Tag Archives: Propositional Logic

Rules for Proofs

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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 →

Dilemmas in Stories

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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?

What comes after Logic? Rhetoric!

Introductory and Intermediate Logic together provide a complete foundational logic curriculum. Informal, categorical, and modern propositional logic are all included. The next step in your student’s classical education is to begin to apply what he has learned in logic to effective speaking and writing. This means your student should move on to the study of formal rhetoric, the capstone of a classical education. Rhetoric applies the tools of logic – defining terms, declaring truth, arguing to valid conclusions, and refuting invalid ones – to the persuasion of people. Rhetoric puts flesh onto the bones of logical analysis, that we may breathe arguments into life through the wise use of fitting words.

Fitting Words: Classical Rhetoric for the Christian Student is a complete formal rhetoric curriculum. Presented from a thoroughly Christian perspective, Fitting Words provides students with tools for speaking that will equip them for life. Drawing from Aristotle, Quintilian, Augustine, and others, and using examples from the greatest speeches from history and scripture, this robust curriculum guides Christian students in the theory and practice of persuasive communication.

The complete curriculum includes:

  • Student text with 30 detailed lessons
  • Student workbook with exercises for every lesson
  • Answer key for the exercises and tests
  • Test packet with nine tests, review sheets for every test, and speech judging sheets
  • Video course in which the author introduces and teaches through every lesson

After Intermediate Logic?

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What is recommended after Intermediate Logic? The short answer is: Rhetoric! But let me give you a bit more than that.

Introductory and Intermediate Logic together provide a complete foundational logic curriculum. Informal, categorical, and modern propositional logic are all included. The next step in a student’s classical education is to begin to apply what they have learned in logic to effective speaking and writing. This means that the student should move on to study formal rhetoric. Rhetoric applies the tools of logic: defining terms, declaring truth, arguing to valid conclusions, and refuting invalid ones. Indeed, of the modes of rhetorical persuasion – ethos, pathos, and logos – one-third is applied logic.

With this in mind, Roman Roads has released a new curriculum, Fitting Words: Classical Rhetoric for the Christian Student. I am the author of this text, and in Fitting Words I work to apply in rhetoric much of what the student has learned in logic. I am very excited about this project, because one significant reason that I wrote this text was to provide a satisfying answer the question of where to go next!

Take a look HERE for the most up-to-date information about Fitting Words.

Equivalence w/ Shorter Truth Tables

Mr. Nance,

Within Intermediate Logic Lesson 11, what would keep us from setting up the propositions both being true at the same time, and if there were a contradiction they would not be equivalent? Instead of setting them up one true and one false and if there’s a contradiction then they are equivalent?

That would be checking for consistency, not equivalence. If you set them both as true, and get a contradiction, then they are not consistent (which of course also means they are not equivalent, nor related by implication, per the chart in Introductory Logic, p. 71). But if you get no contradiction, all you have shown is that they can both be true, which is the meaning of consistency. To show equivalence, you have to show that they cannot have opposite truth values: the first cannot be true while the second is false, and vice versa.

Blessings!

Introductory Logic Prerequisite for Intermediate Logic?

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It is certainly possible for a student who has not taken (or not completed) Introductory Logic to take and successfully complete Intermediate Logic. Though the Intermediate Logic text is designed as a continuation to Introductory Logic, it does not assume a mastery of the concepts in it. Almost all of the concepts from Introductory Logic that are essential for Intermediate Logic are re-taught (the only exceptions being the definitions of logical argument, premise, and conclusion; definitions assumed in Intermediate Logic, Lesson 7, but taught explicitly in Introductory Logic, Lesson 19).

That being said, a new Intermediate Logic student who is familiar with Introductory Logic will have an advantage over a student who is not. The following concepts from Introductory Logic are repeated and re-taught in Intermediate Logic (the concepts are first taught in the respective given lesson numbers): Continue reading Introductory Logic Prerequisite for Intermediate Logic? →

The ambiguous OR

Intermediate Logic

Logic is a symbolic language. It is also a very precise language, every term well defined and unambiguous. English, on the other hand, is a somewhat ambiguous language. The same word can have multiple meanings: a pen is a writing utensil and an enclosure for livestock.

One key term in logic is the disjunction “or”. In English, the word “or” has two meanings. The first is the inclusive or, which means basically “this, or that, or both.” If someone said, “Most Bible students read the King James or the NIV,” this statement is still true for a student who reads both the King James and the NIV. The “or” includes both possibilities.

The exclusive or basically means “this or that, but not both.” This is the sense used in this classic argument for the deity of Christ: “Jesus was either God or a bad man.” If Jesus was God, then He was not a bad man. If He was a bad man, then He was not God.

Symbolic logic deals with the ambiguous “or” this way. The logical operator OR is taken in the inclusive sense. “A or B” is true if A is true, B is true, or both A and B are true. To represent the exclusive or, we use the compound proposition “A or B, but not both A and B.”

Question about Conditionals

Mr. Nance,

My student has a question on Exercise 4 number 14.  Her answer for was ~C ⊃ S instead of S ⊃ ~C. Can the statements “I will go swimming only if the water is not cold” be considered logically equivalent to “If the water is not cold, I will go swimming”?

Also, how can I explain the difference between “If the water is not cold I will go swimming” and “I will go swimming unless the water is cold”?

Thank you! Continue reading Question about Conditionals →

What will I learn in Intermediate Logic?

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intermediate-logic-complete-program-dvd-course[1]Logic gives us standards and methods by which valid reasoning can be distinguished from invalid reasoning. It teaches students to think in a straight line, and to justify each step of their thought. Intermediate Logic does this using a symbolic language to represent the reasoning inherent in the language of argument. It is more flexible than syllogistic logic, and can thus apply to more real-life arguments.

Intermediate Logic Unit One teaches the powerful method of truth tables to determine the validity of propositional arguments. Unit Two takes these methods and teaches students how to deduce a conclusion from a set of premises, so they are able not only to show that an argument is valid, but also prove why it is valid. Unit Three teaches these same concepts using the modern method of truth trees. Unit Four applies these methods to the analysis of real-life arguments from 1 Corinthians 15, Hebrews 2, Boethius’ The Consolation of Philosophy,  Augustine’s City of God, and more (including a scene from the movie “Get Smart”). Unit Five teaches the fascinating application of these methods to the logic of digital electronics.

Audit Intermediate Logic

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Would you like to be a fly on the wall in my logic class? Want to improve your understanding and/or teaching of logic by watching me teach and interact with my students, discussing the lesson after the class, and having the recorded class sessions available? If so, click HERE to audit Intermediate Logic for the 2017 school year!

What’s included for Auditors? First, you have access to all the live classes. During the discussion, you will not be called upon as I do with my regular students. You are free to watch in the background by muting your mic and camera, but you also have the option of appearing to ask a question or make a comment if you’d like.

After the regular class time has ended, students leave the virtual classroom while auditors are invited to stick around for a few minutes to ask “Teacher Questions”! This is when you would have me all to yourselves as teachers. Turn on your webcams and mics, and discuss the lesson, teaching logic in general, or whatever questions you might have.

We will meet together live for online recitations Monday/Thursday from 8:00-9:30 AM (PST), or Tuesday/Friday from 8:00-9:30 AM (PST). The spring semester starts January 5/6, 2017, and goes to May 18/19, with a Winter Break in mid-February and an Easter Break in mid-April.

I hope to see you there!