Grant Edwards <grant.b.edwa...@gmail.com>:
> On 2016-08-17, Marko Rauhamaa <ma...@pacujo.net> wrote:
>
>> Somewhat analogously, I remember how confusing it was to learn formal
>> logic in college. I was having a hard time getting the point of
>> definitions like:
>>
>> (x ∧ y) is true iff x is true and y is true
>>
>> That's because I had learned in highschool that "x ∧ y" was just an
>> abbreviation of "x and y".
>
> It is. The expression "x ∧ y" is the same as "x and y". And that
> expression is true "iff x is true and y is true". It's just a sligtly
> more explicit way of writing the expression...
Well, not quite.
Notice the word "and" after "iff". That word is on a different plane
than "∧". The word "and" is on the semantic plane while "∧" is part of
the syntax. (Of course, that would be true even if "∧" were written
"and".)
The formal sentence template
(x ∧ y)
contains the symbols "(", "∧" and ")". However, "x" and "y" are not part
of the formalism; rather, they are semantic placeholders for arbritrary
formal sentences.
The rest of the definition:
is true iff x is true and y is true
is plain-English semantics.
In particular, the definition is *not* identical with the formal
sentence:
(x ∧ y) ↔ (x ∧ y)
Marko
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