Given below are two statements followed by two conclusions. Assuming these statements to be true, decide which one logically follows?

Question

Statements:

1. No manager is a leader.
2. All leaders are executives.

Conclusions:

1. No manager is an executive.
2. No executive is a manager.

Which conclusion is true ?

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How to solve by truth table or first order logic ?

Answer 1

No one of conclusions is true:

1. Ad Conclusion 1: If there is a manager who is an executive, too, it doesn't contradict neither statement 1 nor 2.

2. Ad Conclusion 2: If there is an executive who is a manager, too, it again doesn't contadict statements 1 and 2.

Answer 2

Typically, you can't solve first-order logic problems using a truth-table, since truth-tables can only explore finitely many types of possible worlds, and with quantifiers, you typically need to consider infinitely many types of possible worlds. However, as Noah pointed out, we can still do something like this:

'L': 'is a leader'

'E' : is an executive

'M' : is a manager

so now we can symbolize the premises as $M \rightarrow \neg L$ and $L \rightarrow E$, and the conclusions as $M \rightarrow \neg E$ and $E \rightarrow \neg M$ respectively, and using a truth-table you'll find that neither conclusion follows.

This kind of truth-functional analysis will also work for this argument:

1. Some managers are leaders ($M \land L$)

2. All are leaders are executives ($L \rightarrow E$)

$\therefore$ 3. Some managers are executives ($M \land E$)

as 3 can be inferred from 1 and 2 using a truth-table, and the argument is indeed valid.

But if the argument is:

1. Some managers are leaders ($M \land L$)

2. Some executives are leaders ($E \land L$)

$\therefore$3. Some managers are executives ($M \land E$)

then the truth-table will say it is valid, but the argument is in fact not valid.

In fact, even this argument:

1. Some managers are leaders ($M \land L$)

$\therefore$ 2. All executives are leaders ($E \rightarrow L$)

would be truth-functionally valid, but it is certainly not a valid argument.

So the moral is: you want to be super careful in using truth-functional analysis for arguments involving quantifiers, since you can easily draw the wrong conclusion!

Fortunately, there are still plenty of other methods to solve these kinds of problems. If the inference is invalid (as in this case), you can try and construct a counterexample to demonstrate its invalidity, which is exactly what MarianD and Theophile did. And as Joffan points out, the two conclusion are equivalent, so one counterexample will serve to show that neither conclusion follows.

Why are the 2 conclusions equivalent? For this, we can use certain equivalence laws. Conclusion 1 symbolizes to $\forall x (M(x) \rightarrow \neg E(x))$, and conclusion 2 is: $\forall x (E(x) \rightarrow \neg M(x))$, and you now see how the formula's after the quantifiers are really just each other's contrapositive, so they are equivalent.

We could also symbolize conclusion 1 as $ \neg \exists x (M(x) \land E(x))$ and conclusion 2 as $ \neg \exists x (E(x) \land M(x))$, and now you see the difference uis just a commutation, so once again they are equivalent.

And to see that $ \neg \exists x (M(x) \land E(x))$ is equivalent to $\forall x (M(x) \rightarrow \neg E(x))$ there are two important general equivalences for quantifiers that say that $\neg \forall x P(x) \Leftrightarrow \exists x \neg P(x)$ and that $\neg \exists x P(x) \Leftrightarrow \forall x \neg P(x)$ , so:

$ \neg \exists x (M(x) \land E(x)) \Leftrightarrow$ (Quantifier Negation)

$ \forall x \neg (M(x) \land E(x)) \Leftrightarrow$ (DeMorgan)

$ \forall x (\neg M(x) \vee \neg E(x)) \Leftrightarrow$ (Implication, i.e. $P\rightarrow Q \Leftrightarrow \neg P \vee Q$)

$ \forall x (M(x) \rightarrow \neg E(x))$

If any of the inferences were valid, you could try any of a number of methods to demonstrate this validity, e.g. formal derivations.

But this particular argument is an interesting one, since it is an example of a categorical syllogism: its only predicates are 1-place predicates, reflecting categories of things, in this case managers, executives, and leaders. For categorical logic, a number of additional technoiques are available to analyze arguments, and one of my favorites is to use a Venn Diagram. Here is the Venn diagram analysis for this particular argument:

The areas that are shaded reflect areas that are claimed to be empty. That is since premise 1 says that no managers are leaders, we can shade the intersection between managers and leaders: there is nothing that is both a manager and a leader!

To reflect that all 'all leaders are executives', we shaded the area inside the Leaders circle but outside the Executives circle, since we want to rule out any leader that is not an executive.

So now the diagram represents the truth of the premises; it contains all the information contained in the premises. Now, is that enough to draw any of the supposed conclusions? Well, to say that No managers are executives, you would need the intersection between Managers and Executives to be empty. But, we don't know if it is empty: there xcould be something in that bottom part of the intersection between Managers and Executives, i.e. there could be a Manager that is also an Executive, but not a Leader. And so we see that the conclusion (whether 1 or 2, since as we already saw: they make the same claims really) could be false, meaning that either inference is invalid. Indeed, we have once again generated a counterexample, but in a systematic fashion, and in fact we now see that MarianD's counterexample was a little incomplete, since she should have specified that the manager that is an executive is not a leader (Theophile did this better).