Question 1

1. (∀x)[Ex ⊃ (∀y)(Fy • Gxy)]
2. (∃x)(Ex • Hxb)
-Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

Accepted Answer

The second premise (∃x)(Ex • Hxb) implies there exists some 'a' such that Ea and Hab are true, making B a direct consequence.

Question 2

1. (∀x)[Ex ⊃ (∀y)(Fy • Gxy)]
2. (∃x)(Ex • Hxb)
-Which of the following propositions is derivable from the given premises in F?

Accepted Answer

From the given premises, we can derive that there exists at least one x such that Ex and Hxb are true simultaneously due to the second premise. The first premise implies that for all x, if Ex is true, then for all y, Fy and Gxy are true. Combining these, we can infer the existence of at least one x for which Gxy and Hxy are true for some y, specifically for y=b, leading to the conclusion E) (∃x)(∃y)(Gxy • Hxy).

Question 3

1. (∀x)(∃y)Axy ⊃ (∀x)(∃y)Bxy
2. (∃x)(∀y)∼Bxy
-Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

Accepted Answer

From premise 2, we know that there exists an x such that for all y, ~Bxy. This means that there does not exist a y such that Bxy. Therefore, the proposition (∃x)∼(∃y)Bxy is true. This is the only option that can be immediately deduced from the premises.

Question 4

1. (∀x)(∃y)Axy ⊃ (∀x)(∃y)Bxy
2. (∃x)(∀y)∼Bxy
-Which of the following propositions is derivable from the given premises in F?

Accepted Answer

BFrom the first premise, we understand that for every x, there exists a y such that Axy implies that for every x, there exists a y such that Bxy. The second premise states that there exists an x for which all y do not satisfy Bxy. This situation makes it impossible for every x to have a corresponding y that satisfies Bxy, as required by the implication of the first premise, unless there is an x for which all y do not satisfy Axy as well, making it impossible to trigger the condition of the implication. Therefore, both A and B are derivable because they express the existence of an x for which all y do not satisfy Axy or Ayx, respectively, which is a necessary condition to prevent a contradiction with the second premise.

Question 5

1. (∃x)[Dx • (∀y)(Ey ⊃ Fxy)]
2. (∀x)(Dx ⊃ Ex)
-Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

Accepted Answer

The first premise states that there exists an x such that Dx and for all y, if Ey then Fxy. The second premise states that for all x, if Dx then Ex. From these, we can derive that there exists at least one specific instance (let's call it 'a') where Da is true and for all y, if Ey then Fay, which matches option C. This follows because the first premise guarantees an x that satisfies both Dx and the conditional relation between Ey and Fxy, and the second premise ensures that whatever x satisfies Dx also satisfies Ex, making the specific instance 'a' satisfy both conditions.

Question 6

1. (∃x)[Dx • (∀y)(Ey ⊃ Fxy)]
2. (∀x)(Dx ⊃ Ex)
-Which of the following propositions is derivable from the given premises in F?

Accepted Answer

The answer of 1. (∃x)[Dx • (∀y)(Ey ⊃ Fxy)]
2. (∀x)(Dx...

Question 7

1. (∀x)[(Cx • Exa) ⊃ Dx]
2. Cd • ∼Dd
-Which of the following propositions is an immediate (one-step) consequence in F of the given premise?

Accepted Answer

The proposition "∼Dd • Cd" is an immediate consequence of premise 2, as it is simply a rearrangement of the conjunction given in premise 2.

Question 8

1. (∀x)[(Cx • Exa) ⊃ Dx]
2. Cd • ∼Dd
-Which of the following propositions is derivable from the given premises in F?

Accepted Answer

From premise 2, we know that Cd is true and ∼Dd is true. From premise 1, we know that for any x that is both a C and an E of a, Dx is true. Since d is not a D, we can conclude that d is not an E of a, or ∼Eda is true.

Question 9

1. (∀x)[(Px $ \lor $ Qx)] ⊃ Rxx]
2. (∀x){Qx ⊃ [(∃y)Rxy ⊃ Sxx]}
3. Pn • Qn
-Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

Accepted Answer

The answer of 1. (∀x)[(Px $ \lor $ Qx)] ⊃...

Question 10

1. (∀x)[(Px $ \lor $ Qx)] ⊃ Rxx]
2. (∀x){Qx ⊃ [(∃y)Rxy ⊃ Sxx]}
3. Pn • Qn
-Which of the following propositions is derivable from the given premises in F?

Accepted Answer

From the premises, we can derive Rnn and Snn. The first premise, when applied to 'n' (given Pn • Qn, meaning both Pn and Qn are true), ensures Rnn is true. The second premise, given Qn, allows for the derivation of Snn if there exists some y such that Rny, which is confirmed by Rnn. Thus, Rnn • Snn is derivable.

Question 11

1. (∀x)[Ax ⊃ (∃y)(By • Cxy)]
2. (∃x)(Ax • Dx)
3. (∀x)(Bx ⊃ Ex)
-Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

Accepted Answer

The answer of 1. (∀x)[Ax ⊃ (∃y)(By • Cxy)]
2. (∃x)(Ax...

Question 12

1. (∀x)[Ax ⊃ (∃y)(By • Cxy)]
2. (∃x)(Ax • Dx)
3. (∀x)(Bx ⊃ Ex)
-Which of the following propositions is derivable from the given premises in F?

Accepted Answer

Starting from the first premise:
(∀x)[Ax ⊃ (∃y)(By • Cxy)]
Using universal instantiation:
Aa ⊃ (∃y)(By • Cay)
Looking at the second premise:
(∃x)(Ax • Dx)
Using existential instantiation:
Aa • Da
From here, we can use conjunction elimination to obtain Aa:
Aa
Then, we can use modus ponens to obtain:
(∃y)(By • Cay)
Using existential instantiation:
By • Cay
Using conjunction elimination, we obtain:
Cay
Finally, we can use existential generalization to obtain:
(∃x)[Dx • (∃y)(Ey • Cxy)]
Which matches answer choice A.

Question 13

1. (∀x)[Ax ⊃ (∀y)(By ⊃ Cxy)]
2. (∃x)[Ex • (∀y)(Hy ⊃ Cxy)]
3. (∀x)(∀y)(∀z)[(Cxy • Cyz) ⊃ Cxz]
4. (∀x)(Ex ⊃ Bx)
-Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

Accepted Answer

The answer of 1. (∀x)[Ax ⊃ (∀y)(By ⊃ Cxy)]
2. (∃x)[Ex...

Question 14

1. (∀x)[Ax ⊃ (∀y)(By ⊃ Cxy)]
2. (∃x)[Ex • (∀y)(Hy ⊃ Cxy)]
3. (∀x)(∀y)(∀z)[(Cxy • Cyz) ⊃ Cxz]
4. (∀x)(Ex ⊃ Bx)
-Which of the following propositions is derivable from the given premises in F?

Accepted Answer

From premise 1, we can use Universal Instantiation to get:
(Aa ⊃ (∀y)(By ⊃ Cay))
Then, we can use Existential Instantiation on premise 2 to obtain:
Ea and (∀y)(Hy ⊃ Cay)
We can then use Simplification to get:
Hy ⊃ Cay
Using Universal Instantiation on premise 3, we can obtain:
(Cab • Cbc) ⊃ Cac
We can then use Universal Instantiation on our previous results to get:
(Cay • Cby) ⊃ Cax
Using this and our simplified version of premise 1, we can use Universal Generalization to get:
Ax ⊃ (∀y)(Hy ⊃ Cxy)
which matches option D.

Question 15

(∀x)[Px ⊃ (∃y)Rxy] ⊃ [(∀x)(∀y)∼Rxy ⊃ ∼(∃x)Px]
-Consider assuming '(∀x)[Px ⊃ (∃y)Rxy]' for a conditional proof of the above logical truth. Which of the
Following propositions is a legitimate second step in that proof?

Accepted Answer

Assuming '(∀x)(∀y)∼Rxy' for a nested conditional proof allows us to derive the negation of the consequent of the conditional statement we are trying to prove. From there, we can use conditional proof to conclude the entire statement. Assuming any of the other options does not seem to be directly relevant to the proof.

Question 16

(∀x)[Px ⊃ (∃y)Rxy] ⊃ [(∀x)(∀y)∼Rxy ⊃ ∼(∃x)Px]
-Which of the following propositions is also derivable in F?

Accepted Answer

We can first simplify the main implication as follows: 
(∀x)[Px ⊃ (∃y)Rxy] ⊃ [(∀x)(∀y)∼Rxy ⊃ ∼(∃x)Px] 
≡ (∀x)(∀y)[(Px ⊃ Rxy) ⊃ (∼Rxy ⊃ ∼Px)]

Then we apply the rule of contraposition to the antecedent of the consequent: 
(∀x)(∀y)[(Px ⊃ Rxy) ⊃ (Px ⊃ ~∼Px)] 
≡ (∀x)(∀y)[(Px ⊃ Rxy) ⊃ Px]

Finally, we use existential instantiation on x to obtain: 
(∀x)[Px ⊃ (∃y)Rxy] ⊃ [(∃x)Px ⊃ (∃x)(∃y)Rxy]

Question 17

(∀x)[Px ⊃ (∃y)Qxy] ⊃ [(∃x)Px ⊃ (∃x)(∃y)Qxy]
-Consider assuming '(∀x)[Px ⊃ (∃y)Qxy]' for a conditional proof of the above logical truth. Which of the
Following propositions is a legitimate second step in that proof?

Accepted Answer

Assuming '(∃x)Px' for a nested conditional proof allows us to derive the consequent '(∃x)(∃y)Qxy' of the given logical truth. This is because if we can assume the antecedent of the conditional (which we have done with the assumption of '(∀x)[Px ⊃ (∃y)Qxy]'), then we can derive the consequent by using existential instantiation and modus ponens. Specifically, we can instantiate '(∃y)Qxy' with some y for which 'Px' is true (which we can do since we assumed '(∃x)Px') to get 'Qxy.' Then, we can use modus ponens with the assumption of '(∀x)[Px ⊃ (∃y)Qxy]' to derive 'Px ⊃ (∃y)Qxy.' Finally, we can use modus ponens again with 'Px' and 'Px ⊃ (∃y)Qxy' to derive '(∃y)Qxy.' We can repeat this process for various x's until we have instantiated all the existentials and derived '(∃x)(∃y)Qxy.'

Question 18

(∀x)[Px ⊃ (∃y)Qxy] ⊃ [(∃x)Px ⊃ (∃x)(∃y)Qxy]
-Which of the following propositions is also derivable in F?

Accepted Answer

Starting with the given formula, we can apply the following deduction:

(∀x)[Px ⊃ (∃y)Qxy] ⊃ [(∃x)Px ⊃ (∃x)(∃y)Qxy]

= (∀x)[~Px \lor (∃y)Qxy] ⊃ [~(∃x)Px \lor (∃x)(∃y)Qxy]

= (∀x)[~Px ⊃ ~(∃y)Qxy] ⊃ [(∀x)~Px \lor (∃x)(∃y)Qxy]

= (∀x)[~(∃y)Qxy ⊃ ~Px] ⊃ [(∀x)~Px \lor (∃x)(∃y)Qxy]

= (∀x)[~(∃y)Qxy ⊃ ~Px] ⊃ [(∀x)∼Px \lor (∃x)(∃y)Qxy]

Therefore, option B is derivable in F.

Question 19

[(∀x)Pax • ∼(∃x)Pxa] ⊃ (∀x)(Pax • ∼Pxa)
-Consider assuming '(∀x)Pax • ∼(∃x)Pxa' for a conditional proof of the above logical truth. Which of the
Following propositions is a legitimate second step in that proof?

Accepted Answer

The answer of [(∀x)Pax • ∼(∃x)Pxa] ⊃ (∀x)(Pax • ∼Pxa)
-Consider...

Question 20

[(∀x)Pax • ∼(∃x)Pxa] ⊃ (∀x)(Pax • ∼Pxa)
-Which of the following propositions is also derivable in F?

Accepted Answer

The answer of [(∀x)Pax • ∼(∃x)Pxa] ⊃ (∀x)(Pax • ∼Pxa)
-Which...

Quiz 5: Full First-Order Logic

1. (∀x) [Ex ⊃ (∀y) (Fy • Gxy) ] 2. (∃x) (Ex • Hxb) -Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

1. (∀x) [Ex ⊃ (∀y) (Fy • Gxy) ] 2. (∃x) (Ex • Hxb) -Which of the following propositions is derivable from the given premises in F?

1. (∀x) (∃y) Axy ⊃ (∀x) (∃y) Bxy 2. (∃x) (∀y) ∼Bxy -Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

1. (∀x) (∃y) Axy ⊃ (∀x) (∃y) Bxy 2. (∃x) (∀y) ∼Bxy -Which of the following propositions is derivable from the given premises in F?

1. (∃x) [Dx • (∀y) (Ey ⊃ Fxy) ] 2. (∀x) (Dx ⊃ Ex) -Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

1. (∃x) [Dx • (∀y) (Ey ⊃ Fxy) ] 2. (∀x) (Dx ⊃ Ex) -Which of the following propositions is derivable from the given premises in F?

1. (∀x) [(Cx • Exa) ⊃ Dx] 2. Cd • ∼Dd -Which of the following propositions is an immediate (one-step) consequence in F of the given premise?

1. (∀x) [(Cx • Exa) ⊃ Dx] 2. Cd • ∼Dd -Which of the following propositions is derivable from the given premises in F?

1. (∀x) [(Px $\lor$ Qx) ] ⊃ Rxx] 2. (∀x) {Qx ⊃ [(∃y) Rxy ⊃ Sxx]} 3. Pn • Qn -Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

1. (∀x) [(Px $\lor$ Qx) ] ⊃ Rxx] 2. (∀x) {Qx ⊃ [(∃y) Rxy ⊃ Sxx]} 3. Pn • Qn -Which of the following propositions is derivable from the given premises in F?

1. (∀x) [Ax ⊃ (∃y) (By • Cxy) ] 2. (∃x) (Ax • Dx) 3. (∀x) (Bx ⊃ Ex) -Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

1. (∀x) [Ax ⊃ (∃y) (By • Cxy) ] 2. (∃x) (Ax • Dx) 3. (∀x) (Bx ⊃ Ex) -Which of the following propositions is derivable from the given premises in F?

1. (∀x) [Ax ⊃ (∀y) (By ⊃ Cxy) ] 2. (∃x) [Ex • (∀y) (Hy ⊃ Cxy) ] 3. (∀x) (∀y) (∀z) [(Cxy • Cyz) ⊃ Cxz] 4. (∀x) (Ex ⊃ Bx) -Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

1. (∀x) [Ax ⊃ (∀y) (By ⊃ Cxy) ] 2. (∃x) [Ex • (∀y) (Hy ⊃ Cxy) ] 3. (∀x) (∀y) (∀z) [(Cxy • Cyz) ⊃ Cxz] 4. (∀x) (Ex ⊃ Bx) -Which of the following propositions is derivable from the given premises in F?

(∀x) [Px ⊃ (∃y) Rxy] ⊃ [(∀x) (∀y) ∼Rxy ⊃ ∼(∃x) Px] -Consider assuming '(∀x) [Px ⊃ (∃y) Rxy]' for a conditional proof of the above logical truth. Which of the Following propositions is a legitimate second step in that proof?

(∀x) [Px ⊃ (∃y) Rxy] ⊃ [(∀x) (∀y) ∼Rxy ⊃ ∼(∃x) Px] -Which of the following propositions is also derivable in F?

(∀x) [Px ⊃ (∃y) Qxy] ⊃ [(∃x) Px ⊃ (∃x) (∃y) Qxy] -Consider assuming '(∀x) [Px ⊃ (∃y) Qxy]' for a conditional proof of the above logical truth. Which of the Following propositions is a legitimate second step in that proof?

(∀x) [Px ⊃ (∃y) Qxy] ⊃ [(∃x) Px ⊃ (∃x) (∃y) Qxy] -Which of the following propositions is also derivable in F?

[(∀x) Pax • ∼(∃x) Pxa] ⊃ (∀x) (Pax • ∼Pxa) -Consider assuming '(∀x) Pax • ∼(∃x) Pxa' for a conditional proof of the above logical truth. Which of the Following propositions is a legitimate second step in that proof?

[(∀x) Pax • ∼(∃x) Pxa] ⊃ (∀x) (Pax • ∼Pxa) -Which of the following propositions is also derivable in F?