INTPs are pensive, analytical folks. They may venture so deeply into thought as to seem detached, and often actually are oblivious to the world around them.
Precise about their descriptions, INTPs will often correct others (or be sorely tempted to) if the shade of meaning is a bit off. While annoying to the less concise, this fine discrimination ability gives INTPs so inclined a natural advantage as, for example, grammarians and linguists.
INTPs are relatively easy-going and amenable to almost anything until their principles are violated, about which they may become outspoken and inflexible. They prefer to return, however, to a reserved albeit benign ambiance, not wishing to make spectacles of themselves.
A major concern for INTPs is the haunting sense of impending failure. They spend considerable time second-guessing themselves. The open-endedness (from Perceiving) conjoined with the need for competence (NT) is expressed in a sense that one's conclusion may well be met by an equally plausible alternative solution, and that, after all, one may very well have overlooked some critical bit of data. An INTP arguing a point may very well be trying to convince himself as much as his opposition. In this way INTPs are markedly different from INTJs, who are much more confident in their competence and willing to act on their convictions.
Mathematics is a system where many INTPs love to play, similarly languages, computer systems--potentially any complex system. INTPs thrive on systems. Understanding, exploring, mastering, and manipulating systems can overtake the INTP's conscious thought. This fascination for logical wholes and their inner workings is often expressed in a detachment from the environment, a concentration where time is forgotten and extraneous stimuli are held at bay. Accomplishing a task or goal with this knowledge is secondary.
INTPs and Logic -- One of the tipoffs that a person is an INTP is his or her obsession with logical correctness. Errors are not often due to poor logic -- apparent faux pas in reasoning are usually a result of overlooking details or of incorrect context.
Games NTs seem to especially enjoy include Risk, Bridge, Stratego, Chess, Go, and word games of all sorts. (I have an ENTP friend that loves Boggle and its variations. We've been known to sit in public places and pick a word off a menu or mayonnaise jar to see who can make the most words from its letters on a napkin in two minutes.) The INTP mailing list has enjoyed a round of Metaphore, virtual volleyball, and a few 'finish the series' brain teasers.
INTPs in the main are not clannish. The INTP mailing list, with a readership now in triple figures, was in its incipience fraught with all the difficulties of the Panama canal: we had trouble deciding:
whether or not there should be such a group,
exactly what such a group should be called, and
which of us would have to take the responsibility for organization and maintenance of the aforesaid group/club/whatever.
"Clever" is the word that perhaps describes ENTPs best. The professor who juggles half a dozen ideas for research papers and grant proposals in his mind while giving a highly entertaining lecture on an abstruse subject is a classic example of the type. So is the stand-up comedian whose lampoons are both funny and incisively accurate.
ENTPs are usually verbally as well as cerebrally quick, and generally love to argue--both for its own sake, and to show off their debating skills. ENTPs tend to have a perverse sense of humor as well, and enjoy playing devil's advocate. This sometimes confuses, even angers, those who don't understand or accept the concept of argument as a sport.
ENTPs are as innovative and ingenious at problem-solving as they are at verbal gymnastics; on occasion, however, they manage to outsmart themselves. ENTPs can be prone to "sharp practice" – especially cutting corners without regard to the rules if it's expedient – or, their juggling acts may simply be so over-ambitious they collapse.
Both at work and at home, ENTPs are very fond of "toys" -- physical or intellectual, the more sophisticated the better. Once these have been "solved" or become too familiar, however, they’ll be replaced with new ones.
ENTPs are basically optimists, but in spite of this (perhaps because of it?), they can become petulant about small setbacks and inconveniences. (Major setbacks they regard as challenges, and tackle with determination.) ENTPs have little patience with those they consider wrongheaded or unintelligent, and show little restraint in demonstrating this. In general, however, they are genial, even charming, when not being harassed by life.
In terms of their relationships with others, ENTPs are capable of bonding very closely and suddenly with their loved ones. Some appear deceptively offhand with their nearest and dearest; others are so demonstrative that they succeed in shocking co-workers who've only seen their professional side. ENTPs are also quick to spot a kindred spirit, and good at acquiring friends of similar temperament and interests.
ENTPs may sometimes give the impression of being largely oblivious to the rest of humanity except as an audience: good, bad, or potential. In general this is unfair – but it can be difficult to get an ENTP’s attention when they’re not immediately aware of you, especially for an Introvert.
The best approach in communicating with an ENTP is to be straightforward. No games – they’ll win. No "pulling rank" – they’ll just want to put you in your place. No apologies – you’ll undermine yourself. Try "I need/want to talk to you."