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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