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philosophy note

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Department
Philosophy
Course
1002
Professor
James Hildebrand
Semester
Spring

Description
1. a. A branch consists of all the sentences that are picked out by starting with a sentence on the bottom of a tree and following a path up to the top. b. A branch is a completed open if and only if every non-literal sentence on the branch has been checked off and there is no atomic sentence that occurs together with its negation on the branch. c. A tree is open if and only if it is not closed, that is, if and only if at least one branch does not contain an atomic sentence together with its negation. d. A tree is unfinished if and only if at least one branch has an unchecked non-literal sentence on it and does not (yet) contain an atomic sentence and its negation. e. A non-literal sentence is a sentence that is neither an atomic sentence nor the negation of an atomic sentence. 2. a. P itself. b. ~(P ≡ Q) 3. The negation of the corresponding material conditional. 4. a. A completed open branch proves that P cannot be t-f false. This is because it proves that there is at least one tva on which P is true. b. A completed open branch proves that P and Q are not t-f equivalent. This is because it proves that there is at least one tva on which ~(P ≡ Q) is true, meaning that there is at least one tva on which P ≡ Q is false, meaning that there is at least one tva on which P and Q have different truth values. c. A completed open branch proves that the argument is t-f invalid. This is because there is at least one tva on which the negation of the corresponding material conditional is true, which means that there is at least one tva on which the corresponding material conditional is false, which means that there is at least one tva on which the corresponding material conditional has a true antecedent and a false consequent. Since the antecedent is the iterated conjunction of the premises and the consequent is the conclusion, and since we know that an iterated conjunction is only true if all its components are true, we know that there is at least one tva on which all premise
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