One practical method of organizing arguments is to identify relationships between terms. Terms may be related as different parts of a whole (including different steps in a process) or as different species of a genus. In his “Letter from Birmingham Jail” Martin Luther King Jr. uses both methods of relating terms to organize and clarify his arguments. Continue reading Relating Terms from Birmingham Jail
“Without the aid of trained emotions the intellect is powerless against the animal organism… In battle it is not syllogisms that will keep the reluctant nerves and muscles to their post in the third hour of bombardment. The crudest sentimentalism (such as Gaius and Titius would wince at) about a flag or a country or a regiment will be of more use. We were told it all long ago by Plato. As the king governs by his executive, so Reason in man must rule the mere appetites by means of the ‘spirited element.’ The head rules the belly through the chest.” — C. S. Lewis, The Abolition of Man
Have you ever read through the Declaration of Independence and thought to yourself, “What a clear presentation of reasoning. Why can’t our leaders today argue in such a straightforward way?” If you read the Declaration carefully, you can discern this pattern of reasoning:
If P then Q
P
Therefore Q.
This pattern of reasoning is called modus ponens, and the three parts of the Declaration can be understood in this way.
If any form of government becomes destructive of our unalienable rights of life, liberty, and the pursuit of happiness, then it is the right of the people to alter or to abolish it, and to institute new government.
The government of the king of Great Britain has shown itself to be destructive of these rights (as shown in a long list of grievances).
Therefore, these United Colonies are, and of right ought to be free and independent States.
One wishes that such clear reasoning would once again rule in the hearts and minds of our nation, its people and its rulers.
Two types of rules can be used to justify steps in formal proofs: rules of inference and rules of replacement. In order to use these properly, you should understand the differences between them.
The main difference is that rules of inference are forms of valid arguments (that’s why they have a therefore ∴ symbol), but rules of replacement are forms of equivalent propositions (which is why they have the equivalence sign ≡ between the two parts). This fundamental distinction is the cause of all other differences in how they are applied in proofs. Continue reading Rules for Proofs
Great stories often owe their greatness in part to dilemmas that confront the protagonist, who must make some difficult choice. Below, I have summarized several example dilemmas from stories I love. As you read through them, try to figure out which method is (or could be) used to escape the dilemma in the story: going between the horns, grasping the horns, or rebutting the horns with a counter-dilemma.
The Odyssey
If Odysseus sails close to the rocks then he will lose some men to Scylla, but if he sails close to the whirlpool then he will lose his entire ship to Charybdis. He must either sail close to the rocks or close to the whirlpool. Thus he will either lose some of his men to Scylla or lose the entire ship to Charybdis.
The Aeneid
If Aeneas stays in Carthage then he will not fulfill his destiny to found Rome, and if he flees to Italy then he will lose the pleasures of a kingdom. He will either stay or flee, therefore he will either lose Rome or lose Carthage.
The Fellowship of the Ring
If Frodo goes to Mordor alone, then he will likely fail in his quest, but if he goes to Mordor with the fellowship then he endangers his friends. He will either go alone or with the fellowship. Therefore he will either endanger his friends or he will likely fail in his quest.
The Lion, the Witch, and the Wardrobe
If the Narnians release the traitor Edmund to the Witch then he will be killed, and if they do not let the Witch have him as her rightful kill for treachery then Narnia will perish in fire and water. The Narnians must either release Edmund, or not let the Witch have her rightful kill. Therefore either Edmund will be killed, or Narnia will perish.
The Adventures of Tom Sawyer
If Tom Sawyer confesses that Injun Joe killed Dr. Robinson, then Injun Joe will kill him. If he doesn’t confess, then Muff Potter will be falsely accused. He will either confess or he won’t. Hence, either Injun Joe will kill him, or Muff Potter will be falsely accused.
Watership Down
If Hazel and his rabbits again ask the Efrafans for some does then they will be imprisoned. If they try to fight the Efrafans then they will lose. They either ask them or fight them. Therefore they will either be imprisoned or defeated in battle.
The Princess Bride
If Westley and Buttercup enter the Fire Swamp then they will be killed by flame, quicksand, or R.O.U.S. If they do not enter the Fire Swamp then they will be captured by Humperdinck. They enter the Fire Swamp or they do not, so they will either be killed or captured.
Harry Potter and the Sorcerer’s Stone
If Harry seeks the Sorcerer’s Stone then he will be expelled, but if he does not seek the Stone then Voldemort will return. Harry will either seek the Sorcerer’s Stone or he will not, so he will either be expelled or Voldemort will return.
Can you think of dilemmas that the protagonists face in other stories you have read?
Shorter truth tables can help us find if an argument is valid, or a set of propositions are consistent, or if two propositions are equivalent. However, when completing a shorter truth table, we must sometimes guess a truth value for a variable. This occurs when there are no “forced” truth values — that is, when there exists more than one way to complete the current truth value for every remaining proposition.
Here are two rules to keep in mind when you must guess a truth value:
Take a look at this post for a flowchart for guessing with validity.
I have read that the Apostle Paul was well educated in classical literature, and it is fun to find indications of that fact. In 2 Corinthians 3:3 he wrote, “you are an epistle of Christ, ministered by us, written not with ink but by the Spirit of the living God,
not on tablets of stone but on tablets of flesh, that is, of the heart.
This is an apparent allusion to Pericles’ Funeral Oration (431 BC), when that great statesman told the Athenians,
in foreign lands there dwells also an unwritten memorial of them, graven not on stone but in the hearts of men.
The Apostle Paul knew his Pericles, just as he elsewhere echoed Aristotle.
Mr. Nance,
In Copi’s 14th edition of Introduction to Logic, one problem reads, “Iran and Libya both do not raise the price of oil.” The symbolic translation is ~I • ~L. I thought it might also be translated as ~(I • L). However, using a truth table to check for equivalence, I found the two are NOT equivalent.
Later in the exercise there is a problem that reads, “Either Iran raises the price of oil and Egypt’s food shortage worsens, or it is not the case both that Jordan requests more U.S. aid and that Saudi Arabia buys five hundred more warplanes.” The symbolic translation is (I • E) ∨ ~(J • S). I’m confused by reading “…it is not the case both that Jordan requests more U.S. aid and that Saudi Arabia buys five hundred more warplanes” as ~(J • S). That seems a lot like saying “It is not the case both that Iran and Libya do not raise the price of oil,” which I thought might be translated ~(I • L).
Can you explain how to read this correctly? That is, why are they not logically equivalent? Or did I just mess up royally?
Thanks so much.
You are correct in saying that ~(p • q) is not equivalent to ~p • ~q. How then do we determine the correct form for statements that use “both” and “not”?
Fundamentally, we must use the forms that reflect the meaning of the statements. The form ~(p • q) means “not both p and q”, as in “Tom and Jim are not both from Idaho.” The form ~p • ~q means “both not p and not q” which is equivalent to “neither p nor q”, as in “Tom and Jim are both not from China.”
Practically, the first thing to ask when symbolizing statements like this is, “Which comes first in the statement, the ‘not’ or the ‘both’?” If it is ‘not both’ then it is probably the form ~(p • q). If it is ‘both not’ then is is probably the form ~p • ~q. Let’s apply this to the statements in question.
1. “Iran and Libya both do not raise the price of oil.” This is correctly symbolized ~I • ~L. The meaning is that neither Iran nor Libya raise the price of oil.
2. “It is not the case both that Jordan requests more U.S. aid and that Saudi Arabia buys five hundred more warplanes.” This is correctly symbolized ~(J • S).
You have too many nots in your second to last paragraph, which is confusing the issue. But I trust that my explanation clears things up.
For more on this issue, read this EARLIER POST.
Blessings!
Before studying categorical syllogisms, students learn to translate statements into standard categorical form. The first step is translating the statement such that it uses only the “to-be” verb, so the form becomes [Subject] [to-be verb] [Predicate nominative]. This standardizes the statements so that the arguments are more easily analyzed, which is beneficial when the arguments themselves get more complicated.
But it can result in some very strange statements, e.g. translating “The Apostle Paul rebuked Peter at Antioch” into
The Apostle Paul was a Peter-at-Antioch rebuker.
Most spell-checkers will mark “rebuker” with that squiggly red underline, and some students might balk at the goofy compound noun.
Also, if one is not careful to keep the meaning the same, some of the translations can get rather awkward, such as turning “Susan works hard to resist temptation” into (ahem),
Susan is a hard-to-resist temptation worker.
Most of my students have found the awkwardness of such translated categorical statements to be merely funny, and have just taken it in stride. But occasionally a student will be bothered by it, perhaps thinking that their answers (and thus they themselves) will be thought of as strange or weird. In a larger classroom setting, when everyone is saying the same strange statements, they get used to it pretty fast, but it might be different in a home school setting, or among a small set of students.
The awkwardness of the translations can often be reduced by simply adding a normal noun in a normal place, trying to make the statement sound as normal as possible. For example, rather than translating “The forests will echo with laughter” into
The forests will be with-laughter echoers,
an acceptable translation would be
The forests will be places that echo with laughter.
This requires the addition of a new noun (“places”), but it is perfectly correct. The two rather awkward statements from above could also be correctly translated
The Apostle Paul was a man who rebuked Peter at Antioch.
Susan is a girl who works hard to resist temptation.
This method usually results in long predicates, but more ordinary sounding statements. For more on this topic, read my earlier post, Common errors to avoid: The “to be” verb.
