Category Archives: Logic

Why Define?

We define terms in order to make their meaning understood, of course. But we might ask, what does understanding the meaning of a term give us? Let’s dig a little deeper into this question.

There are at least six purposes for defining terms.

  1. To show relationships between terms. A good example is Aristotle, in his Rhetoric, defining indignation as “a feeling of pain caused by the sight of undeserved good fortune” and pity as “a feeling of pain cause by the sight of undeserved bad fortune.” This helps us to see how these terms are related, and that a just man would tend to feel both indignation and pity, because he has an understanding of what is deserved.
  2. To remove ambiguity. An ambiguous word has more than one meaning, like the word bar. “A tall man walked into a bar. He said ‘ouch!'” We define terms in order to deal with equivocations like this, making it clear which meaning we have in mind when we use the word.
  3. To reduce vagueness. A vague word is unclear as to its extent. The instructions for constructing a model, instead of saying “For adults,” will give a more precise definition for adult, e.g. “For people 18-years old and up.” Such definitions will usually not be found in dictionaries,  applying only to particular situations.
  4. To increase vocabulary. Much teaching consists in giving students new meanings for new terms, thus enlarging their vocabulary and adding to their storehouse of knowledge. Thus, when we learn the definitions of ambiguous and vague, our personal vocabulary has increased, and we can begin to understand how these words relate.
  5. To explain theoretically. Often a definition will be theoretical in a way that accepting the definition implies that one is buying into a particular theory. If you agreed that light is “visible electromagnetic waves,” this is the same as accepting the wave theory of light (as opposed to the particle theory).
  6. To influence attitudes.  Sometimes we give a definition for a term in order to make people feel a certain way, either good or bad, about the thing being defined. When you father calls the television “a one-eyed brain sucker,” he is trying to influence your attitude about watching it.

There are no doubt additional purposes for defining terms, but this list of six perhaps goes a little ways toward expanding our understanding.

Help Solving Proofs

If you are in Intermediate Logic and learning about proofs for the first time, or struggling through them again for the second or third time, here are some helpful suggestions for justifying steps in proofs, constructing proofs, or just getting better at proofs.

  1. Think about what a proof does. Recognize that the conclusion of a previous step becomes a proposition to use as a premise for a new step. Proofs are a series of connected arguments, conclusion of previous arguments becoming premises for new ones.
  2. If you are learning how to justify steps in proofs (that is, you are working on Exercise 14a:10-16, or 15a:1-6, or 16:11-18) and you are in the middle of a proof, ask yourself which steps you have not yet used. If you are trying to justify step 6, and the previous lines already used steps 1, 3 and 4, then you will probably use steps 2 and 5. Try reading them aloud, and listen for familiar patterns from the premises of the rules of inference.
  3. Rewrite the argument that you are trying to prove. This will help you more clearly see the premises you have and the conclusion you are aiming at. You also might recognize patterns for rules of inference and replacement that you need to use. Often, a proof is built around a single rule of inference or replacement, and the other steps are just needed to set the premises up. For example, if you read Exercise 17a problem 7 aloud, you might recognize the modus tollens. But it takes a couple of steps to set up the second premise of the modus tollens, and one step afterward to fix the double negation.
  4. In general, find the premises you have available to you (e.g. if you’re on step 5, the available premises are from steps 1-4), read them aloud, and listen for rule patterns. In fact, get used to the patterns of the rules by reading them aloud, using something other than p’s and q’s (e.g. for Disjunctive Syllogism say to yourself, “This or that, not this, therefore that.”
  5. See if later proofs use procedures from earlier proofs. Exercise 17a problem 9 is built around a hypothetical syllogism, but you need to modify the proposition is line 1 to turn the conjunction into a disjunction so that the middles match for the H. S. To do that, you follow the procedures you used in problems 3 and 4.
  6. If you’re stuck, consider whether the next step might use the rules of Addition or Absorption. These are the rules that are often difficult to see when you need to use them. This is why in Exercise 14a:10-16, five of the seven proofs use one of these two rules, and in Exercise 15a, half of them use one of these rules. You need the practice.
  7. Another hint for if you are stuck constructing a proof is to try writing down every possible conclusion you can make from the available premises, and see if any of them help.
  8. You may have struggling through the assignment, succeeded writing some proofs but needed to look at the answer key for others. That’s okay. But I would suggest that you then go back and do the assignment again without looking. Practice makes habit.
  9. If it is still hard for you, if you are still not quite getting it, don’t sweat it. Take your time. Go drink some coffee and come back. Don’t say to yourself, “Well, I don’t get this, but I’ll just go on to the next lesson.” No. The lessons build on each other. If you are worried about getting through the entire text, stop worrying about it. You don’t need to cover it all. Better to learn a small amount of material well then a large amount of material poorly.

Think about proofs like solving a puzzle, rather than thinking of it like homework. Make it a fun challenge.


The ambiguous OR

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.”

Quick negation rules

Here are some quick rules to help you symbolize propositions that use negation:

Not both p and q  =  ~(p ⋅ q)
Either not p or not q  =  ~p v ~q
Both not p and not q  =  ~p ⋅ ~q
Neither p nor q  =  ~(p v q)

Truth tables can be used to show that the first two proposition forms are equivalent, and the last two forms are equivalent. The meaning of the sentences also help to show this.

The Biblical Biconditional

Symbolic logic has five standard logical operators, each of which has a standard translation in English:

negation is “not”
conjunction is “and”
disjunction is “or”
conditional is “if/then”
biconditional is “if and only if”

While the translations of the first four logical operators are frequent in English, the phrase “if and only if” is used very infrequently, and then only occasionally among mathematicians, philosophers, and lawyers.

For instance, while it is easy to find hundreds of nots, ands, ors, and if/thens in the Bible, the phrase “if and only if” is completely absent. However, for those who look carefully, biconditional reasoning is used several times in scripture. Keeping in mind that p if and only if q means if p then q and if q then p — and remembering other equivalences we have learned — the following verses all reflect biconditional reasoning:

Genesis 43:4-5, “If you will send our brother along with us, we will go down… But if you will not send him, we will not go down.”

2 Kings 7:4, “If they spare us, we live; if they kill us, then we die.”

John 6:53-54, “Unless you eat the flesh of the Son of Man and drink his blood, you have no life in you. Whoever eats my flesh and drinks my blood has eternal life.”

The first example could be translated as “We will go down if and only if you send our brother with us.” The second, “We will live if and only if they spare us.” The third, “You have eternal life if and only if you eat Christ’s flesh and drink His blood.”

Can you find any other biblical examples of statements that could be translated with the “if and only if”?

If/Then Truth Table

One of the difficulties new students of symbolic logic must overcome is understanding the defining truth table for the conditional, the “if/then” logical operator. The defining truth table tells us what the truth value of the proposition is, given the truth value of its component parts. For the conditional, it looks like this:

p    q     p ⊃ q
T    T         T
T    F         F
F    T         T
F    F         T

One way to defend this is to look at real-life conditional propositions with known truth values, for which we also know the truth value of the component parts. We will take our examples from the Bible.

The first row of the defining truth table states that a conditional with a true antecedent and a true consequent is true. In Genesis 44:26, Judah says about Benjamin, “If our youngest brother is with us, then we will go down.” The antecedent “Our youngest brother is with us” is true, and the consequent, “We will go down” was also true. We also know this is a true statement; Judah is speaking truthfully. There is the first row: If true then true is true.

The second row says a conditional with a true antecedent and a false consequent is false. In Judges 16:7 Samson says to Delilah, “If they bind me with seven fresh bowstrings, not yet dried, then I shall become weak, and be like any other man.” In verse 8 they bind him with seven fresh bowstrings (i.e. the antecedent is true) , but in verse 9 he breaks them easily (the consequent is false). So in verse 10, Delilah recognizes that Samson had lied to her; that is, she knows the conditional was false. Thus, if true then false is false.

The third row says a conditional with a false antecedent and a true consequent should be considered true. In Genesis 24:41, the servant quotes Abraham: “You will be clear from this oath when you arrive among my family; for if they will not give her to you, then you will be released from my oath.” The antecedent of the conditional (they will not give her to you) is false (they do give her), but the consequent (you will be released from my oath) is true (given the statement in verse 41 prior to the conditional). And we know the conditional was true; Abraham was speaking the truth. So if false then true is true. (See also Genesis 34:17).

Finally, the fourth row says conditionals with false antecedents and false consequents are also true. Here is one example: “If I find in Sodom fifty righteous within the city, then I will spare all the place for their sakes” (Genesis 18:26). The conditional is true, since it is God speaking. But note that the antecedent (God finds in Sodom fifty righteous people) is false, and the consequent (God spares the city) is also false. (See also Genesis 42:38). Thus, if false then false is true.

We see that it is not difficult to find examples of if/then statements in the Bible that support the traditional defining truth table for the conditional logical operator.


Not both v Both not, again

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.


Those weird categorical statements

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.

Common errors to avoid: The “to be” verb

Introductory Logic Lesson 11, “The One Basic Verb,” teaches the first step in translating categorical statements into standard form. This step is to translate the statement so that the main verb in the sentence is a verb of being: is, are, was, were, will be, and so on. Thus a statement like “Stars twinkle at night” gets translated into something like

Stars are nighttime twinklers. 

To do this correctly, the subject and predicate must both be nouns, and the verb must be the proper ‘to-be’ verb. The procedure outlined in the lesson is generally clear, but there are two errors I want to help you avoid.

One common error not mentioned in the textbook is the problem of the helping verb. Some students might try to translate the above sentence this way:

Stars are twinkling at night.

The student thinks, “I used the word are, which is a ‘to-be’ verb, so it must be correct.” The problem is that the whole verb here is “are twinkling,” the are being merely a helping verb. The way to fix this is to make sure that the predicate is a noun, usually formed by turning the main verb into a noun (e.g. twinkle –> twinklers).

Secondly, it is sometimes best to make the predicate a noun by adding a new noun, usually a genus of the subject. For example, you could translate the above statement as

Stars are bodies that twinkle at night.

For clarity’s sake, you may want to use a different noun than the one implied by the verb. For example, in translating “She’s got electric boots” it would be overly awkward to say,

She is an electric boots getter.

Much better to translate this as

She is an owner of electric boots


She is an electric-boot wearer.

Happy translating!