Is Second Order Logic Logic?
First-order predicate calculus, a logician might argue, is an effective way of symbolically representing those aspects of sentences in natural language pertinent to their use in arguments and related to the preservation of truth value when considered together. By first-order predicate calculus I mean that logical system involving the standard connectives – &, v, , –> – name letters – a, b, c, etc. – and quantifiers – for all and there is an. There is of course some debate as to whether identity – = – is to be included in first-order logic, but that is the subject for another debate, and I shall simply assume that it can be included, but that this assumption does not have a vast effect on the current discussion1. I shall argue, however, that it is clear that there are groups of sentences which first-order logic is unable to correctly represent, and we use them quite happily in natural language. I shall go on to discuss whether extending our logic, allowing quantification of predicates
