Well formed formulas

Appendix A contains a brief summary of sane relevant ideas of satisfiability and validity of well-formed formulas in the predicate calculus. Using these ideas it gives a definition of partial and total correctness of a scheme with respect to a well-formed formula as output criterion. The treatment is cursory and nonrigorous. Readers who have not seen these ideas before should examine this appendix before we return to the treatment of correctness and program verification in Chapter V, and finally conclude this treatment in Chapjter VII. 

COROLLARY 5.4 If P is an always halting program scheme we can construct a quantifier-free well-formed formula p(P) such that P is totally correct with respect to TRUE and p(P). ... 

From the paths in TECS) we can select the set (a, ...,a containing all and only consistent paths which start at a tagged point, end at a tagged point and otherwise do not pass through any tagged point. Suppose each cm starts at t and ends at r (in some cases we have t = r ). Then we define W(P) as the well-formed formula ... 

We extend the definition of terms to give us (extended) functional terms, atomic formulae, and well-formed formulae. All these are defined inductively. 

DEFINITION A well-formed formula (wff) is defined as follows. 

To decide if a formula is a well formed formula (wff) of the PC, it must conform to the following definition ... 

What follows then depends entirely on how each of these assumptions are made, and various formulations communicate various ideas. We must remember that none of it has any meaning except in the context of the four assumptions above. Only when well-formed formulas are interpreted can the system be related to the real world and a label such as true or false given to an individual well-formed formula. 

A function or mapphig from the well formed formulas of a language to the truth space. 

Automatic Theorem Proving Proving a theorem from axioms, using a mechanistic procedure, represented as well-formed formulas. 

Definition 4-2 A logic algorithm defining a predicate r/n, denoted LA r), is a closed well-formed formula of the form ... 

First, let us consider Classical Propositional Logic (CPL). Denoting by For the set of well formed formulas of CPL, and using Greek letters as metavariables for formulas, a CPL valuation is a function v For 0,1 subject to the following conditions (considering For over the alphabet A, V, -> ) ... 

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