Question

Determine if the following arguments are valid or invalid by analyzing them and identifying if Universal Modus Ponens, Universal Modus Tollens, Universal Transitivity or one of the fallacies were used. Carefully define all predicates and universe of discourse you are using. Write your comments directly on this sheet. Also give a counter example for the invalid arguments using a complete sentences to explain why it is invalid. Universal Modus Ponens: If given that for all x in the universe of discourse P(x) implies Q(x) and that P(a) is true then we can conclude that Q(a) is also true. ?x (P(x) ? Q(x)) P(a) ? Q(a) Universal Modus Tollens: If given that for all x in the universe of discourse P(x) implies Q(x) and that ¬Q(a) is true then we can conclude that ¬P(a) is also true. ?x (P(x) ? Q(x)) ¬Q(a) ? ¬P(a) Universal Transitivity: If given that for all x in the universe of discourse P(x) implies Q(x) and Q(x) implies R(x) then we can conclude that P(x) implies R(x) is also true. ?x (P(x) ? Q(x)) ?x (Q(x) ? R(x)) ? ?x (P(x) ? R(x)) (1) A ? B ? C and x ? A. Therefore x ? C. (Hint: do not introduce predicates here, just use the definition and math expressions)

          Determine if the following arguments are valid or invalid by analyzing them and identifying if Universal Modus Ponens, Universal Modus Tollens, Universal Transitivity or one of the fallacies were used. Carefully define all predicates and universe of discourse you are using. Write your comments directly on this sheet. Also give a counter example for the invalid arguments using a complete sentences to explain why it is invalid. 

Universal Modus Ponens: If given that for all x in the universe of discourse P(x) implies Q(x) and that P(a) is true then we can conclude that Q(a) is also true. 

?x (P(x) ? Q(x)) 
P(a) 
? Q(a) 

Universal Modus Tollens: If given that for all x in the universe of discourse P(x) implies Q(x) and that ¬Q(a) is true then we can conclude that ¬P(a) is also true. 

?x (P(x) ? Q(x)) 
¬Q(a) 
? ¬P(a) 

Universal Transitivity: If given that for all x in the universe of discourse P(x) implies Q(x) and Q(x) implies R(x) then we can conclude that P(x) implies R(x) is also true. 

?x (P(x) ? Q(x)) 
?x (Q(x) ? R(x)) 
? ?x (P(x) ? R(x)) 

(1) A ? B ? C and x ? A. Therefore x ? C. (Hint: do not introduce predicates here, just use the definition and math expressions)

Determine if the following arguments are valid or invalid by analyzing them and identifying if Universal Modus Ponens, Universal Modus Tollens, Universal Transitivity or one of the fallacies were used. Carefully define all predicates and universe of discourse you are using. Write your comments directly on this sheet. Also give a counter example for the invalid arguments using a complete sentences to explain why it is invalid.

Universal Modus Ponens: If given that for all x in the universe of discourse P(x) implies Q(x) and that P(a) is true then we can conclude that Q(a) is also true.

?x (P(x) ? Q(x))
P(a)
? Q(a)

Universal Modus Tollens: If given that for all x in the universe of discourse P(x) implies Q(x) and that ¬Q(a) is true then we can conclude that ¬P(a) is also true.

?x (P(x) ? Q(x))
¬Q(a)
? ¬P(a)

Universal Transitivity: If given that for all x in the universe of discourse P(x) implies Q(x) and Q(x) implies R(x) then we can conclude that P(x) implies R(x) is also true.

?x (P(x) ? Q(x))
?x (Q(x) ? R(x))
? ?x (P(x) ? R(x))

(1) A ? B ? C and x ? A. Therefore x ? C. (Hint: do not introduce predicates here, just use the definition and math expressions)

Added by Trinidad M.

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