Translating Sentences You now know how to translate words like “and”, “or”, and “if … then.” Let’s look at some words that are, perhaps counter-intuitively, are synonyms of these. “But”: As it turns out, “but” means the same thing as “and” in logic. If this isn’t immediately apparent, we can figure this out by taking an example and constructing a truth table for it. For instance, consider: “Albert had wine, but Brenda had beer.”

Translation: A, but B.

Now imagine that A and B are both true. Albert DID have wine and Brenda DID have beer. In this case, “A, but B” is true. Now imagine that A is true and B is false. Albert DID have wine, but Brenda DID NOT have beer. In this case, “A, but B” is false. Similarly, if A is false and B is true, the sentence is false. Also, if A and B are both false, the sentence is also false. We may now construct the truth table for “but” as follows: P T T F F

Q T F T F

P, but Q T F F F

Look familiar? It’s the same as the truth table for conjunction. It turns out that “but” operates on statements in the same was as “and” does. So, in the realm of logic, we will treat “but” as a synonym for “and”. “If”: Sometimes, conditionals are stated in a reversed order. For instance: “I will eat pepperoni, if there is any pepperoni left.” The important thing here is to look for the word “if”. Everything after the “if” is STILL the antecedent EVEN IF the statement after the word “if” COMES SECOND. So, this statement ALSO gets translated as: “If there is any pepperoni left, then I will eat pepperoni.” Translation: P  E 1

“Only If”: However—and this can be confusing so pay close attention—when the term “ONLY if” is used, everything after the “if” is the CONSEQUENT, and NOT the antecedent. For instance, “You will win the lottery only if you’ve bought a ticket.”

(“W only if B”)

Gets translated as: WB This may seem counter-intuitive, but the first sentence is really just saying that you MUST buy a lottery ticket in order to win the lottery. That is, having bought a ticket is a NECESSARY pre-condition for winning. You might have been tempted to translate “W only if B” as “B  W”, but, consider the difference between the following two sentences: “If you will win the lottery, then you’ve bought a ticket.”

(W  B)

“If you’ve bought a ticket, then you will win the lottery.”

(B  W)

“W only if B” was obviously true. Here, W  B is also obviously true. However, B  W is not obviously true. After all, lots of people buy lottery tickets but never win. So, we translate “W only if B” as “W  B”. It turns out that these two statements mean the same thing in logic. Note #1: These last two explanations on “if” and “only if” should help to make sense of the bi-conditional, “if and only if”. For, we have just learned that: “P if Q”

gets translated as:

QP

gets translated as:

PQ

and “P only if Q”

So, when we combine them, “P if and only if Q” gets translated as: (P  Q) and (Q  P)

which we abbreviate as:

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PQ

Note #2: You may have been thinking that “You will win the lottery only if you’ve bought a ticket” should be translated as: “If you have NOT bought a ticket, then you will NOT win the lottery” instead. Symbolize this as ‘¬B  ¬W’. That seems right. So, why did we say that it should be translated as ‘W  B’? As it turns out, these two strings of symbols mean the same thing. In logic, we say that they are logically equivalent. We’ll learn more about this later. W  B  ¬B  ¬W “Unless”: “Unless” turns out to be a synonym for “or”. This is not immediately obvious. But, let’s take a closer look. Consider this claim: “You will fail to win the lottery unless you have a ticket.” i.e., “You will lose the lottery unless you have a ticket.”

(“L unless T”)

This seems sort of like a conditional. It might mean something like one of the following: (1) “If you have a ticket, then you will win the lotto.”

Translation: T  ¬L

(2) “If you don’t have a ticket, then you will fail to win the lotto.” Translation: ¬T  L (3) “If you fail to win the lotto, then you don’t have a ticket.”

Translation: L  ¬T

(4) “If you will win the lotto, then you have a ticket.”

Translation: ¬L  T

We can rule out (1) and (3). Having a ticket doesn’t GUARANTEE that you’ll win the lotto. (you might have a ticket, but not a winning ticket), and similarly, failing to win doesn’t entail that you didn’t have a ticket. So, we’re left with (2) and (4). As it turns out, (2) and (4) mean the same thing. And, they BOTH mean this: (5) “Either you have a ticket, or you will fail to win the lotto (and maybe both).” But, as we have learned, this is just to say “T  L” (or, “L  T” if you prefer). So, the simplest way to translate “L unless T” is just to write: “L  T”. Note: We’ll do more to prove that (2), (4), and (5) mean the same thing soon.

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Creating Compound Statements So far, we have (mostly) only looked at statements where there is ONE operator (for instance, “¬A”, “A  B”, “A  B”, “A  B”, and “A  B”). Each of these has only ONE operator. But, often, statements can be much more complicated than this, and require TWO or more operators. For instance, we already saw one such statement: (F  S)  ¬(F  S) Recall that this meant, “He is either five or six, and he is not both five and six.” Notice the words in bold (“or”, “and”, “not”, and “and”). Each of these four words are operators, and so the symbolic translation requires FOUR operator symbols. But, we can’t write the statement above just ANY old way. The statement above is a “well-formed formula”. A well-formed formula is a formula which does not violate any of the rules for symbolic formulas. Here are several rules of thumb for formulas: 1) Two letters can never appear side by side; they must always be separated by an operator (and they cannot be separated ONLY by a “¬”). The following are NOT well-formed formulas: AB ¬AB A  BC A¬B To see that these sentences are counter-intuitive, consider how poorly they would look if we translated them into English: Albert likes cheddar Brenda likes swiss. It is not the case that Albert likes cheddar Brenda likes swiss. Either Albert likes cheddar or Brenda likes swiss Charlie likes mozzarella. Albert likes cheddar not Brenda likes swiss. 2) All symbols except “¬” must have something on either side of them, and the “¬” must have something on its right side. The following are NOT well-formed formulas:

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AB A¬ A

( “” needs something on both sides of it) ( “¬” must go on the left) ( “” needs something on both sides of it)

3) Two operators can never appear side by side unless the second is a “¬”. The following are NOT well-formed formulas: AB AB ¬A ¬A

(remember that the “” needs something on both sides of it)

But note that the following IS a well-formed formula: A  ¬B 4) Statements with three or more letters must have parentheses or brackets. Brackets are sort of like commas in English sentences. For instance, the following sentence is ambiguous: “Buddy likes cheese and Peggy likes pepperoni or Sue likes Hawaiian.” We might be tempted to symbolize this statement as the following: BPS But, which of the following is the speaker saying? “Either Buddy likes cheese and Peggy likes pepperoni … OR Sue likes Hawaiian.” OR “Buddy likes cheese … AND Either Peggy likes pepperoni or Sue likes Hawaiian.” There is an important difference. Commas would help to clarify things. If we add commas to these two interpretations, we get the following: “Buddy likes cheese and Peggy likes pepperoni, or Sue likes Hawaiian.” “Buddy likes cheese, and Peggy likes pepperoni or Sue likes Hawaiian.” The commas represent separators, just like parentheses do. Here are the symbolizations: 5

(B  P)  S B  (P  S) In the first sentence, Buddy and Peggy occur together, while Sue is separated by a comma. This indicates that the claim about Buddy and Peggy come together as a single unit. We indicate this by putting parentheses around them. Meanwhile, in the second sentence, Buddy occurs alone, and Peggy and Sue occur together after the comma. So, we put parentheses around Peggy and Sue in the second formula. But, there won’t always be a comma. The placement of the word “either” can be helpful in these cases. For instance, notice the difference between the following two sentences: “Either Harry orders juice and Mark orders beer or John orders soda.” “Harry orders juice and either Mark orders beer or John orders soda.” These get symbolized as the following: (H  M)  J H  (M  J) Here are a few more sentences and their translations: “Harry likes juice or both Mark and John like soda.” H  (M  J) “Harry likes juice and Mark or John like soda.” H  (M  J) “If Harry orders juice, then if Mark orders beer, then John will order soda.” H  (M  J) “If Harry will order juice provided that Mark orders beer, then John will order soda.” (M  H)  J “If Harry and Mark order juice or John orders beer, then Larry will order water.” [(H  M)  J]  L 6

Identifying the Main Operator Before moving on, we must learn how to identify the “main operator” within a formula. The main operator determines which KIND of statement a statement is (e.g., negation, conjunction, disjunction, etc.). Here are two rules for finding the “main operator”: 1) If there ARE NO parentheses, then the main operator will be the ONLY operator—unless there is more than one operator, in which case the main operator is the operator that is the one that is not a “¬”. 2) If there ARE parentheses or brackets, then the main operator will be the ONLY operator which is outside of all of the parenthesis or brackets—unless there is more than one operator outside of the parenthesis/brackets, in which case the main operator is the one that is not a “¬”. Here are some examples. Negation: In the following formulas, the main operator is the “¬”. So the following compound statements are all negations: ¬A ¬(A  B) ¬[(A  B)  (C  D)] Conjunction: In the following formulas, the main operator is the “”. So the following compound statements are all conjunctions: AB ¬A  ¬B (A  B)  (C  D) ¬A  [(B  C)  (D  E)] Disjunction: In the following formulas, the main operator is the “”. So the following compound statements are all disjunctions: AB ¬A  ¬B (A  B)  (C  D) ¬A  [(B  C)  (D  E)]

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Conditional: In the following formulas, the main operator is the “”. So the following compound statements are all conditionals: AB ¬A  B (A  B)  ¬(C  D) [(A  B)  (C  D)]  E Bi-Conditional: In the following formulas, the main operator is the “”. So the following compound statements are all bi-conditionals: AB A  ¬B ¬(A  B)  (B  C) [A  (B  C)]  [D  (E  F)] Once we have found the main operator, we know what KIND of statement something is. This becomes very clear if you note that, we can represent all of the complicated symbolizations above as very simple ones with a single operator. If Δ and □ are wff’s: ALL of the negations above have the form, “¬□” ALL of the conjunctions above have the form, “Δ  □” ALL of the disjunctions above have the form, “Δ  □” ALL of the conditionals above have the form, “Δ  □” ALL of the bi-conditionals above have the form, “Δ  □” For instance, take the very last equation under the bi-conditionals above. Now: Let ‘Δ’ = [A  (B  C)]

Let ‘□’ = [D  (E  F)]

and

In that case… [A  (B  C)]  [D  (E  F)] …is really just an instance of a well-formed formula of the form: Δ□ i.e., this is really just a bi-conditional with a wff on either side of the double-arrow. 8

Determining the Truth Values of Compound Statements If I give you an easy statement, such as “A  B” and tell you that “A” is true and “B” is false, then you would easily be able to tell me that “A  B” is false. (Have a look at the truth table for “” and see that “P  Q” ONLY comes out true if BOTH conjuncts are true. Example #1: But, let’s try a harder one: “If Either Obama or Romney wins the election, then the world will end in 2012.” In symbolic form, this becomes: (O  R)  W

But we know that: O=True, R=False, and W=False (Obama DID win, Romney DIDN’T, and the world DID NOT end)

Using “T” for “True” and “F” for “False”, we can re-write this as follows: (T  F)  F The main operator in the formula above is the “”. We get rid of one operator at a time, and save the main operator for last. So, first let’s focus on the “T  F” portion of the formula. What is the truth value of “P  Q” when “P” is true and “Q” is false? It’s true! So, we know that “T  F” is true. We can replace the entire disjunction with a “T”, like this: TF Now, what is the truth value of “P  Q” when “P” is true” and “Q” is false? It’s false! So, the WHOLE sentence that we originally started with is false. In other words, “If either Obama or Mitt Romney wins the election, then the world will end in 2012” is FALSE. Example #2: Here is another example: ¬[(A  B)  (C  D)]

A=true, B=true, C=true, D=false

Is this sentence true or false? Referring to our truth tables, we get the following steps: ¬[(T  T)  (T  F)] ¬( T  F ) ¬ T F

(we replace the letters with “T” or “F”) (the first conjunction is true; the second one is false) (the disjunction is true) (the negation of a truth is a falsehood) 9

Answer: So “¬[(A  B)  (C  D)]” is false when A=true, B=true, C=true, and D=false. Example #3: Let’s try one more example: [A  (B  C)]  [D  (E  F)] A=false, B=true, C=false, D=false, E=true, F=false Is this sentence true or false? Using our truth tables above, we get the following steps: [F  (T  F)]  [F  (T  F)] (F  F )  (F  F ) F  F T

(we replace the letters with “T” or “F”) (both of the conditionals in the parentheses are false) (both of the disjunctions are false) (a bi-conditional is TRUE when both sides are false)

Answer: True

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