In the first scene of the play, Bassanio asks Antonio for a loan so that he can travel to Belmont and woo Portia. Bassanio already owes Antonio money, but he offers an analogy to convince Antonio to give him the loan (even though Antonio has already offered to help his friend):
In my school-days, when I had lost one shaft,
I shot his fellow of the selfsame flight
The selfsame way, with more advised watch,
To find the other forth; and by adventuring both,
I oft found both.
Bassanio claims that when he lost an arrow as a child, he would shoot another arrow, follow its flight, and find both arrows at once. His logic here is that he will do the same with Antonio's loan: though he has already lost the money that his friend initially gave him, he will be more careful with this next round of money and ultimately be able to repay Antonio in full. The fact that Bassanio used this logic as a child helps to support his argument: he suggests that he has always been clever and has solved problems like this before, even at a young age.
This logic does, however, prove somewhat ridiculous given Bassanio's track record. Bassanio owes money to others besides Antonio—so why should this next loan be any different? While the analogy seems fitting, Bassanio doesn't have any concrete evidence that he will get the money back, as he never actually has in the past. These lines therefore speak to Bassanio's nature: he appears charming and innocent, but is, in actuality, immature and possibly knowingly manipulative.
Portia's suitors use logos as they attempt to deduce which casket holds her portrait—and while they all make convincing cases, each of their arguments proves troublesome.
In Act 2, Scene 7, the Prince of Morocco is the first suitor to take a guess. He begins by reading the lead casket, which states “Who chooseth me must give and hazard all he hath.” He reasons that he would not risk everything for dull lead. Next, he reads the silver casket—“Who chooseth me shall get as much as he deserves”—and reasons that he does not want to undervalue all that he deserves by choosing silver. Lastly, he reads the gold casket: “Who chooseth me shall gain what many men desire.” Morocco views Portia as the object of that sentence, as men travel from around the world to woo her. He ultimately chooses the gold, offering one last explanation of his logic:
One of these three contains her heavenly picture.
Is ’t like that lead contains her? ’Twere damnation
To think so base a thought. It were too gross
To rib her cerecloth in the obscure grave.
Or shall I think in silver she’s immured,
Being ten times undervalued to tried gold?
O, sinful thought! Never so rich a gem
Was set in worse than gold.
But as Morocco chooses incorrectly, his reasoning proves flawed. He is, overall, too focused on the material of the caskets themselves; he fixates on appearances in a way Portia's father has tried to discourage. Worse yet, he uses his logic to equate Portia with the casket material. Though he attempts to honor her by choosing the gold, he ultimately ends up objectifying Portia. This moment also alludes to Merchant's idea of love as a form of currency.
The Prince of Aragon is next to guess in Act 2, Scene 9. Aragon rejects the lead as he, like Morocco, doesn't wanting to risk everything for something so plain-looking. He also rejects gold, as he does not wish to be like the "many men" that judge solely based on appearance. The main flaw in Aragon's argument arises here: he claims that he does not want to give undue weight to looks, and yet he has just rejected the lead casket because he considers it unattractive. Choosing silver and remarking that all men should get what they deserve, Aragon also ends up wrong.
Last is Bassanio, who guesses in Act 3, Scene 2. He makes a sweeping argument that substance is more important than appearance, remarking that "The world is still deceived with ornament." This mindset allows him to correctly choose the lead casket and win Portia's hand. But while Bassanio seemingly defies the other men's appearance-centric logic, his argument has holes, too: he still values Portia for her beauty, and his speech is less than convincing, as its elaborate language suggests his focus on appearing intelligent.
In court, the law seems to be on Shylock's side. But as he is about to cut into Antonio and collect his bond, Portia (disguised as the lawyer Balthazar) tells him,
The words expressly are "a pound of flesh."
Take then thy bond, take thou thy pound of flesh,
But in the cutting it, if thou dost shed
One drop of Christian blood, thy lands and goods
Are by the laws of Venice confiscate / Unto the state of Venice.
Portia uses logos—appealing to an audience's sense of logic—to interpret the law even more literally than Shylock, reasoning that the bond allows Shylock to collect Antonio's flesh, but not his blood; if Shylock sheds any of Antonio's blood, Shylock will be charged with murder. Shylock can neither refute Portia's claim nor collect the flesh without the blood, and Portia thus saves Antonio from certain death.
And Portia doesn't stop there—she continues to use logos to punish Shylock. She states,
If it be proved against an alien
That by direct or indirect attempts
He seek the life of any citizen,
The party ’gainst the which he doth contrive
Shall seize one half his goods; the other half
Comes to the privy coffer of the state,
And the offender’s life lies in the mercy
Of the Duke only, ’gainst all other voice.
[...] Thou hast contrived against the very life
Of the defendant, and thou hast incurred
The danger formerly by me rehearsed.
Portia traps Shylock again, asserting that he—an "alien"—has conspired to kill Antonio, a Venetian "citizen." As punishment, per Venetian law, Shylock's property will be confiscated; Antonio can seize half, while the state will seize the other. Shylock's life, meanwhile, lies at the mercy of the Duke.
Shylock demanded the law, and that is exactly what he received. This moment aptly illustrates how, in Merchant, powerful people can manipulate the law—as well as each other—to achieve the outcomes they desire.
In court, the Duke asks Shylock how he can expect to be treated mercifully if he doesn't exhibit any mercy. Shylock responds with a speech using logos:
You have among you many a purchased slave,
Which, like your asses and your dogs and mules,
You use in abject and in slavish parts
Because you bought them. Shall I say to you
"Let them be free! Marry them to your heirs!"
[...] You will answer
"The slaves are ours!" So do I answer you:
The pound of flesh which I demand of him
Is dearly bought; ’tis mine and I will have it.
If you deny me, fie upon your law.
Shylock demonstrates that the law is on his side by grounding his argument in a logical analogy. This speech also exposes Christians' hypocrisy: Shylock's ownership of Antonio's flesh is essentially the same as Christians' ownership of slaves, and yet people interpret the two situations differently, for no logical reason. Shylock also, once again, alludes to the fact that his cruelty is the result of Christians' prior cruelty to him: he acknowledges that his bond is inhumane, but Christians' own lack of humanity sets the precedent for what he now seeks to exact for himself.