1. Intermediate Logic is a challenging course, especially trying to complete it in one semester. Is all of it equally important?
No. Unit 1 on Truth Tables is foundational to propositional logic, as is Unit 2 on Formal Proofs. Both of these are essential and must be completed by every student. Unit 3 teaches the Truth Tree method. A truth tree is another tool that does the same job as a truth table: determining consistency, equivalence, validity, etc. Some people like truth trees more than truth tables, since they are more visual. But Unit 3 could be considered an optional unit. Unit 4 covers Applying the Tools to Arguments. This is where the rubber meets the road for propositional logic, showing how to apply what has been learned up to this point to real-life reasoning. Consequently, Unit 4 should be completed by every student. Note that if you skip Unit 3, one question in Unit 4 will have to be skipped (namely, Exercise 28c #1). Unit 5 on Digital Logic – the logic of electronic devices – is entirely optional. Like Unit 4, this unit covers a real-life application of the tools of propositional logic, but one that is more scientific (though ubiquitous in this age of computers and smart phones). Though optional, many students find that they really enjoy digital logic.
2. How do you take formal logic and apply it to real life? In other words, how do you move from the theory into practical application of what has been learned?
By its very nature logic is applicable; every area of study in which reasoning occurs employs the tools of logic. These tools include defining terms, identifying formal and informal fallacies, determining assumed premises in enthymemes, and developing and answering dilemmas. Students who fully grasp the tools of logic will make their own application, just as students who understand mathematics, grammar, or the sciences naturally find applications of those subjects in their daily lives.
To be sure, as a symbolic language, logic, like mathematics, has a certain elegance or even beauty. But God has given us logic in order to apply it to the world around us, and we must lead students in showing how this is done. Point out examples of informal fallacies in the newspaper, analyze enthymemes in the Bible to uncover the assumptions, show how Jesus uses and gets out of dilemmas, analyze Aquinas’s arguments with shorter truth tables.
In response to requests for more applications of propositional logic, the most recent edition of Intermediate Logic includes two new units. Unit 4 teaches how to apply the tools of propositional logic to real-life arguments, including arguments from the epistle to the Hebrews, Boethius’ Consolation of Philosophy, 1 Corinthians, Augustine’s City of God, and more. The new Unit 5 on Digital Logic then teaches how to apply these same tools to the logic of electronic devices. Many students really enjoy learning digital logic.
Also, we must continually review and apply the tools of logic. Courses like Geometry and Rhetoric will explicitly re-teach many of the techniques learned in the logic class, but every subject will use some of the tools. Do not all subjects require the defining of terms? Do not all draw premises from conclusions? Situations like these are good opportunity for review.
3. Do you apply formal logic to other subjects? If so, can you give an example?
Formal logic, as the key exercise of the dialectic stage, clearly applies to other subjects. For example, when I taught Doctrine, we would discuss classical arguments for and against the existence of God. One such argument is the Problem of Evil and goes something like this: “If God exists, then he is both perfectly good and infinitely powerful. If he is perfectly good, then he is willing to prevent evil. If he is infinitely powerful, then he is able to prevent evil. But if evil exists, then God is either unwilling or unable to prevent it. Evil does exist. Therefore God does not exist.” We would use a truth table or write a proof to show that this argument is valid. Logic students know that if an argument is valid, but the conclusion is false (as in this case), then at least one of the premises must be false. This leads to a fruitful discussion about which premise is false, and why. Does evil exist? Is God infinitely powerful but not able to prevent evil? Does God’s perfect goodness require that He is always willing to prevent evil?
4. How do you take the facts learned in the grammar stage and use them within the logic stage? In other words, how do you form a bridge between the two?
The facts of the grammar stage are the ham and cheese for the logic sandwich, the premises that are to be used to demonstrate the tools and techniques of logic. For example, in the grammar years the students learn that a platypus lays eggs. In the logic years this fact can be used to show that universal statements such as “No mammals are egg layers” can be proven to be false by a single counterexample. In the grammar stage students are introduced to variables representing numbers. In the logic years they then can grasp the idea of variables representing statements. Examples such as these could be multiplied endlessly.
But keep in mind the fact that God has made the minds of logic-age students to naturally awaken to the desire for the dialectic. They will seek to justify statements with which they agree, and identify the fallacies made in statements with which they disagree, before they ever learn the proper techniques for doing so. We don’t need to build the bridge for them between these stages; God does it within our students for us. We just point to the bridges, and help our students cross them when they get there.
5. Why learn modern propositional logic? Isn’t it enough to learn categorical logic?
Propositional (or symbolic) logic provides powerful methods by which students can learn how to learn, beyond the methods of traditional categorical logic. Tools such as formal proofs of validity teach students how to reason in a straight line, while providing them with standards and methods by which they can judge and correct their own arguments and the arguments of others. The study of propositional logic can help them understand the history of thought, while giving them insight into the modern digital age. Many Christian thinkers have found the study of propositional logic to be interesting and valuable, and have contended that an inquiry into modern logic can aid us in understanding the nature and character of God. Propositional logic is flexible and very applicable to analyzing arguments; in fact, the example of applying logic from question 2 above uses a tool of propositional logic.
6. What Logic 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 fundamental logic curriculum. Informal, categorical (Aristotelian), and modern propositional (symbolic) logic are all covered, and I have tried to make these as practical as possible for the junior-high and high-school student. Where to go from there depends in large part on the interest of the student, as well as his and his parents’ desires for furthering his education. The next best step for most students is to move on to applying what they have learned in logic. Much of formal logic is specifically applied in other subjects, most notably Geometry and Rhetoric. Geometry especially applies the lessons of Intermediate Logic, such as equivalence and proofs. Rhetoric especially applies the tools of informal and categorical logic, though the exercise of proof prepares rhetoric students to reason in a straight line. One-third of the modes of rhetoric persuasion – ethos, pathos, and logos – is applied logic.
With this in mind, Roman Roads has published Fitting Words: Classical Rhetoric for the Christian Student. Take a look HERE for the most up-to-date information about Fitting Words.
But if a student, having successfully completed Intermediate Logic, desires to move deeper into formal logic (in a more theoretical or philosophical way), there are some college-level texts that would be appropriate. Here are my suggestions:
- The Art of Reasoning by David Kelley. Kelley takes a fresh approach to the lesson learned from texts like Introductory and Intermediate Logic, but also includes further branches of formal logic, namely predicate logic, term logic, and formal inductive logic.
- Introduction to Logic by Irving Copi. This is a standard college text (it was the text I first learned from), covering the basics of my texts in good detail, along with predicate logic, analogical reasoning, and probability.
- Logic: The Right Use of Reasoning in the Inquiry After Truth by Isaac Watts. This text goes a different direction, emphasizing informal logic, going into significant detail on concepts, terms, definitions, judgment, truth value, and statements. He doesn’t get to formal logic (syllogisms and validity) until page 271 out of 352. Solidly Christian, of course, but a bit of a slog at times, with no exercises.
- Socratic Logic by Peter Kreeft. Subtitled “a logic text using Socratic method, Platonic questions, and Aristotelian principles,” this appears to be a solidly Christian text seeking to make categorical logic applicable. It has some exercises with an answer key. Kreeft is a Roman Catholic.
- Logic: A God-centered Approach to the Foundation of Western Thought by Vern Poythress. If a student wants to understand the Christian worldview behind what they have studied in logic, this 708 page book should satisfy! It is not a logic text, and has no exercise beyond questions for further reflection at the end of each chapter. Poythress is a solid Reformed Christian thinker.
There are many other good logic texts available at the college level, but this should give you an idea of where you could go next to continue logic studies after Intermediate Logic.