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程序代写案例-106B

时间：2021-03-12

Philosophy 106B Problem Set 1

Due: Monday February 22 at 6pm

Please submit your solutions to the course Gradescope page (link on Latte),

and remember the course collaboration policy :)

1. Devise formation rules for a language whose alphabet consists of the three

letters ‘a’, ‘b’, and ‘c’ and whose formulas are palindromes (that is, the

strings that read the same forward and backward).

Hint: Don’t forget that palindromes can be of odd and even length!

2. Rewrite the following assertion so that the formulas are mentioned in

pedantic style, without use-mention or shorthand conventions:

for every formula F of L=, the formula ∼ (F. ∼ F ) is derivable in Σ=.

3. Prove the assertion of the previous problem by showing how to obtain a

formal derivation using axioms (T1)–(T5) and modus ponens. (Here you

may avail yourself of the conventions, i.e. use abbreviations etc.)

4. Show using axioms (T1) and (T2) and modus ponens that for any formulas

F,G and H,

(a) if ` F ⊃ G and ` G ⊃ H then ` F ⊃ H

(b) if ` F ⊃ (G ⊃ H) then ` G ⊃ (F ⊃ H)

5. Using the result of problem (4a) and other axioms of (T1)–(T5), show

that for any formulas F and G, ` F ⊃ (∼ F ⊃∼ G).

6. Show that if Σ is a formal system that is truth-functionally complete and

contains modus ponens as a rule of inference, then Σ is consistent if and

only if it is consistent*.

7. Using the derivability of ∼ x = 0 ⊃ ∃z(x = Sz) but no further invocation

of the induction axiom, show that the following formulas are derivable in

PA:

(a) y + x = 0 ⊃ x = 0 . y = 0

(b) x ≤ 0 ⊃ x = 0

1

Extra Credit. Let F be a formula of LS containing no quantifiers and

only the one variable ‘x’. Show that F is true in the intended interpreta-

tion for either a finite number of values of ‘x’ or for all but a finite numbers

of values of ‘x’.

Hint: Do this first for atomic formulas, then show the property is pre-

served by negation and conditional.

2

学霸联盟

Due: Monday February 22 at 6pm

Please submit your solutions to the course Gradescope page (link on Latte),

and remember the course collaboration policy :)

1. Devise formation rules for a language whose alphabet consists of the three

letters ‘a’, ‘b’, and ‘c’ and whose formulas are palindromes (that is, the

strings that read the same forward and backward).

Hint: Don’t forget that palindromes can be of odd and even length!

2. Rewrite the following assertion so that the formulas are mentioned in

pedantic style, without use-mention or shorthand conventions:

for every formula F of L=, the formula ∼ (F. ∼ F ) is derivable in Σ=.

3. Prove the assertion of the previous problem by showing how to obtain a

formal derivation using axioms (T1)–(T5) and modus ponens. (Here you

may avail yourself of the conventions, i.e. use abbreviations etc.)

4. Show using axioms (T1) and (T2) and modus ponens that for any formulas

F,G and H,

(a) if ` F ⊃ G and ` G ⊃ H then ` F ⊃ H

(b) if ` F ⊃ (G ⊃ H) then ` G ⊃ (F ⊃ H)

5. Using the result of problem (4a) and other axioms of (T1)–(T5), show

that for any formulas F and G, ` F ⊃ (∼ F ⊃∼ G).

6. Show that if Σ is a formal system that is truth-functionally complete and

contains modus ponens as a rule of inference, then Σ is consistent if and

only if it is consistent*.

7. Using the derivability of ∼ x = 0 ⊃ ∃z(x = Sz) but no further invocation

of the induction axiom, show that the following formulas are derivable in

PA:

(a) y + x = 0 ⊃ x = 0 . y = 0

(b) x ≤ 0 ⊃ x = 0

1

Extra Credit. Let F be a formula of LS containing no quantifiers and

only the one variable ‘x’. Show that F is true in the intended interpreta-

tion for either a finite number of values of ‘x’ or for all but a finite numbers

of values of ‘x’.

Hint: Do this first for atomic formulas, then show the property is pre-

served by negation and conditional.

2

学霸联盟