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Truth Tables Are Stipulations

A student emailed me about `... some confusion in regards to the ∨ symbol representing `or’ ...’

‘... some confusion in regards to the ∨ symbol representing ‘or’...’

I think the source of the confusion is this mistake.
It's a mistake to think that ∨ represents or!
Let me explain.
Here's the truth table for v ...
Oops, there's a mistake in this truth table. Let's fix it ...
TTT<-- ‘surely this is illogical’
Now we've got the right truth table.
The truth table is a stipulation about the meaning of v, the disjunction symbol.
We're not trying to capture the meaning of any English word.
The student goes on to ask this:

‘What makes the ∨ inclusive or exclusive, and is it always one of the two, or can it mean different things in different situations?’

It ‘is clearer with the example: ‘The pig went down the left fork or the right fork.’ Clearly this is a case where it can only be one or the other. The pig cannot go down both forks.’