A surprise exam

Intermediate Logic

Instructor: Alexander R. Pruss

Date: An unexpected day between February 17 and February 21, 2014, inclusive.

Name: _________________________

Closed book.  Answer all questions.  Be careful not to follow any of these directions.

Section A. Multiple choice. Circle exactly one option for each of these questions.

1. This sentence is not true.

(a) false

(b) true

(c) neither true nor false

2. Which of the following is not the answer you are circling?

(a) Elephant

(b) Frog

3. The sentence displayed at Question 4 is true.

(a) false

(b) true

4. The sentence displayed at Question 3 is not true.

(a) false

(b) true

5. Which of the following is the capital of China?

Section B. Short essay. Write a one page essay.

6. Give a valid deductive argument that dialetheism is incorrect without using premises or rules of inference that can together yield the law of noncontradiction.

Section C. Reflection. No writing necessary.

7. Reflect on the implications of this sentence for your grade: "If this sentence is true, you failed."

The ontological argument and the semantic paradoxes

I've been feeling that there is some kind of an analogy between Anselm's version of the Ontological Argument (OA) and semantic paradoxes like the Liar or Curry's. Here is one analogy. I've argued in an earlier post that when the consequent in material-conditional Curry sentences is true, the Curry sentence is true, and when the consequent is false, the Curry sentence is nonsense. (The Curry sentence with consequent p is: "If this sentence is true, then p." There is a cool argument from the meaningfulness of the sentence to p.) If this is right, then we have a valid way of arguing from meaning to truth: We have sentences that are true if and only if they are meaningful (for when the consequent is true, the whole sentence is true). Now, I've always thought that Anselm's argument went through as soon as it were granted that one had a concept of that than which nothing greater can be conceived. However, as St. Anselm himself notes but does not make enough of, to have a concept is more than just have a sequence of words in one's head. Thus, it may well be that we have the sequence of words without them expressing a concept.

Just as the Curry sentence is true iff it expresses a proposition, so too the Anselmian predicate has a satisfier (i.e., God) iff it expresses a property. At the same time, this suggests a caution. It would be mistaken to try to figure out by introspection whether a Curry sentence with empirical consequent expresses a proposition, and likewise it may not be appropriate to figure out by introspection whether the Anselmian predicate expresses a property.

Liar and Curry

Consider this sentence:

1. No true sentence (i.e., sentence that is true) satisfies F.

Observe that for some F's, this is quite unproblematic, and even true. For instance, no true sentence is written on the surface of Io, no true sentence is self-contradictory, no true sentence is such that it is both original and was uttered in the Honorable Member's speech. Now, if (1) is the one and only sentence that satisfies F, then (1) is a liar sentence. So some instances of (1) are paradoxical, and some are not.

In general, as in the Io case, it seems that:

2. When F does not have semantic vocabulary and no sentence satisfies F, then (1) is unproblematically true.

If we accept (2), we get an interesting result. Suppose that F is the predicate "is identical to (1) and not-p", where "p" is any non-semantic statement. Then, (1) says that no true sentence is identical to (1) and is such that not-p. This is equivalent to the claim that if (1) is true, then p. In other words, in this case, (1) is equivalent to a Curry sentence just like:

3. If (3) is true, then p.

Now suppose that p is in fact true. Then no sentence satisfies F, and by (2), it follows that (1) is unproblematically true. Therefore, it seems that (3) is also unproblematically true. But if (3) is unproblematically true, then we have established something quite interesting: Curry sentences with true non-semantic consequents are true.

No Curry sentence can be false. For if it's false, then it's true, because any material conditional with false antecedent is true. But whenever a Curry sentence is true, its consequent is true as well, by modus ponens (the antecedent is true because the Curry sentence is true!). So a Curry sentence is true when its consequent is true (assuming the consequent lacks semantic vocabulary) and is nonsense when its consequent is false.

Or so this argument shows. My own intuition is that (3) is nonsense even when p is true. But then I need to get out of the above arguments...