- Suppose the conclusion of an argument is a tautology. What can you conclude about the validity of the argument?
My answer: Valid, because this argument is impossible to have true premises and false conclusion.
- What if the conclusion is a contradiction?
My answer: Invalid when an argument is possible to have true premises and false conclusion; otherwise, Valid.
- What if one of the premises is either a tautology or a contradiction?
My answer: Invalid when an argument is possible to have true premises and false conclusion; otherwise, Valid. (It is the form that makes arguments valid. Plus we could asume a premise were true even this premise is a contradiction.)
ADDENDUM 1 to add my new answers after reading the accepted answer
- Suppose the conclusion of an argument is a tautology. What can you conclude about the validity of the argument?
My new answer: Valid. Because $\forall$x C(x), it is impossible to have $\exists$x[ $\forall$y P(x,y) $\land$ $\neg$ C(x) ].
- What if the conclusion is a contradiction?
My new answer:It depends on the premises.
Valid when $\forall$x [$\forall$yP(x,y) $\to$ C(x)]
Invalid when $\exists$x[ $\forall$y P(x,y) $\land$ $\neg$ C(x) ]
- What if one of the premises is either a tautology or a contradiction?
My new answer:
Valid when $\forall$x [$\forall$yP(x,y) $\to$ C(x)]
Invalid when $\exists$x[ $\forall$y P(x,y) $\land$ $\neg$ C(x) ]
ADDENDUM 2
Valid
$\forall$x [$\forall$yP(x,y) $\to$ C(x)]
Invalid
meaning: Negate the above formula
$\neg$ { $\forall$x [$\forall$yP(x,y) $\to$ C(x)] }
$\iff$
$\exists$x[ $\forall$y P(x,y) $\land$ $\neg$ C(x) ]
x $\in$ {1,…,$\mathrm{2}^{n}$}
y $\in$ {1,…,n}
P(x, y) A premise in (x,y) enty of the truth table.
C(x) The conclusion in the row x of the truth table.
Best Answer
Every conditional with a tautological consequent is a logical validity.
In other words, every argument with a tautological conclusion is valid.
A conditional with a contradiction as consequent is a logical validity precisely when its antecedent is unsatisfiable.
In other words, an argument with a contradiction as conclusion is valid precisely when its premises are inconsistent.
A conjunction's truth value is not altered by eliminating one of its tautological conjuncts; therefore, a tautological premise confers no information about the argument's validity.
Every conjunction with a contradiction as conjunct is a contradiction; therefore, every argument with a contradiction as one of its premises is valid.