On Formalizing Logical Modalities

This paper is in the scope of the philosophy of modal logic; more precisely, it concerns the semantics of modal logic, when the modal elements are interpreted as logical modalities. Most authors have thought that the logic for logical modality—that is, the one to be used to formalize the notion of logical truth (and other related notions)—is to be found among logical systems in which modalities are allowed to be iterated. This has raised the problem of the adequacy, to that formalization purpose, of some modal schemes, such as S4 and S5 . It has been argued that the acceptance of S5 leads to non-normal modal systems, in which the uniform substitution rule fails. The thesis supported in this paper is that such a failure is rather to be attributed to what will be called “Condition of internalization.” If this is correct, there seems to be no normal modal logic system capable of formalizing logical modality, even when S5 is rejected in favor of a weaker system such as S4, as recently proposed by McKeon.

Crystal Structure Studies on p-Substitutedbenzenesulphonamides 4-X-C6H4SO2NH2 (X = CH3, NH2 F, Cl or Br)

Effect of ring substitution on the crystal structures of p-substitutedbenzenesulphonamides, p-XC6H4SO2NH2 (X = F, Cl, Br, CH3 or NH2) has been studied by determining the crystal structures of 4-chlorobenzenesulphonamide (4-ClC6H4SO2NH2) and 4-bromobenzenesulphonamide (4-BrC6H4SO2NH2) and analyzing the results along with the structures of 4-methylbenzenesulphonamide (4-CH3C6H4SO2NH2), 4-fluorobenzene-sulphonamide (4-FC6H4SO2NH2) and 4-aminobenzenesulphonamide (4-NH2C6H4SO2NH2). The crystal type, space group, formula units and lattice constants in Å of new structures are: (4-ClC6H4SO2NH2); monoclinic, P21/n, Z = 4, a = 6.6276(10), b = 16.219(3), c = 7.5716(10), β = 93.387(14)°; (4-BrC6H4SO2NH2): monoclinic, P 21/n, Z = 4, a = 6.5660(10), b = 16.4630(10), c = 7.6900(10), β = 92.760(10)°. Orientation of the amine group with respect to the phenyl ring is given by the torsion angles C(2)-C(1)-S-N: 70.9° and C(6)-C(1)-S-N: −108.5°. Similarly, the orientation of S, O(1) and O(2) with respect to the ring are given by torsion angles. The comparison of bond lengths and bond angles of 4-fluoro-, 4-chloro-, 4-bromo-, 4-methyl- and 4-amino-benzenesulphonamides reveal that the S-N and C-S bond lengths decrease with the introduction of electron-withdrawing substituents such as F, Cl or Br, while these groups do not have significant effects on the S-O distances. The effect on ring C-C distances was not uniform. Substitution of F, Cl or Br decreases the O-S-N bond angle, but increases the O-S-N, N-S-C(1) and C(3)-C(4)-C(5) bond angles.