SUBJECTIVE:existing in the mind; belonging to the thinking subject rather than to the object of thought (opposed to objective ).
OBJECTIVE:being the object of perception or thought; belonging to the object of thought rather than to the thinking subject (opposed to subjective ).
OBJECT: anything that may be apprehended intellectually:objects of thought.
THOUGHT: the act or process of thinking; mental activity:a consideration or reflection:
PROOF (informal): if you admit one can think of their own thought, then it follows that your thoughts can be the object of your thoughts and belong to the thinking subject at the same time.
also another person can think of your thoughts making it the object of their thought and belong to the thinking subject at the same time.
thus proving that subjective and
objective are not mutually exclusive
now this also applies to ALL thought, therefore, subjective is a subset of objective
in math the longer you use words or a wordy rationalization to prove what you want to show, the more likely the chance that is fallacious. so any weaknesses in the proof(s) i present here will be the portions in which i use rationale instead of mathematical set law. since i am confident in my reasoning skill i will highlight these portions in pink. so if you want to attack this proof, your best bet will be the portions in pink.)
in set notation:
S = the set of all subjective things
O = the set of all objective things
O′ = the set of all objects
T = the set of all thoughts
to prove a set A⊆ B (A is a subset of another set B)
Proof. Suppose a ∈ A. . . . Therefore a ∈ B. Thus a ∈ A implies a ∈ B, so it follows that A ⊆ B.
PROOF 1 ( T ⊆ O′):
since all t∈T may be apprehended by the thinking subject (original thinker) or by an external mind.
then all t∈O′
to prove a set A = B (A and B are the same set)
Proof. [Prove that A ⊆ B.] [Prove that B ⊆ A.] Therefore, since A ⊆ B and B ⊆ A, it follows that A = B
PROOF 2 (O = O′):
since any o∈O is being an object of thought
then all o∈O’
since any o′∈O′ is capable of being an object of thought or perception (proof by contradiction : consider an o’∈O′ that cannot be the object of thought or perception. the fact that its under consideration makes it an object of thought. alternate version: try thinking of something that cannot be objective, by doing so youve just made it objective. LOL)
then all o′∈O
PROOF 3 ( T⊆O):
since T⊆O′ (by proof 1)
then all t∈O′
and since all t∈O′ and O′=O (by proof 2)
PROOF 4 ( T = S ):
since all t∈T exist in the mind and belong to the thinking subject
then all t∈S
and since all s∈S are an act of thinking, mental activity,sensory perception, consideration, or reflection
then all s∈T
PROOF 5 ( S⊆O):
since T=S (by proof 4) and T⊆O (by proof 3)
then S⊆O (by transitivity)
PROOF 6 (S ≠ Ø):
thoughts exists, i could not be writing this or you could not perceive this if they didnt
therefore S ≠ Ø
PROOF 7 ( S∩O ≠ Ø):
since S⊆O (by proof 5)
then S∩O = S
and since S∩O = S and S≠Ø (by proof 6)
therefore S∩O ≠ Ø
this makes it incumbent upon the person relying on this definition set to either redefine the set or find another set that is logically consistent. now even though there may exist an alternate definition set that presents the term objective as mutually exclusive, i claim if its logically consistent it will match my new definition set only the terms have different names.