Part IV: Exercises

Explain the difference between the principle of excluded middle and the principle of bivalence (recall that they are closely connected. One is about pairs of contradictory opposites, the other is about individual propositions).

In the following sketch proofs, mark by what principle (PEM, PNC, or the principle of identity) each of the bolded steps follows (Do not worry over whether or not the proofs are certainly sound).

e.g.: Apples are not oranges. 

Proof: 

1. Either apples are oranges, or apples are not oranges. (by PEM)
2. Suppose apples are oranges.
3. Then, all apples must taste like oranges. (by the principle of identity)
4. But no apple tastes like an orange.
5. Therefore, apples are not oranges. (by PNC, b) leads to a contradiction, so it must be false)

1.There is no largest number.

Proof:

1. Either there is a largest number, or there is no largest number.
2. Suppose there is a largest number, L.
3. Then, there is no number that is larger than L.
4. We can add numbers together.
5. Then, there is a number Q that is equal to L + 1.  
6. But then, there exists a number Q that is larger than L. 
7. Therefore, there is no largest number.

2.PEM is true.

Proof:

1. Either PEM is true, or PEM is not true.
2. Suppose PEM is not true.
3. Then, all instances of PEM are not true.
4. Then, “Either PEM is true, or PEM is not true” is itself not true.
5. Then, PEM is true and PEM is not true simultaneously.
6. But e) is a contradiction.
7. Therefore, PEM is true.

3.Apples are nutritious.

Proof:

1. Food is nutritious if and only if it somehow promotes health.
2. Foods that contain healthy nutrients are things that promote health.
3. Either apples promote health, or apples do not promote health.
4. Apples are a kind of food that contains healthy nutrients.
5. Therefore, apples promote health.
6. Therefore, apples are nutritious.

(Challenge) Think of proofs by contradiction more generally, or what Aristotle calls “deductions through impossibility”. How is PEM used in making such proofs? How is PNC used? [Hint: There might be an implicit assumption that we make right before we negate the conclusion and try to deduce a contradiction from it].

State whether the following example is an instance of PEM, the principle of bivalence, or neither (i.e. it is merely an assertion).

e.g. “The sky is blue” is either true, or false – this is an instance of the principle of bivalence, because it is a statement about the truth-value of an individual assertion.

Either the sky is blue, or the sky is not blue.

State whether the following example is an instance of PEM, the principle of bivalence, or neither (i.e. it is merely an assertion).

There are two possible truth-values for “The sky is blue.”

State whether the following example is an instance of PEM, the principle of bivalence, or neither (i.e. it is merely an assertion).

 The sky is either blue, or not blue.

State whether the following example is an instance of PEM, the principle of bivalence, or neither (i.e. it is merely an assertion).

There is a unique truth-value – truth or falsehood – for “The sky is blue.”

State whether the following example is an instance of PEM, the principle of bivalence, or neither (i.e. it is merely an assertion).

“Either the sky is blue, or the sky is not blue” is true.

State whether the following example is an instance of PEM, the principle of bivalence, or neither (i.e. it is merely an assertion).

“Either the sky is blue, or the sky is not blue” is either true, or false.

(Extra Challenge) Recall double negation, the logical principle which states that any proposition is equivalent to itself negated twice, from our introductory lesson of propositions. For example, “Chalk is white.” is equivalent to “It is not the case that chalk is not white.” and “All trees are human.” is equivalent to “It is not the case that all trees are not human.” In terms of truth-value, negation works as a switch, a toggle: if we take a true proposition and negate it once, we get a false one; if we negate that again, we get a true proposition. Use double negation to get from PEM to PNC [Hint: think about what happens if you negate PEM twice and look at how we would assign truth-values to the “nested” propositions].