For Monday, read Chapter 4, sections 1 and
2.
Nongraded Homework: Problems at the end of section 2.

P Q S P → Q, S → Q, ~ Q \
~ P & ~ S
T T T T T
^ ^ ^ ^
T T
^
T T
^
T
^
T
^ ^
T
^ ^
T
^ ^ ^
T
^ ^ ^
T T
^ ^
T
^
T T T T
^
^
T
^
T T
^
T
^ ^
T T T
^ ^
T T
^
T T
^ ^
^ ^ ^
T T T T T T

Answer? Valid. There is no line showing all true premises and a false conclusion. Thus, it is impossible for the argument to have all true premises and a false conclusion. So, the reasoning can ’t go wrong. Whenever the premises are all true, the conclusion is as well.
Remember, when an argument is valid, there is no particular line that proves validity.

Each row on the truth-table represents a relevant possibility (an interpretation), and taken together the rows represent all of the relevant possibilities
(all possible interpretations).
So, if there is a row on the truth-table with all true premises and a false conclusion, then it is possible to have all true premises and a false conclusion —thus the argument is invalid; and if there is no row with all true premises and a false conclusion, it is impossible to have all true premises and a false conclusion, and by definition, the argument is valid.

A B G A → (B & G), ~ B
\
A

~ B
T T T T T
^
T T
^
T
^
T
T
^ ^
^
^
^
^
^
^
^
^
T T
T
^
^ ^
T
^ ^ ^
T
T
T
^
^
T
^ ^
T
Answer: Invalid, proven by line seven

Try going straight to an interpretation that makes all of the premises true and the conclusion false:
A B C (A v ~ B) ↔ C, ~ C \ A → C
T
^ ^ T ^ T ^
The only possible way to make the conclusion false and the second premise true automatically makes the first premise false.
So, the argument is valid.

A B C (C v B) → ~ A , ~ C \ A → C
T
^ ^ ^ ^ ^ T ^ T ^
Invalid
--Make A true and C false: False conclusion
--C ’s being false automatically makes the second premise true.
--A ’s being true makes the consequent of the first premise false; so we must assign
^ to B to make the antecedent false (by making the wedge false), so that the entire premise is true.

Assign truth-values to statement letters so as to make the conclusion false.
If it ’s possible to assign truth-values to the remaining statement letters in a way that makes all of the premises true, then the argument is invalid.
If this cannot be done, the argument is valid.

If there ’s more than one way to make the conclusion false, then, before you conclude the argument is valid, be sure to check all of the possibilities (reaching a dead end in each case).
Partly for this reason, when your answer is
‘valid’, we want you to explain in English the process by which you arrived at your answer.

A B D A → B, D → B, ~ B \ A ↔ D
If A is true and D false, the conclusion is false.
Then, because A is true, B has to be true (to make the first premise true). That, however, makes the third premise false.
BUT, we have to check the other way of making the conclusion false before we answer ‘valid’. If D is true and A false, then B has to be true to preserve the truth of the second premise. But this makes the third premise false. So, the argument is valid.