Prove the one point rule, Prove the one point rule-

Prove the one point rule

Reference no: EM13918624

Metatheorem. (Auxiliary Constant Metatheorem) Let c be a constant that does not appear in the formulae A or B. Assume that I |- (∃x)A. Moreover, let I' +A[x :=c] |- B, with a proof where the formulae invoked from I' do not contain the constant c. Then I'|- B as well.

ADDMONAL EXERCISES-

1. Show that |- (∀x)(A → B) → (∃x)A→ (∃x)B.

2. Show that |- (∀x) ((A ν B) →C) → (∀x) (A → C).

3. Show that |- (∀x)((A ν B) → C) → (∀x)(A → C).

4. Show that |- (∀x)(A → (B Λ C)) → (∀x)(A → B).

5. Prove the following version of the relativized ∀-monotonicity,

|-(∀x)_A(B →C) →(∀x)_A B→ (∀x)_AC

while in standard notation it reads

|- (∀x)(A→ B→ C) → (∀x)(A →B) → (∀x)(A → C)

6. Prove |- (∃x)_AΛB C ≡ (∃x)_A(B Λ C)

Hint Translate first to standard notation.

7. Prove

|- AV (∀x)_BC = (∀x)_B(A ν C), as long as x not free in A, Hint. Translate first to standard notation

8. Prove

|- A Λ (∃x)BC ≡ (∀x)_B(A Λ C),as keg attract free is A, Hint Translate first to standard notation.

9. Prove

|- (∃x)_A B v (∃x)_AC ≡ (∃x)_A(B v C)

Hint Translate first to standard notation.

10. Prove that if x is not free in A, then |- A ≡ (∃x)A.

11. Prove the one point rule- ∃-version: |- (∃x)(x = t Λ A) ≡ A[x:= t] if x is sot free in t.

12. Prove |- (∃x)(A Λ (∃y)(B Λ C)) ≡ (∃y)(B Λ (∃x)(A Λ C)), on the condition that y is not free in A and x is not free in B.

13. Prove |- (∃x)_AvB C ≡ (∃x)_AC v (∃x)_BC. Hint. Translate to standard notation first.

14. Prove |- (∃x)( ∃y)(A Λ B Λ C) ≡ (∃x)(A Λ (∃y)(B Λ C)), on the condition that y is not free in A.

15. Prove dummy renaming for ∃: If z does not occur in A, then |- (∃x) A ≡ (∃z) A[x := z].

16. Here is a suggested proof of

|- (∀x)(∃y)A→ (∃y)(∀x)A (*)

We split the → and go via the deduction theorem:

(1) (∀x)(∃y)A (hypothesis)

(2) (∃y)A ((1) + spec)

(3) A[y :=z] (auxiliary hypothesis associated with (2); z is fresh)

(4) (∀x)A[y :=z] ((3) + gen; Okay: x is not free in hypothesis line (1))

(5) (∃y)(∀x)A

17. Prove using the auxiliary variable metatheorem |- (∃x)(A→B) → (∀x)A→ (∃x)B.

18. Prove using the auxiliary variable metatheorem: |- (∃x)B → (∃x)(A v B).

19. Prove |- (∃x)_AC →(∃x)_AvBC.

20. Let Φ be a predicate of arity 2.

Reference no: EM13918624

Questions Cloud

Attributed to the value of land : Terry purchased and placed into service a building that costs $2 million. An appraisal determined that 25% of the total cost was attributed to the value of land. The bottom floor of the building is leased to a retail business for $32,000.

Should one or two employees handle machine repair operation : Each employee is paid $20 per hour. Machine downtime is valued at $80 per hour. From an economic point of view, should one or two employees handle the machine repair operation? Explain.

Financial statements for sammie enterprises : amounts have been taken from the recent financial statements for Sammie Enterprises

Analysis of the three theories of morality : Application Morality and Social Responsibility Philosophical perspectives and theories on morality contribute to an understanding of the deepAnalysis of the three theories of morality Application Morality and Social Responsibility Philosophical pe..

Prove the one point rule : Prove the one point rule- ∃-version: |- (∃x)(x = t Λ A) ≡ A[x:= t] if x is sot free in t. Prove |- (∃x)(A Λ (∃y)(B Λ C)) ≡ (∃y)(B Λ (∃x)(A Λ C)), on the condition that y is not free in A and x is not free in B

Divide the class into groups of four or five students : Verizon Communications, Inc., is one of the world's largest providers of communication services.

A short-term investment : June 10 Sold 400 shares of Wild Goose Marina Corporation common stock that has been held as a short-term investment. The stock was sold for 19 1/4, less a commission of $240.00. The stock was originally purchased at a total cost of $5,200.00.

Arenas of health care politics : There are four policy arenas in health care politics subsidy, financing, reorganization, and regulatory.Arenas of Health Care Politics There are four policy arenas in health care politics subsidy, financing, reorganization, and regulatory.Arenas o..

What is competitive advantage and how do you measure it : What does "value chain" mean and how does it relate to competitive advantage? Identify strategies that can be used for cost advantage.

User Account

All Pages