Exercise 1.4 - Answers

 

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".

2.

(i) Easily! For instance:

 


Socrates is a man.
So, Socrates is a philosopher.

Or:

Socrates is a Roman.
So, Socrates is a philosopher.


(ii) No. All arguments with inconsistent premises are valid.

(iii) Yes.

 


2+2=4.
So Socrates is a philosopher.

(It could have been that 2+2=4 and that Socrates was 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+24.
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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