Monthly Archives: January 2017

Parentheses in Conditionals

Mr. Nance,

Could you please explain how the placement of the parentheses is determined in Test 1, Form A, #12 of Intermediate Logic? My student wrote “(M ⊃ P) ⊃ ~C,” but the answer key says “M ⊃ (P ⊃ ~C).”

The original proposition is,

“If we see a movie then if we eat popcorn then we do not eat candy.”

This proposition has the overall form p ⊃ q, where p is the antecedent, “We see a movie” (abbreviated M) and q is the consequent. This consequent is another complete conditional: “If we eat popcorn then we do not eat candy.” This is the (P ⊃ ~C). Because it is a complete proposition in itself, this consequent gets placed in parentheses.

It will be important later to note that propositions of the form p ⊃ (q ⊃ r) are equivalent to propositions of the form (p • q) ⊃ r. The given proposition could be understood in this way. “If we see a movie and eat popcorn, then we do not eat candy.” Notice in this form, “We see a movie and eat candy” is the antecedent, and it is a complete proposition in itself, and thus gets placed in parentheses.

Blessings!

Shorter Truth Tables for Validity

Mr. Nance,

As I am teaching shorter truth tables for validity, I noticed that sometimes (on a valid argument) I get the contradiction in a different place than the answer key does. Is that okay, or am I making a mistake?

You are probably not making a mistake.

The shorter truth table method used on a valid argument will always result in a contradiction, but where that contradiction appears depends on the order of the propositions you work with, which can certainly vary.

For example, on Exercise 8, problem #1, the answer key shows the contradiction in two places, which happens if you find all of the truth values in the conclusion first, before going back to the premises.

STT1

But you might start by getting the truth values for S and W from the antecedent of the conclusion first, and then going directly to the premises. That would bring you to this point:STT2

Now, for the premises to be true, the consequents of each (P and F) must be true as well. That gives the contradiction in the conclusion, instead of in the premises as before:STT1

This is a perfectly legitimate answer. In the answer key, I tried to place the truth values in the positions I thought most likely for other who did the problem correctly. Typically, after making the premises true and the conclusion false, I try to start on the right side (the conclusion) and work my way left.

Here are a few more thoughts.

Shorter truth tables take some time to learn. Do not rush through them. Students need lots of examples to see how they work. Also, make sure you and they understand the concept behind them. You are assuming that the argument is invalid (by making the premises true and the conclusion false). If this assumption leads to an unavoidable contradiction, then the argument cannot be invalid, so it must be valid. But if you assume the argument is invalid and can fill out all the truth values without any contradiction, you have shown that the premises can be true and the conclusion false, i.e. you have shown it to be invalid.

Keep this in mind also: For each proposition (premise or conclusion), you must place the truth values under the main logical operator. The main logical operator is the operator in the column that would be the last to be filled out in the larger truth table. For example, consider this compound proposition:

~(p • q) ⊃ r

If this were a premise of an argument, the T would be placed under the conditional. But for the proposition

~[(p • q) ⊃ r]

the T would be placed under the negation. Working the truth values all the way out for this proposition would result in the truth values shown here:

~[(p • q) ⊃ r]
T   T T T  F  F

Feel free to comment if you have any questions.

Introductory Logic Prerequisite for Intermediate Logic?

Intermediate Logic, Introductory Logic, , , ,

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? →

Truth Tables for Validity

Intermediate Logic, Introductory Logic, ,

Truth tables can be used to determine the validity of propositional arguments. In a valid argument, if the premises are true, then the conclusion must be true. The truth table for a valid argument will not have any rows in which the premises are true and the conclusion is false. For example, here is a truth table of a modus tollens argument, with the final columns, showing it to be valid:

TT1

The fourth row down is the only row with true premises, and in that row it also has a true conclusion. So this argument is valid.

An argument is invalid when there is at least one row with true premises and a false conclusion, such as in this affirming the consequent truth table: Continue reading Truth Tables for Validity →

King’s Grand Style

Rhetoric, , ,

It has been maintained that Martin Luther King Jr. was the last American orator to use the grand level of style appropriately. In my rhetoric text Fitting Words, I define the grand level as that “in which the stylistic devices are intended to be dramatic, apparent, and impressive. Its purpose is not only to inform the mind and persuade the will, but to grip the emotions and heart. It is most appropriate for speeches delivered on formal occasions.”

Anyone who has listened to (or at least read) some of his speeches – especially his most famous “I Have a Dream” – is aware that MLK uses stylistic devices in a dramatic and impressive way, a way that can grip the mind and heart of his hearers.  Here are some quotes from my text which shows his skill in using the grand level of style. Continue reading King’s Grand Style →

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 →