Logical Equivalence

LOGICAL EQUIVALENCE

We have talked about the error of mixing up a statement and its converse, and the error of thinking that just because a statement is true its converse must also be true. However, sometimes both a statement and its converse happen to be true. In this case we have a situation of logical equivalence. That is, if A implies B and also B implies A, then A and B are logically equivalent – whenever A is true, B must be true, and also whenever A is false, B must be false.

This means that A and B are logically interchangeable, and are usually just different viewpoints on the same thing. Crucially, this doesn’t mean they’re exactly the same, as illustrated in the example of Obamacare and ACA above. Logically those are the same, but emotionally they are very different to some people, who feel fine supporting something with the calm and compassionate name “Affordable Care Act”, but can’t bear the idea of supporting something referring to Obama. Some other people have the opposite response, where the reference to Obama makes them more positively disposed towards it. We will later come back to the logical fallacy of false equivalence, where two things are assumed to be logically equivalent when they are not, such as when having a degree is taken as being equivalent to being intelligent. However, the Obama/ACA example is a case of “false inequivalence”, where some people take things to be different, when in fact they’re logically equivalent. Still, we should accept that they are not emotionally equivalent, and work with this fact rather than simply deny it on the grounds that it contradicts logic. We will come back to these issues in Chapter 15 on emotions.

When two things are logically equivalent, the implication flows in both directions, so we use this symbol: A B. In words there are several ways of saying this, and they come in symmetrical pairs as the logic flows both ways:

• A is true if and only if B is true.

B is true if and only if A is true.

• A is a necessary and sufficient condition for B.

B is a necessary and sufficient condition for A.

• A is logically equivalent to B.

B is logically equivalent to A.

• If A is true B is true, and if A is false B is false.

If B is true A is true, and if B is false A is false.

This last pair sheds some light on the fact that “A implies B” tells us nothing about the case when A is false: if we want to deduce something from A being false we need the converse although, on the face of it, the converse gives us a way to deduce something from B being true rather than from A being false. We’ll come back to this in the next chapter, in which we are going to explore what it means for things to be false.

Footnotes: [1] The question of whether or not men experience sexism comes down to the definition of sexism. According to some theories of prejudice, sexism and racism should only be used for instances that fit into a bigger picture of systematic oppression. The idea is that when an oppressed group is biased against their oppressors, this is not the same as when the oppressors are biased against the oppressed group as a form of control. Whether or not we agree with these definitions I think it is important to notice the difference between oppressed people and oppressors, and we will come back to this in Chapter 13 on analogies. Thinking of it in these abstract terms clears up something about the difference; giving it a name would help us think about it.

[2] This approach of looking for white men would make sense if you knew that the gang had five people and you also know there are exactly five white men in the country. It might even be reasonable if you know there are exactly ten white men in the country, because if you round up five of them the odds are quite high that you’ll get some of the actual criminals. This approach becomes gradually less like logic and more like racism as the country’s population of white men grows. In reality this profiling is likely to be done against non-white people, not white people. How big does the country or city’s population of the minority have to be before this veers from rational into racist? This is a question of grey areas that we will come back to in Chapter 12, and we have to be very careful that incremental arguments do not give us logical justifications of prejudice.

[3] I recently discovered that, technically, diagrams like the ones above aren’t supposed to be called “Venn diagrams” but rather “Euler diagrams”. To qualify as a Venn diagram, it is apparently supposed to exhibit all possible logical combinations of the sets in question. The ones above don’t qualify because one circle is completely contained within the other, so there isn’t a region where you can be in the small circle without being in the big one. Of course, this is the entire point of that particular diagram. Personally I think the distinction between Venn diagrams and Euler diagrams is more like pedantry than precision, so I shall go on referring to these as Venn diagrams, especially as that term is much more well known.

[4] Note that the logical equivalence of these statements doesn’t mean they are true and it certainly doesn’t mean they are morally justifiable. It just means that logically they mean the same thing. White people are killed by police too, including in traffic stops. The questions of whether black people are killed disproportionately and whether this is because of racism are different, much more complex questions.