ece190_rec5 - What is a propositional formula Periklis A...

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What is a propositional formula? Periklis A. Papakonstantinou ? University of Toronto We introduce the concept of recursive definition by defining propositional formulas. Then, we use induction to prove that {∧ , ∨} is not a complete set of connectives. It’s kind of weird but even when we do mathematics we may not be in a position to precisely describe what the main mathematical objects are. For example, since elementary school you were using arithmetic expressions. Everybody is able to say that (1 + 2) - (3 · 4) is an arithmetic expression whereas 1 + +(+( - 42 is not. Despite this if somebody asks you whether you can define what an arithmetic expression is you probably would have trouble defining it. Note that this certainly does not mean that you do not understand what an arithmetic expression is, how to handle it or use it to prove statements. A similar thing holds for propositional formulas 1 . You may be dealing with propositional formulas for a couple of months and still not being able to define them. And this is absolutely not a problem, unless you want to prove something regarding infinitely many propositional formulas. In this case if we don’t know exactly what it is, then we simply can’t prove anything about it. This is the case in this handout. We would like to prove something about an infinite fragment of possible propositional formulas by talking “abstractly” about them. As we will see just defining propositional formulas is an interesting issue by itself. This is the first among the tenths of “recursive definitions” and appearances of “recursion” in your studies. Recursive definitions The main intuitive difference between a recursive definition and other definitions you may have encountered up to now is that a recursive definition seems to “require our involvement”. That is, a recursive definition specifies some very simple objects that it defines and is like telling us: “now go ahead, apply again and again the rules in order to construct more and more objects among those that I define”. And you can build them all by combining what you know for simpler objects so as to construct more complicated ones. Definition 1. A propositional formula is one of the following: 1. Each variable (propositional atom) alone e.g. x, y, z, . . . is a propositional formula.
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