2

Here is the question I am attempting to express in propositional logic:

No man is weak, unless their name is Bob.

From this expression I see that every man who is weak must be called Bob (W => B), but the expression does not equate to all Bob's are weak (B => W), as being named Bob is simply the criteria for allocating everyone who isn't called Bob to the set of "all men who are not weak", meaning that the set of "all weak men" is a subset within the set of "all men named Bob".

I just want to make sure that my reasoning for why it can't be (B => W) is correct, I know that this might be obvious but I just found this confusing initially.

The Amplitwist
  • 795
IAhmed12
  • 47
  • 3

1 Answers1

5

Yes, your line of reasoning is entirely correct. The assertion

  • no man is weak, unless their name is Bob

just means that

  • if the person is not Bob, then they are not weak,

and says nothing about whether Bob is weak or strong.

As such, the correct translation is: $$\text{for each $x\;$ ($x$ is not Bob $\implies x$ is not weak)},$$ that is (by contrapositive), $$\text{for each $x\;$ ($x$ is weak $\implies x$ is Bob)};$$ thus, the set of weak men is a subset of the set of men named Bob—just as you've explained.

In general, $$P\text{ unless }Q$$ means $$P\text{ if not }Q,$$ i.e., $$(\text{not }Q)\text{ implies }P\tag1$$ or, equivalently, $$P\text{ or }Q.\tag2$$

Translation $(2)$ is nicely symmetrical and more succinct; on the other hand, Translation $(1)$ is more intuitive as it emphasises that fulfilling $Q$ leads to no certain consequence. (Thanks, @Bram28.)

ryang
  • 38,879
  • 14
  • 81
  • 179