The following five passages contain
arguments. For each argument formulate it as a standard form categorical syllogism. Indicate the Major
Term, Minor Term, and Middle Term of each categorical syllogism. Indicate the mood and figure of each
categorical syllogism. Assuming the Boolean Standpoint, determine the validity of each categorical
syllogism by producing a Venn Diagram for the syllogism. Provide a brief statement about the Venn
Diagram that indicates what in the diagram shows invalidity or validity. If the categorical syllogism is
invalid explain which of our text’s rules of validity the categorical syllogism violates.
- No red kings are red queens, for no red queens are knaves, and no red kings are knaves.
- Since no red kings are queens of hearts, and some queens of hearts are playing cards, it follows that
some playing cards are red kings. - Since some coins desired by coin collectors are circulating coins, and no coin coveted by a coin
collector is common, we may conclude that a few rare coins are found is circulation. - Anyone who should be invited to a pun festival enjoys playing with language, but no one who writes
logic problems should be invited to a pun festival, so no one who enjoys pl - Construct a truth table for the following statement. Indicate whether the statement is a tautology, a
contradiction, or a contingent statement. Do not use any shortcuts in the table. After completing your truth
table explain how it indicates your answer. (10 marks)
[A v (A º ~B)] º ( ~B É A) - Construct a truth table to determine whether the following pair of statements are logically equivalent.
Do not use any shortcuts in the table. After completing your truth table explain how it indicates your
answer. (10 marks)
~ (A • B) v ~ (B v C)
(~ A v ~ B) v (~ B • ~ C) - Construct a truth table to determine whether the following argument is valid or invalid. Do not use
any shortcuts in your truth table. After completing your truth table explain how it indicates your answer.
(10 marks)
1) ~ (A • B)
2) ~ ( ~ B v C)
~ A
Propositional Logic Proof Questions 13-15: (10 marks each) Provide a formal proof of validity of the
following arguments. You may use CP, or IP, in addition to any of the 18 rules of inference.
13.
1) ~ (A É B)
2) C É B
3) ~ A v D / D • ~ C
14.
1) [ (G • E) • B] v ( ~ G É ~ B)
2) G É E
3) E É B / G º B
15.
1) B º [ (L • P) v W]
2) ~L É W
3) ~ (P • B) • ~ ( ~ P • ~ B) / ~P - Symbolize the following argument, using symbolized statements and the relevant truth–functional
connectives. Indicate the upper case letters you use to symbolize ordinary language simple statements.
Determine whether the argument is valid or invalid. In order to do that, you may construct a complete
truth table, use the indirect method, or construct a formal proof of validity. (10 marks)
Taylor is lonely, and she will either find a mate or remain single.
If Taylor does not remain single, then she will become a mother.
If Taylor is lonely if and only if she remains single, then Taylor will not remain single.
Therefore, Taylor will become a mother.
Predicate Logic Translation Questions 17 – 20: (5 marks each 20 marks total) Translate the
following ordinary language statements into predicate logic symbolic form. You can use the predicate
letters that are provided. Questions #19 – #20 are more advanced, please note the extra instructions for
them. - No coconuts are pink fruit. (C, P)
- Used cars are good if and only if they were well maintained and have low mileage. (U, G, M, L)
Questions #19 – #20 are translations involving relational predicates, overlapping quantifiers, and
identity. You can use the predicate letters provided. - Every grandparent is the parent of a parent of someone. (Gx: x is a grandparent; Pxy: x is a parent
of y) - There is at least one famous custodian. (Fx: x is famous; Cx: x is a custodian)
Predicate Logic Proof of Validity Questions 21-23: (10 marks each) The following 3 arguments are
valid. Provide a proof of validity for each argument. You can use the 18 rules of inference, CP, IP, the
rules for introducing and removing quantifiers and the Change of Quantifier rules. Questions #22 and #23
are more challenging. They involve relational predicates, overlapping quantifiers, and identity.
21. - (x) [Wx É (Xx É Yx)]
- ($x) [Xx • (Zx • ~ Ax)]
- (x) [(Wx É Yx) É (Bx É Ax)] / ($x) ( Zx • ~Bx)
22. - (x) (y) (Fxy É ~Fyx)
- Fba / Fba
23. - (Fb • Gab) • (x) [(Fx • Gax) É x=b]
- ($ x) [(Fx • Gax) • Hx] / Hb
Proving Invalidity Question 24: (10 marks) The following argument is invalid. Show it to be invalid
using the finite universe method.
24. - (x) (Hx É Mx)
- ($x) (Mx• Bx) / ($x) (Hx • Bx)