Aristotle called these types of appeals or proofs "artistic (ἔντεχνοι )" as opposed to "nonartistic (ἄτεχνοί)." Artistic proofs were devised through the art of rhetoric and were in contrast to nonartistic proofs such as what we might call "evidence:" witnesses, contracts, public records, etc.
All three persuasive appeals can be found in the Bible.
Paul gives a great example of an ethos argument in Galatians 1:
1:13 You have heard, no doubt, of my earlier life in Judaism. I was violently persecuting the church of God and was trying to destroy it. 14 I advanced in Judaism beyond many among my people of the same age, for I was far more zealous for the traditions of my ancestors. 15 But when God, who had set me apart before I was born and called me through his grace, was pleased 16 to reveal his Son to me, so that I might proclaim him among the Gentiles, I did not confer with any human being, 17 nor did I go up to Jerusalem to those who were already apostles before me, but I went away at once into Arabia, and afterwards I returned to Damascus. 18 Then after three years I did go up to Jerusalem to visit Cephas and stayed with him fifteen days; 19 but I did not see any other apostle except James the Lord’s brother. 20 In what I am writing to you, before God, I do not lie!Every argument in this passage is meant to build up Paul's character, to portray him as a trustworthy source for the authoritative gospel that he is preaching.
A great example of a pathos argument can be found in Jesus' lament over the city of Jerusalem in Luke 13:34.
Luke 13:34 Jerusalem, Jerusalem, the city that kills the prophets and stones those who are sent to it! How often have I desired to gather your children together as a hen gathers her brood under her wings, and you were not willing!Beginning this sentence with a repetition of the city's name demonstrates the use of a powerful and gripping figure of speech (epanalepsis) and lays Jesus' emotions bare. This figure is then followed up by the touching and evocative simile in which Jesus is compared to a mother hen trying to gather her scattered chicks.
1 Corinthians 15 is an example of a logos appeal as Paul tries to convince his audience of the importance of the resurrection of Jesus.
15:16 For if the dead are not raised, then Christ has not been raised. 17 If Christ has not been raised, your faith is futile and you are still in your sins. 18 Then those also who have died in Christ have perished. 19 If for this life only we have hoped in Christ, we are of all people most to be pitied.This passage is a good example of what Aristotle called an enthymeme, which is also called a rhetorical syllogism. A rhetorical syllogism, which is aimed at persuasion, is a logical syllogism with an unstated premise. The logical syllogism in this passage would go something like this, premise 1: our ultimate hope is resurrection; premise 2: Christ is not raised from the dead; conclusion: Therefore we have no hope. What is left out is the first premise. It is left out because it is a rhetorical assumption of the enthymeme that his audience expresses its ultimate hope in the resurrection.
Try looking through your NT sometime and try and assign given passages to one of these three persuasive appeals. I think you will find it interesting and enlightening in trying to get at the heart of any given passage. Ask yourself, what is the author trying to persuade me of, how are they doing that, how effective is it?