Week 3: Fancy Translations from English to Logic

Last week, we saw that a sentence in a natural language can be represented in the formal language of logic through predicates, implications and quantifiers.  Yet we can be a little fancier with our representations by introducing two new members to our zoo of logical symbols. 

Suppose that we want to represent these English statements in logic:

(1) Some student that is in CSC165 is both cool and likes math.

(2) All students that are CSC165 are either cool or like math.

Let $U$ be the set of all students.  Let $C(x)$ mean $x$ is in CSC165, $K(x)$ mean $x$ is cool and $M(x)$ mean $x$ likes math.  The conjunction symbol, which represents the English word “and”, and the disjunction symbol, which represents the word English “or”, allows the translation of statements (1) and (2) into logic:

                        (1) $\exists x \in U, C(x) \implies (K(x) \wedge M(x))$.

                        (2) $\forall x \in U, C(c) \implies (K(x) \vee M(x))$.

Now let us play a game of I-Spy.   See these statements:

                        (3) $\exists x \in U, C(x) \implies (K(x) \implies M(x))$.

                        (4) $\forall x \in U, C(x) \implies (K(x) \implies M(x))$.

I spy a nuance between statements (3) and (1) and statements (4) and (2).   What is this nuance?  It is the relation between predicates $K(x)$ and $M(x)$.  Replacing either the conjunction or disjunction symbol in a statement with an implication symbol changes the meaning of that statement.  Let us illustrate the difference between statements (1) and (3) and statements (2) and (4) with a CSC165 student named Bobby.

Bobby wants to take up the burden of satisfying statement (1).  Then he needs be cool and he has to like math.  Later Bobby decides that this is too much of a burden for him so he decides to satisfy statement (3) instead.  From last week`s post about implications, we know that he has a choice of either being uncool or keeping his fondness for math.  Hence, the implication is much more flexible than the conjunction.

Now Bobby wants to take up the burden of satisfying statement (2).  He campaigns to convince all his CSC165 classmates to either be cool or like math.  His classmates rebel and decide to neither be cool nor like math.  Yet Bobby is enterprising.  He decides to satisfy statement (4) instead.  Then he needs to get all his classmates to either be uncool or like math.  However, his CSC165 classmates are just as adept at defying Bobby and they decide to be both cool and dislike math.  Poor Bobby!  Hence, the implication and disjunction are different in how they are satisfied but they are the same in their flexibility.