Archived Notes on Grading: Prelim I

In the context of formal logic, you need to be a bit wary when translating formulas into English; this is especially true when the formula involves implication (->).

As an example of the difficulties that can arise when you reason from an informal translation of a proposition, consider...

Exists x(x is odd -> x is even)

...where the universe of discourse is the set of integers. Many people translated the above into something like, "There exists an integer x such that if x is odd then x is even," a statement which struck most people as being false, since "a number that is odd is not required to be even," etc. The formal proposition, however, is true; just take x to be 4. We then have...

4 is odd -> 4 is even

Now, "4 is odd" is false, and "4 is even" is true, so we get...

F -> T

...which is true, by definition.

P  Q  P -> Q
-  -  ------
F  F    T
F  T    T
T  F    F
T  T    T

Likewise, "4 is odd -> 4 is prime," is also true (F -> F).

[Translating a formal proposition into English is often an excellent way to "build intuition" about the formula, but doing so can sometimes lead you astray as well. Just keep in mind that the formal-logic-to-English conversion is inherently lossy.]

Addendum: Apparently, there was also widespread confusion about the meaning of the statement...

"Exists x P(x)" is false.

This confusion gave rise to invalid arguments along the lines of...

"x is odd -> x is even" is false when x = 3,
    therefor "Exists x(x is odd -> x is even)" is false.

[Note that the above is analogous to...

"x < 0" is false when x = 3,
    therefor "Exists x(x < 0)" is false.

...an inference which is obviously flawed.]

In general, the statement "'Exists x P(x)' is false" is equivalent to the formal proposition "! Exists x P(x)," which is logically equivalent to "Forall x(! P(x))" by duality. Thus,

"Exists x P(x)" is false

...if and only if...

"P(x)" is false for every possible x

(i.e., for everything in the universe of discourse).

[...see also pp. 33-34 in Rosen, especially Table 3.]

Please refer to our guidelines for induction proofs.