Tag Archives: Propositional Logic

The Square of Opposition (for logic nerds only!)

Intermediate Logic, Introductory Logic, ,

The Square of Opposition is a useful tool for representing and understanding the relationships between categorical statements with the same subject and predicate:

square-of-opposition

The relationships are defined this way:

Contrariety: The statements cannot both be true, but can both be false.
Contradiction: The statements cannot both be true, and cannot both be false.
Subcontrariety: The statements can both be true, but cannot both be false.
Subimplication: If the universal is true, the particular must be true.
Superimplication: If the particular is false, the universal must be false.

Students might be interested to discover that the square of opposition can be created for non-categorical statements as well. Continue reading The Square of Opposition (for logic nerds only!) →

Applying Intermediate Logic

Intermediate Logic, Why learn logic?

The most recent edition of Intermediate Logic has two new sections, Unit 4 and Unit 5. I’ve included these new units in the text because I wanted to answer the important question, “What are some practical applications of the tools that we are learning about, i.e. truth tables, formal proofs, and truth trees?” I believe that the applications of these tools in the new units will deepen and solidify student understanding of the concepts that, up to that point, have been largely theoretical.

Saint_Augustine_Portrait[1]Unit 4 (Lesson 28) gives logic students the opportunity to analyze chains of reasoning. The arguments to be analyzed are taken from Boethius’s The Consolation of Philosophy, the Apostle Paul’s argument proving the general resurrection of the dead in I Corinthians 15, and a section on angelic will from Augustine’s City of God. I work through this last one in full on the DVD.

An exercise not in the text that may be beneficial would be to have students write their own chains of reasoning, arguing for a conclusion of their choosing, in imitation of these authors. Their arguments must include at least one NOT, AND, and OR, and two IF/THENs. Tell them to include a truth-table or truth-tree analysis of their own chain of reasoning.

Digital display 4Unit 5 (Lessons 29-40) teaches students how to apply what they have learned to the fascinating topic of Digital Logic. Do not be intimidated by the 0’s, 1’s, and new symbols. It’s just the same old true, false, and logical operators that they have already learned about presented in a new way. Students often find this a fun application of what they have learned. It helps them to understand the electronic world around them, and it shows that the tools that they have learned apply not only to philosophy and theology, but to digital clocks and iPhones!

Do these truth tree branches close?

truth tree errorMr. Nance,

I have a student asking me if this would be a valid way of completing this truth tree for consistency. She thinks since she already found inconsistency in the branches, that she doesn’t have to do line 3 (per lesson 24). I’m thinking that this thought process only applies if she finds it consistent, not inconsistent. She’s also asking if it matters the order they are done in (I told her non-branching first, then branching, but that I didn’t think the order mattered if it was all branching that was left). Please help me give her direction! Continue reading Do these truth tree branches close? →

Logic & the Resurrection

Intermediate Logic, Logic, , ,

Intermediate Logic Unit 4 teaches how to apply the tools we have learned in logic to real-life arguments. One such argument is contained in 1 Cor. 15:12-20, in which Paul argues that Christ has been raised from the dead, and as such He is the firstfruits of the general resurrection to come.

There are many points to Paul’s argument, but the main one is from verses 13, 16, and 20:

“If there is no resurrection of the dead, then Christ is not risen…But now Christ is risen from the dead, and has become the firstfruits of those who have fallen asleep.”

This argument can be symbolized as follows (C = Christ is risen, R = There is a resurrection of the dead):

~R ⊃ ~C    C     ∴   R

You can use the tools of truth table, truth tree, or formal proof to demonstrate that Paul’s argument is valid.

There are several other arguments in 1 Cor. 15:12-20, but the others leave premises assumed, so they take additional effort to analyze. But it is a beneficial exercise to work through them. Look at Exercise 28b.

Have a blessed Good Friday!

Where does the CPA come from?

Mr. Nance,

I’m stumped on Logic lesson 18 #5. We got same answers as answer key until line 7…I can see from line 8 why line 7 is important, but how did we deduce a consequent that was not the original consequent of line 1 (from which we assumed the antecedent in line 3)?

Hope that makes sense! Continue reading Where does the CPA come from? →

Must we do every unit?

Intermediate Logic,

Over my 25 years of teaching logic, I have often been asked this question:

“Intermediate Logic is a challenging course, especially trying to complete it all in one semester. Is each unit equally important, or can I skip something if I can’t fit it all in?”

The short answer is “You don’t have to do it all.” 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.

As a teacher I have sometimes skipped either truth trees or digital logic. In fact, only with my best classes have I taught both Unit 3 and Unit 5. The Teacher Edition of the Intermediate Logic text includes two different schedules, one for completing every unit, and another for skipping Unit 5.

For answers to more FAQs, take a look HERE.

The Pattern of T & F in Truth Tables

Mr. Nance,

I was doing the exercises for Intermediate Logic Lesson 6 and got stumped by #2. In the part of the proposition that is ~q ⊃ ~p, when I was doing the defining truth table for the variables, I assumed the first variable, though out of order alphabetically, would get the TTFF pattern. But in the answer key, the letter that comes first in the alphabet (p), though the consequent, got the TTFF pattern. Why is that? Continue reading The Pattern of T & F in Truth Tables →

One Lesson Logic Students Must Learn!

Intermediate Logic, Introductory Logic, Why learn logic?, ,

The most important concepts to understand in Formal Logic is the concept of validity. All logic students should memorize and come to understand these three different (but related) ways of defining validity:

  1. In a valid argument, the premises imply the conclusion
  2. In a valid argument, if the premises are true, then the conclusion must be true.
  3. If an argument has true premises and a false conclusion, then it is invalid.

Continue reading One Lesson Logic Students Must Learn! →