In Exercises 1730, either prove that the wff is a valid argument or give an

All flowers are plants. Pansies are flowers.

All flowers are plants. Pansies are plants

All flowers are red or purple. Pansies are flowers. Pansies are not purple.

Some flowers are purple. All purple flowers are small.

Some flowers are red. Some flowers are purple. Pansies are flowers.

Some flowers are red. Some flowers are purple. Pansies are flowers.

Justify each step in the following proof sequence of (E x)[P(x) S Q(x)] S [(4x)P(x) S (E x)Q(x)] 1. (E x)[P(x) S Q(x)] 2. P(a) S Q(a) 3. (4x)P(x) 4. P(a) 5. Q(a) 6. (E x)Q(x)

Justify each step in the following proof sequence of (E x)P(x) ` (4x)(P(x) S Q(x)) S (E x)Q(x) 1. (E x)P(x) 2. (4x)(P(x) S Q(x)) 3. P(a) 4. P(a) S Q(a) 5. Q(a) 6. (E x)Q(x)

Consider the wff (4x)[(E y)P(x, y) ` (E y)Q(x, y)] S (4x)(E y)[P(x, y) ` Q(x, y)] a. Find an interpretation to prove that this wff is not valid. b. Find the flaw in the following proof of this wff. 1. (4x)[(E y)P(x, y) ` (E y)Q(x, y)] hyp 2. (4x)[P(x, a) ` Q(x, a)] 1, ei 3. (4x)(E y)[P(x, y) ` Q(x, y)] 2, eg

Consider the wff (4y)(E x)Q(x, y) S (E x)(4y)Q(x, y) a. Find an interpretation to prove that this wff is not valid. b. Find the flaw in the following proof of this wff. 1. (4y)(E x)Q(x, y) hyp 2. (E x)Q(x, y) 1, ui 3. Q(a, y) 2, ei 4. (4y)Q(a, y) 3, ug 5. (E x)(4y)Q(x, y) 4, eg

In Exercises 1116, prove that each wff is a valid argument.

In Exercises 1116, prove that each wff is a valid argument.

In Exercises 1116, prove that each wff is a valid argument.

In Exercises 1116, prove that each wff is a valid argument.

In Exercises 1116, prove that each wff is a valid argument.

In Exercises 1116, prove that each wff is a valid argument.

In Exercises 1730, either prove that the wff is a valid argument or give an interpretation in which it is false.

In Exercises 1730, either prove that the wff is a valid argument or give an interpretation in which it is false.

In Exercises 1730, either prove that the wff is a valid argument or give an interpretation in which it is false.

In Exercises 1730, either prove that the wff is a valid argument or give an interpretation in which it is false.

In Exercises 1730, either prove that the wff is a valid argument or give an interpretation in which it is false.

In Exercises 1730, either prove that the wff is a valid argument or give an interpretation in which it is false.

In Exercises 1730, either prove that the wff is a valid argument or give an interpretation in which it is false.

In Exercises 1730, either prove that the wff is a valid argument or give an interpretation in which it is false.

In Exercises 1730, either prove that the wff is a valid argument or give an interpretation in which it is false.

In Exercises 1730, either prove that the wff is a valid argument or give an interpretation in which it is false.

In Exercises 1730, either prove that the wff is a valid argument or give an interpretation in which it is false.

Either prove that the wff is a valid argument or give an interpretation in which it is false. \((\forall x)(P(x) \vee Q(x)) \wedge(\exists x) Q(x) \rightarrow(\exists x) P(x)\)

In Exercises 1730, either prove that the wff is a valid argument or give an interpretation in which it is false.

In Exercises 1730, either prove that the wff is a valid argument or give an interpretation in which it is false.

The Greek philosopher Aristotle (384322 b.c.e.) studied under Plato and tutored Alexander the Great. His studies of logic influenced philosophers for hundreds of years. His four perfect syllogisms are identified by the names given them by medieval scholars. For each, formulate the argument in predicate logic notation and then provide a proof. a. Barbara All M are P All S are M Therefore all S are P b. Celarent No M are P All S are M Therefore no S are P c. Darii All M are P Some S are M Therefore some S are P d. Ferio No M are P Some S are M Therefore some S are not P

Some plants are flowers. All flowers smell sweet. Therefore, some plants smell sweet. P(x), F(x), S(x)

Every crocodile is bigger than every alligator. Sam is a crocodile. But there is a snake, and Sam isnt bigger than the snake. Therefore, something is not an alligator. C(x), A(x), B(x, y), s, S(x)

There is an astronomer who is not nearsighted. Everyone who wears glasses is nearsighted. Furthermore, everyone either wears glasses or wears contact lenses. Therefore, some astronomer wears contact lenses. A(x), N(x), G(x), C(x)

Every member of the board comes from industry or government. Everyone from government who has a law degree is in favor of the motion. John is not from industry, but he does have a law degree. Therefore, if John is a member of the board, he is in favor of the motion. M(x), I(x), G(x), L(x), F(x),

There is some movie star who is richer than everyone. Anyone who is richer than anyone else pays more taxes than anyone else does. Therefore, there is a movie star who pays more taxes than anyone. M(x), R(x, y), T(x, y)

Everyone with red hair has freckles. Someone has red hair and big feet. Everybody who doesnt have green eyes doesnt have big feet. Therefore someone has green eyes and freckles. R(x), F(x), B(x), G(x)

Cats eat only animals. Something fuzzy exists. Everything thats fuzzy is a cat. And everything eats something. So animals exist. C(x), E(x, y), A(x), F(x)

Every computer science student works harder than somebody, and everyone who works harder than any other person gets less sleep than that person. Maria is a computer science student. Therefore, Maria gets less sleep than someone else. C(x), W(x, y), S(x, y), m

Every ambassador speaks only to diplomats, and some ambassador speaks to someone. Therefore, there is a diplomat. A(x), S(x, y), D(x)

Some elephants are afraid of all mice. Some mice are small. Therefore there is an elephant that is afraid of something small. E(x), M(x), A(x, y), S(x)

Every farmer owns a cow. No dentist owns a cow. Therefore no dentist is a farmer. F(x), C(x), O(x, y), D(x)

Prove that [(4x)A(x)] 4 (E x)[A(x)] is valid. (Hint: Instead of a proof sequence, use Example 32 and substitute equivalent expressions.)

The equivalence of Exercise 43 says that if it is false that every element of the domain has property A, then some element of the domain fails to have property A, and vice versa. The element that fails to have property A is called a counterexample to the assertion that every element has property A. Thus a counterexample to the assertion (4x)(x is odd) in the domain of integers is the number 10, an even integer. (Of course, there are lots of other counterexamples to this assertion.) Find counterexamples in the domain of integers to the following assertions. (An integer x > 1 is prime if the only factors of x are 1 and x.) a. (4x)(x is negative) b. (4x)(x is the sum of even integers) c. (4x)(x is prime S x is odd) d. (4x)(x prime S (1)x = 1) e. (4x)(x prime S 2x 1 is prime)