**Paradoxes**

**by Marcus Loane**

There is a story about a landowner on whose land there was a bridge. He had made the decree:

Whoever passes over this bridge from one side to the other, must first take an oath whence he comes and what business he is going about. If he swear true, let him pass, but if he tell a lie, he shall die for it upon the gallows, without any remission.

This worked fine until one day a man came up to the bridge and declared:

By the oath I have taken, I swear I am going to die upon that gallows which stands yonder, and that is my business and no other.

The bridge keepers were left perplexed. If they hang him, he was telling the truth so they should not have hanged him. If they don't hang him he would be lying so they should hang him.

That was an example of a paradox.

There are simpler ones. Consider the following which we will label A:

**"This sentence is false."**

If A is true then according to A it is false. If A is false then it is true. We are unable to say if it is true or false. Yet it cannot be both true and false. Perhaps it is just meaningless?

We cannot get off that lightly. Consider this sentence B:

**"This sentence is false or meaningless."**

If B is true then according to B it cannot be true. If B is false then it is neither false nor meaningless so it must be true. If B is meaningless then according to B it must be true so it cannot be meaningless! So B is neither false, true or meaningless. So what is it? Perhaps it is meaningful but not true or false like a question or a command is. However the structure of it appears to be making a claim which suggests it must be true or false.

**Why tall people cannot exist**

A] **"A person who is short, is still short after
growing one millimeter."**

If A] is true then it does not matter how many times the short
person grows by one millimeter, according to A they will still be
short. Does that mean that A] is false? This is a much easier
paradox to figure out - I'll leave it up to you.

And for the mentally agile:

**Why 1+1=3**

B]** " IF this very proposition is true THEN 1+1=3
"**

Now for a conditional proposition (if-then) to be true, all that is needed is for the conclusion to follow from the condition (the way the conditional proposition "IF the moon is made of cheese THEN moon rocks are made of cheese" is true). In B] the conclusion does indeed follow from the condition therefore B is true. Now for the conclusion to be true the condition also has to be true, but we have already shown that the condition is true since it refers to the whole conditional proposition. Therefore the conclusion 1+1=3 is true.

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Marcus Loane