

Introduction to Logic

Tools for Thought


Propositional Logic allows us to talk about relationships among individual propositions, and it gives us the machinery to derive logical conclusions based on these relationships. Suppose, for example, we believe that, if Jack knows Jill, then Jill knows Jack. Suppose we also believe that Jack knows Jill. From these two facts, we can conclude that Jill knows Jack using a simple application of Implication Elimination.
If Jack knows Jill, then Jill knows Jack. 
Jack knows Jill. 
Therefore, Jill knows Jack. 

Unfortunately, when we want to say things more generally, we find that Propositional Logic is inadequate. Suppose, for example, that we wanted to say that, in general, if one person knows a second person, then the second person knows the first. Suppose, as before, that we believe that Jack knows Jill. How do we express the general fact in a way that allows us to conclude that Jill knows Jack?
If one person knows another, then the second person knows the first. 
Jack knows Jill. 
Question: Does Jill knows Jack? 
Here, Propositional Logic is inadequate; it gives us no way of succinctly encoding this more general belief in a form that captures its full meaning and allows us to derive such conclusions.

Relational Logic is an alternative to Propositional Logic that solves this problem. The trick is to augment our language with two new linguistic features, viz. variables and quantifiers. With these new features, we can express information about multiple objects without enumerating those objects; and we can express the existence of objects that satisfy specified conditions without saying which objects they are.
∀x.∀y.(knows(x,y) ⇒ knows(y,x))
This sample sentence in Relational logic expresses the complex fact we just saw. In English, it says that, for every x and every person y, if x knows y, then y knows x.

In this lesson, we proceed through the same stages as in the introduction to Propositional Logic. We start with syntax and semantics. We then discuss evaluation and satisfaction. We look at some examples. Then, we talk about properties of Relational Logic sentences and logical entailment for Relational Logic. Finally, we say a few words about the equivalence of Relational Logic and Propositional Logic and its decidability.

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