What is intuitive for one person is not always intuitive for another, so permit me to make a guess at what you consider to be "intuition," and we'll see how close I am.
Much of philosophy, and indeed all of learning, is tied up in symbols words and phrases, and the manipulation thereof. Linguistically we call these the "syntax" of our language. Syntax describes what the symbols are and how they are allowed to be manipulated. This is contrasted with semantics, which is the study of the meaning of these symbols.
Sometimes we just understand the manipulation of these symbols at an intuitive level. For instance, most of us on this site can manipulate English nouns to make them plural at an intuitive level. We rarely think "The plural of 'cat' is 'cats' because most plural nouns are a concatenation of the singular noun plus the letter 's'." That statement is true, but most of us would simply say "The plural of 'cat' is 'cats'" because one cat is a "cat" and two or more cats is "cats." We have some intuitive sense of what it means to have many of something, and we intuitively understand what the correct word for it is.
This works great until you come across a word like "octopus," whose plural is not obvious (some say "octopi," others say "octopuses," others say "octopodes"). That's when we break out the "rules of grammar" and point out that "octopus" actually borrows from the Greek "oktopous," not Latin, so it should not be given the 2nd declension nominative plural ending.
But what just happened there? I suddenly shifted gears and started talking about syntax -- the construction of 2nd declension nouns in Latin. In Latin, singular nouns ending in -us are 2nd declension nouns, and the nominative plural of those end in -i. What I just described was syntactical. You don't have to know what it means for something to be in the 2nd declension. You don't have to know that the 2nd declension is most commonly used for masculine things. You can simply apply the rule.
This is where I think my linguistic argument starts to tie into your concept of "intuition." You aren't looking for the syntax of logic, you're looking for the semantics of it. You're looking for the meaning of the
→ symbol, not the syntactic manipulations that are allowed around it.
And therein lies the rub which virmaior wrote to. The meaning of
¬p ∨ q. Its a definition of the language. It has no deeper meaning, any more than the made up word "blartlfilthygong" has a deeper meaning.
But there's a bit more to it. Individual people do apply meaning to
p→q. And it has generally been found that philosophers who are aware of the definition of it as
¬p ∨ q generally come to agree on the meaning of
p→q enough for that symbol to facilitate communication.
Accordingly, what I believe you are after is two things:
- How can I arrive at the agreed upon meaning of this symbol?
- Why is this meaning so important that philosophers felt the need to give it a symbol?
I would argue your approach to the former, starting a question with "I pursue only intuition; please do not answer with formal proofs or Truth Tables." is tricky because the syntax is a major feature used to capture the meaning. We use syntax in this way because there is no one way to arrive at an intuitive sense of the meaning of the word. I program computers; there are plenty of arguments which make intuitive sense to me which would not make intuitive sense to an artist. In person, we can tailor our discussion of the symbols to fit what we know about the person we are talking to. Lacking knowledge of the person, the best we can do is fall back on syntax. How one can arrive at a meaning is simply out of scope in many cases, because everyone does it their own way.
The latter half, however, may prove more valuable in a forum such as this. There are typically many answers for why certain meanings resonated better than others. Again, stealing from virmaior's answer, if you are considering bivalence, Aristotle's 3 laws, and a set of operators defined by their truth functions, meanings such as "implication" naturally resonate and are worth capturing in a symbol such as
→. In fact, in this case, it resonates so strongly that propositional logic has two related symbols for this:
⊢. The former is fully defined within propositional logic via the definition above, while the latter is considered a "meta symbol" whose meaning is not fully defined within propositional logic. The meaning of
⊢ is far more subtle, and one way to capture this meaning is to look how it has been used historically, and why philosophers felt it was needed.
I think the latter concept may be something which can be explored in a setting such as StackExchange, because the question inherently has a historical context to it, and StackExchange is very good at topics that are tightly tied to history.