Let's Prove It!
A good argument is hard to find. So when a friend told me his theory about the answer to the "Is the Glass Half
Full of Half Empty?" puzzle, I had to snatch it up. Here we discover the truth of optimism... it also makes a great
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A glass is a glass: Right? Then why has it been
used to represent the dichotomy that is
pessimism and optimism? Imagine this scene
many moons ago: a man is sitting at his table and
his cup is before him. His wife fills his cup with
drink and they sit to eat their dinner. As time
passes, he sees his cup is not as full and thinks
"My cup should always be full." and tells his
wife to pour him some more. She tells him to
finish that wine first. He replies "In my universe,
my cups are not allowed to be half empty. It
must be filled." To which she replies "Well this
universe does not have the funds to support
your universe. Here, any wine in your cup is a
blessing so it is half full. Now eat your dinner"
According to the Merriam-Webster Dictionary,  a glass is "2; something made of glass; esp: tumbler" A tumbler is "2: a drinking glass without foot or stem."
These definitions do not make mention of anything occupying the space of a glass so we can say that a glass has a volume that is intrinsically empty. Thus,
we can state the universal statement that "all drinking glasses are intrinsically empty."

When we say "Is the glass half full or half empty" we are really asking if the volume of the glass is occupied by stuff and is it equal to "a half", and if so, was
it subtracted or added from the properties of the glass?

To do this, let's say that the volume occupied by the glass is
V (the empty set) which is a subset of the universe of the glass, say universe G. We then see if
the element, say
k, does not exist in V (the empty set or "the glass is empty") or does exist in V (there is a volume occupied by stuff). If the element k exists
V, we then see if k is equal to "a half' such that "a half" is a predetermined element of some other universe.

If these condition are met, we can then see if
k was added or subtracted from the properties of the glass.
Since the glass is by definition empty, any solid, liquid, gas or stuff in the plasma state, is in a
conjunction to the properties of the glass and is an addition.

Therefore, since the definition of a glass, as stated above, means the glass is empty, any element
k that exists in V must be an element of another set that is in
conjunction with
V. Hence, the properties of a glass do not allow for anything but V to be empty and thus anything occupying space in V is an addition.
Thus, the glass is always considered "half full".


Let the truth of optimism prevail!

The Merriam-Webster Dictionary, 1997
Suggested Reading

Symbolic Logic 5th ed. Irving Copi. Prentice Hall. 1979
How to Prove It Daniel Velleman. Cambridge. 1994
Elementary Analysis: The Theory of Calculus Kennith A Ross. Springer. 2000
How to Read and Do Proofs Daniel Solow. Wiley. 1982
The Iliad Homer. Long time ago.
Let me just say that this is a great argument when the universe is for the glass and its subset of volume. Maybe if we look at it semantically, the meaning is different.
Perhaps this phrase best makes sense when looked at with the perspective of the element occupying the volume. By removing the volume and making that the universe, we
can say that the volume cannot be the empty set because then it would not exist (as a volume) and thus must contain the element
k. Since this element may be equal to "a
half" we can see if this element is added or subtracted to the properties of the volume itself. This would prove difficult since the element would be the volume and since it
is the volume, it cannot be a half of itself. We would need another universe and that is not something I feel like doing.