Logiweb(TM)

6.7.14 Wellformed formulas

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The Logiweb sequent calculus dictates that certain terms are definitely illformed. As an example, Logiweb sequent calculus does not like terms like 1 + All x : x because it embeds a metaquantifier inside an object term.

Apart from that, the Logiweb sequent calculus does not impose restrictions on terms. If you want to define a theory in which only certain terms are wellformed, then you will have to implement a check for wellformedness and you will have to add wellformedness side conditions to all your rules.

Peano arithmetic PA takes another approach: it considers all terms wellformed but only assign meaning to terms which are built up from object variables and constructors of PA. A term is an 'object variable' if it has no meta definition, no value definition, and no math definition, c.f. the definition of objectvar on the check page.

As an example, 3 :: 4 is no object variable since the pair operator has a value definition. Furthermore, 3 :: 4 does not make sense in Peano arithmetic. Thus, to Peano arithmetic, 3 :: 4 is a constant which denotes some, specific natural number, but Peano arithmetic does not care which one. To Peano arithmetic, 3 :: 4 could denote zero or 117 or a zillion of whatever. You can prove 3 :: 4 = 3 :: 4 in Peano arithmetic if you like because any natural number equals itself, but you cannot do anything useful with 3 :: 4 in that theory.

When taking this approach it is important to ensure that the theory remains consistent. One can do that by ensuring that the resulting system has a model. As an example, one may assign the value zero to all formulas which are neither object variables nor something Peano arithmetic knows about.

One must also check what happens if one applies a quantifier to something which is not an object variable. One can typically interpret that situation as quantification over a fresh variable.

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Copyright © 2010 Klaus Grue, GRD-2010-01-05