Intro to Logic Quzes

T or F: All sound arguments have true premises. 

True

"Fido loves Meridian Hill Park" - How many constants are there in the above sentence?

Two

“Fido loves Meridian Hill Park” - Which part of this sentence is the predicate?

“Loves”

“Fido is a hero” - How many constants are there in the sentence?

One

“Tamara passes the ball to Bob” - What is the relation, predicate, and what place predicate is there?

Relation: Passes - Predicate: Three place: Pass(a,b,c)

What is the truth value of this sentence? ¬P v (¬Q ∧ P)

True

T or F: The following is a valid argument

1. P v Q

2. P

3. Therefore, ¬Q

False

The negation of a tautology is not a contradiction

False

if the antecedent of a conditional is a contradiction, then the conditional is not true

False

If a statement is a contingency, then it is not a possibility

False

T or F: Line (4) is a valid application of  ∧ Intro

1. P

2. R

3. S

4. P ∧ S ∧ R                 ∧ Intro   1, 2

False

Line (2) is a valid application of ¬ Elim

1. P ∧ ¬¬Q

2. P ∧ Q                 ¬ Elim   1

False

If you have proven that a contradiction follows from your premises, then your argument is unsound

True

T or F: In principle, any argument form that is valid in Boole could be used as an inference rule.

True

Any valid argument that you could prove with a truth table could be used as an inference rule, and any collection of rules you create in this way will allow you to prove every valid argument.  

False

These premises are inconsistent:

1. P

2. ¬¬(P v R) --> ¬Q 

3. ¬¬Q

True

Any argument you can prove valid with a truth table, you can prove valid with the rules in Fitch, and any argument you can prove invalid with a truth table, you can prove invalid with the rules in Fitch. 

False

If you have produced the exact and complete content of the "Goal" line in a sub-proof through valid applications of the rules, then you have proven your goal and you should get a checkmark in the "Goal" line. 

False

Here is a possible rule of inference (call it --> Intro Redeux). Note: I am using the ">" to indicate the inference you are drawing. The line above is the form of the premise you are drawing it from.

   ¬P v Q

>  P --> Q

is the following a correct application of the rule (answer "true" if it is)?

1.  ¬(A <--> (B v C)) v ¬(D v (F --> G))

2.  (A <--> (B v C)) --> ¬(D v (F --> G))             --> Intro Redeux    1 

True

There are necessary truths (but not tautologies, aka truth table necessities) that you can prove in Fitch and not with truth tables.

True

Which of the following sentences are logically equivalent to “Not all dogs are mammals”?  (Choose all that apply)

 

No dogs are mammals

All dogs are not mammals

Some dogs are mammals

Some dogs are not mammals

Some dogs are mammals

Which of the following sentences are logically equivalent to “Only cats are awesome pets.” ? (choose all that apply)

 All cats are awesome pets

∀x (¬(A(x) ∧ P(x)) --> ¬C(x))

∀x (¬C(x) --> ¬(A(x) ∧ P(x)))

¬∃x (C(x) ∧ ¬(A(x) ∧ P(x)))

∀x (¬C(x) --> ¬(A(x) ∧ P(x)))

Which is the most plausible interpretation of this sentence?

"If anyone can save earth, Superman can"

(Assume S=x can save y, s=superman, and e=earth)

 

∀x S(x,e) --> S(s,e)

∃x S(x,e) --> S(s,e)

∃x S(x,e) --> S(s,e)

Which of the following translations have the same truth conditions as:

"Not every vampire is evil"

(In other words, I don't care about literal translations here. I want you to select any statement with the same truth conditions as a literal translation. Choose all that apply).

(assume V= x is a vampire, E = x is evil)

 

¬∀x (Vx --> Ex)

∃x (¬Ex ∧ Vx)

¬∀x Vx --> Ex

¬∀x (Vx --> Ex)

∃x (¬Ex ∧ Vx)

Are these statements consistent? (it doesn't matter that one is in English and one is in logic; I am asking if they can be true at the same time. Assume we are evaluating these sentences in Tarski's world)

 

1. Every cube is to the left of every dodecahedron.

2. ¬∃x ∃y RightOf(x,y)

No

Select the best and most literal translation of:

"If Robocop is not a man, then not all cops are men".

(hint: in case you haven’t seen the movie Robocop, the word “Robocop” doesn’t just refer to any robot cop but to a particular individual)

 

¬∃x (Mx ∧ Rx) --> ¬∀x (Cx --> Mx)

¬Mr --> ¬∀x (Cx --> Mx)

¬Mr --> ¬∀x (Cx --> Mx)

T or F: These statements are equivalent:

1. ¬∃x (Ax ∧ Bx) 

2. ∀x (Ax --> ¬Bx)

True

These two statements are equivalent:

1. ¬∀x (Ax --> Bx)

2. ∃x ¬(Ax ∧ Bx) 

False

The following table of equivalencies is correct:

 

∀xAx = ¬∃x¬Ax

¬∀xAx = ∃x¬Ax

∀x¬Ax = ¬∃xAx

¬∀x¬Ax = ∃xAx

True

T or F: "No dog is friendly" could be translated with a "∀x" or a "∃x"

True

¬∃x Cube(x) ∧ ∃x Tet(x)

 

What is the main logical symbol in the above statement (where "logical symbol" includes both connectives and quantifiers)?

∧

Suppose that this was the goal in a valid argument:

Goal: ¬∀x ∃y RightOf(x,y)

Select the best response below

 

One should try to prove it with negation introduction.

One should try to prove it with universal introduction.

One should try to prove it with existential introduction:

One should try to prove it with negation introduction

Suppose you have an argument with the following premise:

1. ∀x (Cube(x) ∧ ¬Cube(x))

Can one know that the argument is valid without knowing the goal?

Yes

Is this a valid application of existential intro? T=yes, F=no

1. Cube(a) ∧ RightOf(a, b) ∧ Between(c, b, a)

2. ∃x (Cube(x) ∧ RightOf(a, b) ∧ Between(c, b, x))       ∃ Intro, 1 

True

Is this a valid application of conditional Elim? T=yes, F=no

1. ∀x Cube(x) --> Cube(a)

2. ∀x Cube(x)

3. Cube(a)             --> Elim, 1,2   

True

Is this a valid application of universal Elim? T=yes, F=no

1. ∀x Cube(x) ∧ ∀x Medium(x)

2. Cube(a) ∧ ∀x Medium(x)             ∀ Elim, 1

False

Can one prove a contradiction from these premises? True=yes, False=no.

1. ∀x (A(x) ∧ B(x))

2. ¬∀x A(x)

True

T or F: You can use Fitch to prove that an argument is invalid.

False

You can prove this argument to be valid in both Fitch and Boole:

1. Cube(a)

2. Cube(b)

Goal: Cube(a) ∧ Cube(b)

True

Which application of existential intro is correct?

1. Between(a, b, c) v Adjoins(a, c)

2. ∃x Between(x, b, c) v Adjoins(a, c)      ∃ Intro, 1

3. ∃x (Between(x, b, c) v Adjoins(x, c))    ∃ Intro, 1

∃x (Between(x, b, c) v Adjoins(x, c))    ∃ Intro, 1