Introduction to SMS and SL

Abstract

SMS is a formal system that employs SL, a notation built upon a set of for- mation rules and a depository of symbols set in a partitioned space. For those who appreciate a distinction between a first-order language and a first-order logic, SL corresponds to the former and SMS to the latter. SMS in some ways resembles restricted FOL, yet it has special features that appear to be extensions of FOL, The syntactical format of SL is similar to infix notation, and as in FOL, well formed formulas (wffs) can be constructed (see section 2.4) and treated as prem- ises from which conclusions can be drawn (see sections 11.3 and 11.4). Deriva- tions in SMS correspond to derivations in FOL but are distinguished by the fact that they are not founded on truth-functional notions of validity. It is in this sense that SMS may be said to fail to achieve status as a logistic system since many people consider the notion of truth to be a necessary component of a logistic sys- tem, although some appear to believe otherwise (e.g. Hogger, 1984; Wojcicki, 1988). Perhaps it could be said that SMS is a logistic system in a mere syntacti- cal sense. It is well known that within a formal logical system, validity can be defined both syntactically and semantically (e.g. Gallaire, Minker and Nicolas, 1977; Haack, 1978; and Levesque, 1986). Given a formal language L and a set of wffs F1, ..., Fn-1, Fn (n - 1), the validity of Fn as a derivation from the axioms of L and F1, ..., Fn-1 can be defined syntactically in terms of whether Fn is derivable by the rules of inference of L. The semantic validity of Fn as a derivation can be defined as follows: Fn is valid in L if it is true in all interpretations in which F1, ..., Fn-1 are true. Semantic validity is thus defined in terms of the truth predicate, whereas syntactic validity need not be understood with reference to the truth predicate. Validity is defined in SMS from both syntactical and semantical points of view; however, the semantics of the system are specified with reference to a many-valued set instead of the truth-predicate. The philosophical justification for adopting this approach is given in section 3.5 and in Chapter 6.