Semantic Contradiction in QL

Test 3 Please complete your work individually without other resources (notes, textbook, internet, etc.). Please write your name only on the back of the final page. 1. Fill in the blanks with the terms from the word bank. Note that there are more terms than spaces and not every word will be used (3 points each): Semantically Contingent Sentence

in QL

Semantic Tautology in QL

Semantic Contradiction in QL

Sentence of QL

Well-formed Formula of QL

Singular Term

Predicate

Variable

Bound Variable

Free Variable

Extension

Referent

Universe of Discourse

Model

i. ______________________________________ An occurrence of a variable, x, that is not within the scope of an x-quantifier ii. ______________________________________ A sentence of QL that is true in every model iii. ______________________________________ An expression that refers to a specific, person, place or thing iv. ______________________________________ The object that a singular term refers to v. ______________________________________ A well-formed formula of QL that contains no free variables vi. ______________________________________ An interpretation in QL, consisting of a set that is the universe of discourse, individual elements of that set that are the referents of the singular terms in the sentences or argument, and subsets of the universe of discourse that are the extensions of the predicates used in the sentences or argument 2. Use the given symbolization key to translate the following English sentences into the version of QL that contains the identity predicate (though you need not use identity in every translation) (3 points each): (i) If Anne Hathaway is an actress, then Meryl Streep is an actress. (ii) Everyone is famous. (iii) There is an actress who is famous. (iv) Every actress has won more awards than Anne Hathaway. (v) Meryl Streep has won more awards than Anne Hathaway, but no one else has. (vi) There are at least two famous people. UD: people Fx: x is famous. Ax: x is an actress. Wxy: x has won more awards than y. h: Anne Hathaway s: Meryl Streep

 

 

3. Put a check mark next to the singular terms. Circle the proper names. Box the definite descriptions. Perform multiple of these operations for each expression where warranted (4 points): ____ Sonia Sotomayor

____ Wahoo!

____ is bumpy

____ the current president of WKU

____ a herd of elephants

____ St. Louis

____ in the woods

____ the smallest positive odd number

4. The following statements comprise the recursive definition of a well-formed formula of QL (where “A” and “B” are metavariables ranging over expressions of QL and “χ” is a metavariable ranging over variables of QL). 1. Every atomic formula is a wff. 2. If A is a wff, then ∼A is a wff. 3. If A and B are wffs, then (A & B) is a wff. 4. If A and B are wffs, then (A ∨ B) is a wff. 5. If A and B are wffs, then (A → B) is a wff. 6. If A and B are wffs, then (A ↔ B) is a wff. 7. If A is a wff, χ is a variable, A contains at least one occurrence of χ, and A contains no χ-quantifiers, then “χA is a wff. 8. If A is a wff, χ is a variable, A contains at least one occurrence of χ, and A contains no χ-quantifiers, then $χA is a wff. 9. All and only wffs of QL can be generated by applications of these rules. For each of the following strings of symbols, identify whether it is a well-formed formula of QL and explain why using the definition above. Then identify whether it is a sentence of QL and explain why (5 points each): i. “z(Hz ↔ Iz) ii. ($yAy ∨ By)

 

 

5. List all of the subsets of the set {Mercury, Venus} (2 points): 6. Are the following sentences true or false, given A = {α, β, γ} and B = {γ, d} (1 point each): i. ______________ A Í B ii. ______________ β Î A 7. State whether the following sentences are true or false in the given model (2 points each): UD = {α, β, γ} extension (P) = {γ, β} extension (Q) = {γ} referent (a) = α referent (b) = β i. ______________ Pa ii. ______________ $xQx iii. ______________ “xPx iv. ______________ “z(Qz ∨ ~Qz) v. ______________ “yQy ∨ “y~Qy 8. Use models to show that the following sentence is semantically contingent in QL (6 points): “yMy 9. Use a model to show that the following is not true (that is, that the argument is invalid in QL) (6 points): {$xFx, $xGx} ⊨ $x(Fx & Gx)

 

 

10. Show that the following is true using a derivation. USE ONLY THE AXIOMS OF QL AND THE RULES FOR IDENTITY. Be sure to follow the proper form and include all line numbers and justifications as well as the content on each line (8 points): {~Ka, “zMz, ~Kb → “zLz, a = b} ⊢ “z(Lz & Mz) 11. Show that the following is true using a derivation. USE ONLY THE AXIOMS OF QL. Be sure to follow the proper form and include all line numbers and justifications as well as the content on each line (8 points): {$yAy, “z(Az → Bz)} ⊢ $xBx

 

 

12. Show that the following is true using a derivation. YOU MAY USE ANY DERIVED RULES (INCLUDING THE RULES OF REPLACEMENT) THAT HAVE BEEN LISTED IN THE TEXTBOOK, AS WELL AS THE AXIOMS OF QL. Be sure to follow the proper form and include all line numbers and justifications as well as the content on each line (8 points): ~”x(Px & Qx) ⊢ $x(~Qx ∨ ~Px) Extra Credit: Is the following true? Use either derivations or a model to support your answer. If you use derivations, you may use only the axioms of QL (3 points, no partial credit): $x”yFxy ⊣⊢ “y$xFxy

 

 

Name:_______________________