Format will be a verse reference in Titus where the word is found, followed by the Greek lemma (root word), a transliteration, English rendering in two translations (NIV, then NRSV), and my own comment.
(NB. about halfway through, I realised the word study was going to be WAY too long, but I started so i'll finish. But pleased be assured, future word studies will be much shorter. :))
1:9 uses four different words related to preaching/teaching,
διδαχὴν, didachḗ, taught/teaching,
παρακαλέω, parakaléō, encourage/preach,
διδασκαλία, didaskalía, doctrine/doctrine,
ἐλέγχω, elénchō, refute/refute,
These words relate to the task of an elder/bishop/overseer/steward. What strikes me about these words is that the role of the teacher is not presented as a passionless task but both parakaléō and elénchō carry tones of urgency and the desire to convince others, either positively of the truth, or negatively by exposing falsehood.
1:11, διδάσκω, didáskō, teaching/teaching, here the verb 'to teach' is used of the false teachers who are damaging the church in Crete. All teaching is therefore not equal and must be judged to some extent on whether it edifies (builds up) or upsets/ruins the church.
1:13, ἐλέγχω, elénchō, rebuke/rebuke, this word from 1:9 reappears but here this term is rendered by both NIV and NRSV as rebuke rather than refute. Perhaps a nuance that is lost in translation is that elénchō carries with a a connotation of effective conviction, not merely empty rhetoric but actually bringing about a new realisation in the one being rebuked. This is born out by the context here where Paul's desire is for the false teachers to become 'sound in the faith' as a result of the rebuke.
λαλέω, laléō, teach/teach, while the word simply means 'speak', in context Paul is clearly detailing for Titus the content of his teaching hence the translation here by the NIV and NRSV. (NB. In 2:2 and 2:3 there is no Greek word where the NIV/NRSV use teach/tell but that is the sense implied.)
διδασκαλία, didaskalía, doctrine/doctrine, makes a second appearance again coupled with ὑγιαίνω, hygiaínō, translated by both NIV and NRSV as 'sound', giving us 'sound doctrine.' 'Sound', as in incorrupted, is is a figurative rendering, literally the two words together mean 'healthy teaching.' Although the word was used as a technical term in philosophical discourse of the day, I still think the idea of 'healthy teaching' has currency in our (health obsessed) culture today. 'Sound doctrine' on the other hand sounds awefully stuffy and boring!
2:4, σωφρονίζω, sōphronízō, train/encourage, is a fascinating word that specifies instruction or encouragement towards sensible or right behaviour. This is reinforced in 2:5 by the use of the adjective σώφρων, sṓphrōn, self-controlled/self-controlled.
2:6, παρακαλέω, parakaléō, encourage/urge, as in 1:9 but translated differently by the NRSV. The sense is still of a strong urgent appeal.
2:7, τύπος, týpos, example/model, shows that Titus is not just to teach with his words but also with his whole life, he is to be the Christian on which the young men can model themselves.
2:12, παιδεύω, paideúō, teaches/training, this word is used to refer to the act of educating or disiplining a child. Here it reminds us that before God's grace we are all children equally in need of being taught constantly how to live rightly. Paul and Titus may be teaching others, but they themselves depend on God's teaching.
2:15 sees another cluster of words related to our subject,
λαλέω, laléō, teach/declare,
παρακαλέω, parakaléō, encourage/exhort,
ἐλέγχω, elénchō, rebuke/reprove,
ἐπιταγή, epitagḗ, authority/authority, this word, which essentially means 'command' and hence implies authority, shows that Titus needs to be willing to take authority in situations and lay down the law in respect to those things which are being falsely taught. Now I am a big fan of conversation and discussion and giving people space to come to their own conclusions but some things cannot be tolerated and must be named for what they are. In particular when destructive views are being pushed on to other vulnerable members of the community. Drawing this line sensitively continues to be one of the hardest jobs a church pastor does.
3:1, ὑπομιμνῄσκω, hupomimḗskō, remind/remind, so here Titus is merely going over old ground, this is not new teachig for the Cretan's but it is important and so needs to be repeated. Teacher's mustn't be afraid to go over the 'basics' time and again. After all doesn't matter how much time you spend on the cutting edge if you dont have the foundation in place you are on a hiding into nothing.
3:8 διαβεβαιόομαι, diabebaióomai, stress/insist, this final word in oour study means to 'maintain strongly' or 'speak confidently' or 'insist', in this case the insistence is on the life of good works that comes in response to the transformation that God gives through Jesus by the Spirit. Paul is leaving no doubt that 'good works' could be some kind of optional extra, they are of vital importance.