1. |
(i)
One can't tell. Some arguments with true premises and a true conclusion
are valid, of course. But some are invalid. For instance:
| |
Socrates
is a man.
So, Socrates is a philosopher. |
(ii)
It must be invalid. Since the premises are actually true and the conclusion
is actually false, it follows that it is not impossible for its premises
all to be true and its conclusion false simultaneously.
(iii)
One can't tell. Here is an example of a valid argument of the required
sort. For example:
| |
Socrates
is a Roman.
All Romans are mortal.
So, Socrates is mortal. |
One would
have an example of an invalid argument of the required sort, if
one replaced the conclusion with "Socrates is a philosopher".
(iv)
One can't tell. Here is an example of a valid argument of the required
sort:
| |
Socrates is a Roman.
All Romans are immortal.
So, Socrates
is immortal. |
One would
have an example of an invalid argument of the required sort, if
one replaced the conclusion with "Socrates is not a philosopher".
|
3. |
(i)
Yes. One cannot, of course, have a valid argument in which the premises
are true but the conclusion is false. If the conclusion is
false, at least one of the premises must be false. But that does not
mean that the premises must be inconsistent. Here is an example of
a valid argument with consistent premises and a false conclusion:
| |
Socrates is a philosopher.
All philosophers are cricketers.
So Socrates is a cricketer. |
(ii)
Yes. If the premises are themselves inconsistent.
| |
For
example:
2+2=5.
So, 2+2=4.
|
In fact any
argument with inconsistent premises will fulfil the conditions as
long as we count count "2+2=5" (say) as being inconsistent
with "Socrates is a philosopher" on the grounds that the
set {2+2=5, Socrates is a philosopher} is inconsistent.
(iii)
No. For suppose the premises were consistent; then it would be possible
for them to be true. But, if the argument is valid, whenever the premises
are true the conclusion must be true too. So if the argument is valid
and the premises are consistent, it must be possible for the premises
to be true and the conclusion true. But in that case the premises
are not inconsistent with the conclusion.
(iv)
Yes.
| |
For
example:
Socrates
is a man.
All men are mortal.
So, Socrates is mortal. |
The negation
of the conclusion is consistent with either of the premises taken
on its own, but not, of course, with the pair of them.
(v)
Yes.
| |
For
example:
2+2¹4.
So,
2+2=4. |
This fulfils
the conditions, since the negation of the conclusion is itself inconsistent,
which is enough to make the argument valid. In fact any argument
whose conclusion is a necessary truth will fulfil the conditions.
But there are also examples where the conclusion is not a necessary
truth. For instance:
| |
Socrates
is not mortal.
Socrates is a man.
All men are mortal.
So, Socrates
is mortal. |
Here the
premises are inconsistent. But there are also examples where the
conclusion is not a necessary truth and the premises are consistent.
| |
Socrates
is not a mortal philosopher. Socrates is a man.
All men are
mortal.
So, Socrates is mortal. |
(vi)
No. But this is not, perhaps, easy to see.
Let us call
the premise in question P, and the conclusion C. And, to make things
simple, let us suppose that there is just one other premise, which
we will call Q. So the argument
is, "P. Q; so, C", and we are going to argue that the
following can't all be true:
(i) The argument
is valid;
(ii) The
argument without P (i.e. Q; so, C) is invalid;
(iii) The
negation of C is inconsistent with the negation of P. (Let
us use "not-P" for the negation of P, and "not-C"
for the negation of C.)
Now, if (ii)
is the case, this means that {Q, not-C} is consistent. But in that
case either {P, Q, not-C} or {not-P, Q, not-C} (or both) must be
consistent. (If {Q, not-C} is consistent, there must be some way
in which both Q and not-C could be true; and that way must be compatible
with P being true or with P being false, since P must be one or
the other.) But if (iii) is the case, {not-P, not-C} is inconsistent;
so {not-P, Q, not-C} must be too. So {P, Q, not-C} will have to
be consistent. But that means that (i) is untrue.
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