Does "positive" mean good and "negative" bad?

Solution 1:

Positive and negative mean exactly what they seem to mean. They are the most straightforward ways to express good or bad when used as adjectives. I am excluding other contexts e.g. mathematics, as that is terminology.

The synonyms listed have additional connotations, so you shouldn't use them unless you want to convey those additional meanings.

• Desirable - Describes a useful, but not mandatory, attribute. Also can be used to describe a positive attribute of sexuality. Use this word only if the additional nuance is helpful, instead of confusing (and possibly embarrassing).
• Eulogy - Used as a noun, specifically, as a posthumous speech of praise to the deceased. If used in the wrong context, e.g. software development, it will cause confusion i.e. as a metaphor for a no longer supported product. Eulogic is NOT a word.
• Panegyric - Yes, that means praiseworthy. Encomium means praise. Both are stilted and inappropriate for everyday usage. In a formal setting, laudatory would probably be best. It is more comprehensible and familiar. If the setting is not formal, positive is a better word choice.

Solution 2:

Usage of positive and negative for good and bad respectively is widespread but pretentious. This goes beyond the commonplace and perverse preference for the trisyllable over the monosyllable, as utilize for use: those who use positive and negative in this way are pretending to be scientifically objective or otherwise "non-judgmental" while in fact expressing a value judgment. My vote goes resoundingly for the unabashed use of good and bad no matter how formal or informal the context.

As for panegyric, it is adjectival in form but most commonly used as a noun, as many Greek adjectives were and are (like rhetoric). Its original meaning is appropriate to a public festival. As applied to discourse, therefore (as in the title of one of Isocrates' written pretend-orations), it means what Aristotle calls epideictic rhetoric, which he declares naturally deals in praise and blame. In English the word has come to mean a speech of praise. By adding the Latin adjectival reflex to the Greek one, we form the adjective panegyrical, which has a somewhat Shandean ring to it.

Solution 3:

I frequently correct people (being a semantics freak) and tell them that a negative statement is a statement that denies, while a positive statement is one that affirms.

You're close. The word negative suffers from multiple meanings and few synonyms. In the sense you are using it, "negative" is correct, but the opposite of negative is not "positive." The opposite of "negative" in this sense is "affirmative."

If I were to restate your sentence then,

...a negative statement is a statement that negates (same meaning as denies in this sense), while an affirmative statement is one that affirms.

In this sense, a negative statement would be "I am not a cat." Meanwhile, an affirmative statement would be "I am a human being."

Negative and positive are opposites in other senses. Mathematically, there are positive (+) numbers, and negative (-) numbers. I can have a positive (good) attitude, or a negative (bad) attitude.

Just as "good" is the opposite of "bad" and of "evil," "negative"s meaning depends on what you contrast it with. I hope this helps!