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A formal semantics provide a direct meaning of the terms in the calculus independently of the syntactic proof rules for manipulating them. Without a formal semantics how can you state whether or not the rules of deduction are correct (soundness) or whether you have enough of them (completeness)?

There have been "laws of thought" proposed before natural deduction came about. Aristotle's syllogisms were one such collection. If we had define them to be sound and complete we'd perhaps still be using them today, rather than developing more advanced logical techniques. The point being, if the syllogisms completely captures the laws of thought, why would we need to devise any further logics. What if they were in fact inconsistent? Having a semantics along with the formal proof calculus and the soundness and completeness proofs connecting them provides a measuring stick for judging the value of such a reasoning system. It would no longer stand in isolation.

Whether or not the proposed semantics corresponds to one's intuitive notion of deduction is a philosophical matter. Consider the difference between classical and intuitionistic logic, the essence of which is whether the law of excluded middle ($X\lor

eg X$) should be considered logically valid. Vast amounts of almost religious work has gone into arguing the validity of intuitionistic logic (see Intuitionism). So we cannot even agree on whether the a single notion of deduction makes sense. Beall and Greg Restall even go as far as arguing that we should accept that there is no one true logic and adopt a pluralistic attitude, using the most appropriate logic for the occasion. Given the plethora of logics available to computer scientists (linear logic, separation logic, higher-order constructive logic, many modal logics, all in classical and intuitionistic varieties), adopting a pluralistic attitude is something most of us probably haven't given a second thought, because logics are a tool to solve a particular problem and we try to select the most appropriate one. A formal semantics is one way of judging the appropriateness of the logic.

Another reason for having a formal semantics is that there are more logics than predicate calculus. Many of these logics are designed to reason about a particular kind of system. (I'm thinking about modal logics). Here the class of systems is known and the logic comes later (although, historically, this is also not true). Again, soundness tells us whether the axioms of the logic correctly capture the "behaviour" of the system, and completeness tells us whether we have enough axioms. Without a semantics, how would we know whether the rules of deduction are sufficient and not nonsense?

One example logic which was defined purely syntactically and work is still ongoing to provide it with a formal semantics is BAN logic for reasoning about cryptographic protocols. The logical inference rules seem reasonable, so why provide a formal semantics? Unfortunately, BAN logic can be used to prove that a protocol is correct, yet attacks on such protocols may exist. The deduction rules are therefore wrong, at least with respect to the expected semantics.