2023F T1 with solutions Q

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University of Toronto *

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PHL245

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Philosophy

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Jan 9, 2024

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1. Suppose you have a consistent set of three sentences. Suppose we construct a new set by adding a fourth sentence that is a contradiction. What semantic property does this new set have? Brie�ly explain your reasoning. Be sure to de�ine your key terms. (4) A set of sentences is consisten t if there is at least one truth - value assignment that makes all the sentences true (at the same time). A sentence is a contradiction if it is false under every truth- value assignment. This new set must be inconsistent. Because the fourth sentence is always false, then there will necessarily never be a truth - value assignment where all 4 sentences can be true at the same time. 2. Suppose φ and ψ are logically equivalent well -for med formulas, and that Z is an atomic statement . Consider the following argument: φ∧Z ∴ ψ . What semantic property or properties does this argument have? How do you know that the argument has this property or these properties? Be sure to de�ine your key terms. (4) If φ and ψ are logically equivalent then they always have the same truth value (under every TVA). The argument φ∧Z ∴ ψ is valid. The only way it could be invalid (i.e. not valid) is if the premises are true and the conclusion is false at the same time. But the only way that φ∧Z is true is if φ and Z are both true. But if φ is true, then ψ must be true as well. So the invalid case can never occur , and thus the argument is valid. 3. Suppose φ is a tautology. Now suppose that the argument φ ∴ ψ is invalid. Do we know that ψ is a contradiction? Circle your answer. (1) YES, WE KNOW THAT ψ IS A CONTRADICTION NO, WE DO NOT KNOW THAT ψ IS A CONTRADICTION
4. Are the following symbolic sentences of�icial, informal, or not well - formed? Circle your answer. If it is of�icial or informal, draw an arrow pointing to the main - connective. If it is not well - formed, draw an arrow (or arrows) pointin g to the mistake (or mistakes) in the sentence. (2 marks each) a. ((~P)→(X∨Y)) O FFICIAL INFORMAL NOT WELL - FORMED b. (P ↔Q)∧X∧~Y O FFICIAL INFORMAL NOT WELL - FORMED c. ~(P→(Q↔(X∨~Z))) O FFICIAL INFORMAL NOT WELL - FORMED d. (X→~Y)∨Z↔P∨~Q O FFICIAL INFORMAL NOT WELL - FORMED 5. Provide a shortened truth-t able that demonstrates that the following set of sentences is consistent. (3) { ~((~Q∨P)∨(~Q↔~(R→P))) } P Q R F T T 6. Provide a shortened truth-t able that demonstrates that the following argument is i nvalid . (4) ~(~Q→(T∧P)). (S→P)∧(T→S). (S∨(P↔R)). ∴ P→(T↔(Q∨R)) P Q R S T T F T T/F F
7. Provide a f ull truth-t able for the following symboli c sentence. (4). What semantic property does this sentence have? Brie�ly explain your answer. (2) P ↔((~P→Q∧R)↔(~Q∨P∨~R)) ANSWER: Tautology . Because this sentence is true under every truth- value assignment. For questions 8-12 , symbolize the English sentence using the abbreviation scheme provided for each question. (4 marks each) 8. Steve will come over to Sarah’s house tonight and help her (Sarah) weed her garden, unless he (Steve) has to work late. P: Steve will come over to Sarah’s house tonight. Q: Steve has to work late. R: Steve will help Sarah weed her garden. (P∧R)∨Q ~Q→(P∧R) 9. Neither Steve nor Sarah enjoys weeding the garden, but weeding their garden is necessary for the plants in their garden to thrive. P: Steve enjoys weeding the garden. Q: Sarah enjoys weeding the garden. R: Sarah or Steve weed their garden. S: The plants in Sarah and Steve’s garden thrive.
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