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Subsection 8.2.1 Inroductiom

In a direct proof, we walk forward from the premises to the desired conclusion, typically relying heavily on modus ponens. We’ll leave out some steps along the way if we believe that a reader of our proof can easily fill them in. In particular, we almost always leave out explicit mention of existential/universal instantiation/generalization.

Exercises Exercises

1.

  1. Consider the following proof of the existence of a white mammal (yes, you saw it as the White Mammal example):

[1] ∀x (Bear(x) → Mammal(x)) Premise

[2] ∃x (Bear(x) ∧ White(x)) Premise

[3] Bear(c*) ∧ White(c*) Existential Instantiation [2]

[4] Bear(c*) Simplification [3]

[5] White(c*) Simplification [3]

[6] Bear(c*) → Mammal(c*) Universal Instantiation [1]

[7] Mammal(c*) Modus Ponens [4], [6]

[8] Mammal(c*) ∧ White(c*) Conjunction [5], [7]

[9] ∃x (Mammal(x) ∧ White(x)) Existential Generalization [8]

Which (one or more) of the following possible English proofs of the same claim (assuming the same premises) is/are valid?

  1. We know that there’s a white bear. Since all bears are mammals, our white bear is thus a white mammal.

  2. Since all bears are mammals, the white bear we know exists must also be a white mammal.

  3. There must be a white mammal because there is a white bear, which is also a mammal.

Which of the following statements is true:

  1. Only I is a valid proof.

  2. Only II is a valid proof.

  3. Only III is a valid proof.

  4. Exactly two of them are valid proofs.

  5. All of them are valid proofs.

Answer.

Correct answer is E.

Solution.

Explanation: All of them are valid. Notice that I explicitly mentions premise [1]. The others just use it. But, if challenged, our proof writers can straightforwardly appeal to it.

2.

Consider the following proof (from one of our videos) of the existence of someone who can participate in our drug test: Assign the following names to basic statements:

W(x): True if x is a Woman.

U(x, y) True if person x Uses medication y.

P(x) True if x may Participate in our drug test.

[1] ∃p (∃q (W(p) ∧ U(p, q))) Premise

[2] ∃q (W(a*) ∧ U(a*, q)) Existential Instantiation [1]

[3] W(a*) ∧ U(a*, b*) Existential Instantiation [2]

[4] ∀x (∀y ((W(x) ∧ U(x, y)) → P(x))) Premise

[5] (∀y ((W(a*) ∧ U(a*, y)) → P(a*)) Universal Instantiation [4]

[6] (W(a*) ∧ U(a*, b*)) → P(a*) Universal Instantiation [5]

[7] P(a*) Modus Ponens [3], [6]

[8] ∃z (P(z)) Existential Generalization [7]

Which (one or more) of the following possible English proofs of the same claim (assuming the same premises) is/are valid?

  1. If there’s a woman who uses a medication, then she can be in our drug test. There is such a woman, so we’ve got someone who may participate in our test.

  2. There’s a woman who uses a medication and anyone who does can be in our test. So we have that person as a participant.

  3. There must be someone who can participate in our drug test since there is a woman who takes a medication.

Which of the following statements is true:

  1. Only I is a valid proof.

  2. Only II is a valid proof.

  3. Only III is a valid proof.

  4. Exactly two of them are valid proofs.

  5. All of them are valid proofs.

Answer.

Correct answer is D.

Solution.

Explanation: I is valid. It explicitly mentions both premises and then states the conclusion. III is also valid although it doesn’t explicitly mention premise [4]. II isn’t valid. It actually asserts that anyone who takes a medication can be in our test and we don’t know that. We know only that any woman who takes a medication can be.