from transitivity if is a proper initial segment of , then is accessible from universal any label is accessible from any label Table 1. conditions on accessibility logic Conditions on Accessibility K, KD general KT general, re exivity KB, KDB general, reverse3 KTB general, re exivity, reverse K4, KD4 general, transitivity S4 (= KT4) general, re exivity, transitivity S5 (= KTB4) universal Table2 . accessibility conditions for various logics label is available on a branch, if it occurs on that branch.... In PAGE 126: ... 402)), namely: L2 X; ( ; 2A) ! Y ` X; ( ; A) ! Y for any accessible from provided (i) for K; KB; and K4; must be available on the branch; (ii) for KD; KT; KDB; KTB; KD4; S4; and S5; must either be available on the branch or must be a simple, unrestricted extension of R2 X ! ( ; 2A); Y ` X ! ( ; A); Y provided is a simple, unrestricted extension of The tableau rules for _, :, 8 and the structural rules mon and cut remain unchanged. If S is any system listed in Table2 , let TQS be the tableau presentation of its constant domain rst-order extension. If we try to reuse the proof of strong cut-elimination for TQS5 in order to establish strong cut-elimination for TQS, we have to be careful, since both R2 and L2 come with complex side conditions.... ..."

### Table 2. Flat Propositional Contrastive Logic: Semantics

"... In PAGE 4: ... The syntax of CL1 is presented in Table 1. I now turn to the semantic de nition of CL1, presented in Table2 . I use `j= apos; to ambiguously denote satisfaction both of a classical propositional formula over one world, and for the satisfaction of a contrastive formula over two worlds, leaving the distinction to context.... ..."

### Table 1. Flat Propositional Contrastive Logic: Syntax

"... In PAGE 125: ... We assume a binary relation of `accessibility apos; between labels. This relation may satisfy certain conditions, and a number of such conditions is de ned in Table1 . K and various extensions of K that can be dealt with by means of labelled tableaux require certain properties of accessibility between labels.... In PAGE 126: ...Strong Cut-Elimination for Constant Domain First-Order S5 name of condition de nition general for every n, n is accessible from reverse for every n, is accessible from n re exivity is accessible from transitivity if is a proper initial segment of , then is accessible from universal any label is accessible from any label Table1 . conditions on accessibility logic Conditions on Accessibility K, KD general KT general, re exivity KB, KDB general, reverse3 KTB general, re exivity, reverse K4, KD4 general, transitivity S4 (= KT4) general, re exivity, transitivity S5 (= KTB4) universal Table 2.... ..."

### Table 2: Institutional Grant Awards at 4-Year Institutions, by Race

"... In PAGE 5: ...nstitutions in the U.S. decreased 3% between 1989 and 1995, from 4,003,992 to 3,892,092. Table2 presents the number of grants, and the average size of each, for all students and for students from each racial group who received: 1) any type of institutional grant; 2) a need grant; or 3) a non-need grant.4 In contrast to the decrease in total enrollment, the number of students receiving any type of institutional grant (shown in panel 1 of Table 2) increased 29% nationally, from 846,583 to 1,089,770, indicating that the proportion of all students who received an institutional grant increased during this period.... In PAGE 5: ...4 In contrast to the decrease in total enrollment, the number of students receiving any type of institutional grant (shown in panel 1 of Table 2) increased 29% nationally, from 846,583 to 1,089,770, indicating that the proportion of all students who received an institutional grant increased during this period. [ Table2 here] The increase in the number of students receiving awards is attributable to a substantial increase in the number of need-based grants awarded, shown in panel 2 of Table 2. While the number of grants for students of all races increased 46% during this period, the number of need-based grants for Asian American students grew the most, and the number for African Americans the least.... In PAGE 5: ...4 In contrast to the decrease in total enrollment, the number of students receiving any type of institutional grant (shown in panel 1 of Table 2) increased 29% nationally, from 846,583 to 1,089,770, indicating that the proportion of all students who received an institutional grant increased during this period. [Table 2 here] The increase in the number of students receiving awards is attributable to a substantial increase in the number of need-based grants awarded, shown in panel 2 of Table2 . While the number of grants for students of all races increased 46% during this period, the number of need-based grants for Asian American students grew the most, and the number for African Americans the least.... In PAGE 5: ... While the size of the mean non-need award to African American students grew only 51%, Asian American students saw a mean award increase of 145%. The last column of Table2 shows the change in the total dollars awarded to each group for each type of grant. Overall, the amount of institutional aid awarded to these students increased 111% from 1989 to 1995, with the amount awarded to each race increasing from a low of 88% for African American students to 220% for Asian American students.... ..."

### Table 2: Institutional Grant Awards at All 4-Year Institutions

"... In PAGE 10: ...our-year institutions in the U.S. decreased 3% between 1989 and 1995, from 4,003,992 to 3,892,092. In contrast, the number of students receiving any type of institutional grant (shown in panel 1 of Table2 ) increased 28%, from 854,042 to 1,089,770, indicating that the proportion of all students who received an institutional grant increased during this period. Table 2 also presents the number of grants, and the average size of each, for all students and for students from each of three income Table 2: Institutional Grant Awards at All 4-Year Institutions... In PAGE 10: ...our-year institutions in the U.S. decreased 3% between 1989 and 1995, from 4,003,992 to 3,892,092. In contrast, the number of students receiving any type of institutional grant (shown in panel 1 of Table 2) increased 28%, from 854,042 to 1,089,770, indicating that the proportion of all students who received an institutional grant increased during this period. Table2 also presents the number of grants, and the average size of each, for all students and for students from each of three income Table 2: Institutional Grant Awards at All 4-Year Institutions... ..."

### Table 2: Institutional Grant Awards at All 4-Year Institutions

"... In PAGE 9: ...in American Colleges and Universities Page 7 In contrast, the number of students receiving any type of institutional grant (shown in panel 1 of Table2 ) increased 28%, from 854,042 to 1,089,770, indicating that the proportion of all students who received an institutional grant increased during this period. Table 2 also presents the number of grants, and the average size of each, for all students and for students from each of three income groups who received: 1) any type of institutional grant; 2) a need grant; or 3) a non-need grant.... In PAGE 9: ...in American Colleges and Universities Page 7 In contrast, the number of students receiving any type of institutional grant (shown in panel 1 of Table 2) increased 28%, from 854,042 to 1,089,770, indicating that the proportion of all students who received an institutional grant increased during this period. Table2 also presents the number of grants, and the average size of each, for all students and for students from each of three income groups who received: 1) any type of institutional grant; 2) a need grant; or 3) a non-need grant.4 The size of the average grant received increased 65%, from $2,641 to $4,345, with the largest increase going to students from middle income families.... ..."

### Table 2: Institutional Grant Awards at 4-Year Institutions, by Race

"... In PAGE 8: ...ear institutions in the U.S. decreased 3% between 1989 and 1995, from 4,003,992 to 3,892,092. Table 2 presents the number of grants, and the average size of each, for all students and for students from each racial group who received: 1) any type of institutional grant; 2) a need grant; or 3) a non-need grant.4 In contrast to the decrease in total enrollment, the number of students receiving any type of institutional grant (shown in panel 1 of Table2 ) increased 29% nationally, from 846,583 to 1,089,770, indicating that the proportion of all students who received an institutional grant increased during this period. 3 For information about institutional aid awards to students from different income groups, see Heller and Nelson Laird (1999).... In PAGE 9: ...Students Receiving Non-need Grants Asian American 9,701 8,405 (13%) $2,408 $5,879 144% 112% African American 20,735 22,950 11% 2,435 3,665 51% 66% Hispanic 12,337 10,961 (11%) 1,935 3,648 89% 68% White 254,716 227,292 (11%) 1,676 3,802 127% 128% All Races 298,541 272,856 (9%) 1,766 3,840 117% 99% Note: Standard errors for the mean grant amount estimates can be found in Appendix Table A-1. The increase in the number of students receiving awards is attributable to a substantial increase in the number of need-based grants awarded, shown in panel 2 of Table2 . While the number of grants for students of all races increased 46% during this period, the number of need-based grants for Asian American students grew the most, and the number for African Americans the least during this period.... In PAGE 10: ... While the size of the mean non-need award to African American students grew only 51%, Asian American students saw a mean award increase of 145%. The last column of Table2 shows the change in the total dollars awarded to each group for each type of grant. Overall, the amount of institutional aid awarded to these students increased 111% from 1989 to 1995, with the amount awarded to each race increasing from a low of 88% for African American students to 220% for Asian American students.... ..."