Lecture Notes 3: Sentential Logic Translations

Translations: First Steps

The able to test the validity of an argument with something like a truth table, the argument first must be put into symbolic notation; we must “translate” the statements it contains.

The first step is to assign a letter to each simple statement. Here, the important thing is consistency.

Often we’ll use the first letter in the statement.

If we are going dancing, then Jessica should change her shoes and Mark should take a shower.

Use: “W” for “we are going dancing”, “J” for “Jessica should change her shoes”, and “M” for “Mark should take a shower”.

The final translation is W → (J & M).

However, you must use different letters for different statements.

Pick something to help remember the difference.

Either I’ll buy a Chevy, or I’ll buy a Ford.

Becomes: C ∨ F

Use the same letter twice only if the same simple statement is repeated.

Either the Belchertown bus is late, or the Belchertown bus is not late and the schedule is outdated.

Becomes: B ∨ (~B & S)

In the homework exercises, certain words are written in ALL-CAPS to make it clear what letter (the first letter of the all-caps word) you should pick so that the computer will recognize it when it checks your answer.

Translating Conjunctions

“&” translates “and”, “but”, “yet”, “although”, “however”, “moreover”, “whereas”, etc.

(These differ in connotation only.)

The “&” always goes in the middle, even if the English word begins the sentence.

Examples:

Peter is intelligent but he voted for Trump.

Translation: I & V

Philosophers love truth whereas rhetoricians love eloquence.

Translation: P & R

Although Quebec is in Canada, the people in Quebec speak French.

Translation: Q & S

Translating Disjunctions

“∨” translates “or” and the phrase “either … or …”.

Examples:

Jenna bought a copy of the book from the bookstore, or she downloaded it online.

Translation: B ∨ D

Either Peter overslept or he forgot about the meeting.

Translation: O ∨ F

If context suggests the exclusive sense of “or”, the translation must be more complex.

Either you can keep dating Sanjukta or you can keep dating Kalinda [but not both].

Translation: (S ∨ K) & ~(S & K)  or S ↔ ~K

Translating Negations

“~” translates anything used to negate a sentence:
“… not …”,
“it is not the case that …”,
“it is not true that …”,
“it is false that …”.

It can be hard to spot if it appears mid-sentence, or in a contracted form as part of a “——n’t” word.

Examples:

It is not true that Boston is boring.

Translation: ~B

It isn’t raining.

Translation: ~R

Kanye West is no genius.

Translation: ~K (where K is “Kanye West is a genius.”)

Translating Conditionals

“→” loosely translates all of
“if … then …”,
“… if …”,
“… only if …”,
“… provided that …”,
“… on the condition that …”,
“… in case …”,
“… implies that …”.

The tricky thing with “→” is that the order matters.

Examples:

If Anna goes out, then Ken babysits.
Anna goes out only if Ken babysits.
Anna’s going out implies that Ken babysits.
Provided that Anna goes out, Ken babysits.
In case Anna goes out, Ken babysits.

ALL translate to: A → K

Anna goes out if Ken babysits.
Anna goes out provided that Ken babysits.
Anna goes out on the condition that Ken babysits.

ALL translate to: K → A

Tips for getting the order of conditionals right:

• The if-part goes before the arrow.
• So if the word “if” (or a synonym) is in the middle, reverse directions.
• “P → Q” rules out P being true while Q is false; think about the truth of which rules out the falsity of the other.
• Although “P → Q” is not equivalent to “Q → P”, it is equivalent with “~Q → ~P”.
• “only if” is like “if” except that it reverses the order (or, equivalently, negates both sides).
• When “only if” is in the middle, it’s like reversing the position back to left-to-right.
• “P only if Q” translates to either “P → Q” or “~Q → ~P”.

Translating Biconditionals

“↔” translates the whole phrase “if and only if”, its abbreviation “iff”, as well as the phrase “just in case”.

Examples:

Annemarie will make it to the conference if and only if her car is running well.

Translation: A ↔ C

Kevin will have a date this weekend just in case hell freezes over.

Translation: K ↔ H

Hidden Conjunctions and Disjunctions

Grammatically non-compound sentences can express molecular propositions when words like “and” or “or” join names or predicates.

Examples:

Massachusetts and Connecticut are in New England.
This means: Massachusetts is in New England (M) and Connecticut is in New England (C).

Translation: M & C

Sean is either Irish or Scottish.
This means: Either Sean is Irish (I) or Sean is Scottish (S).

Translation: I ∨ S

Statements with “and”/“or” between names or predicates cannot always be broken into separate atomic statements.

Consider:

Naomi and Kathy are roommates.

Is this a hidden conjunction?
Does this mean Naomi is a roommate and Kathy is a roommate?

It might, but more likely it means that they are roommates to each other.

It is inappropriate to translate this as N & K, since it does not express two separate thoughts. Make it just R.

You need to consider each case individually.

Other Connectives

Neither … Nor … and Unless

“Neither P nor Q” can be translated as either “~(P ∨ Q)” or “~P & ~Q”.

Notice the first form uses “∨”; the second uses “&”. You cannot translate “neither … nor …” with “~(P & Q)” or “~P ∨ ~Q”

“Unless” roughly means “if not”.

So, “unless P, Q” can be put as “~P → Q”.

“~P → Q” is also equivalent with “P ∨ Q”, so that can also be used for “unless”.

(Like “if”, reverse the order if it’s in the middle!)

Examples:

Neither the sun shone nor the stars twinkled.

Translation: ~(S ∨ T)  or ~S & ~T

Minnesota is neither an Eastern nor a Western state.

Translation: ~(E ∨ W)  or ~E & ~W

Unless you stop starting at me, I’ll throw a taco at you.

Translation: ~S → T  or S ∨ T

Sarah works at the library, unless she’s been fired.

Translation: ~F → W  or W ∨ F

Necessary and Sufficient Conditions

“P is a sufficient condition for Q” means that P guarantees Q, or that if P then Q.

“P is a necessary condition for Q” means that Q can be true only if P is, or that if Q is true, P must be.

(The difference is the same as between if and only if.)

“P is necessary and sufficient for Q” can be translated with “↔”.

Examples:

Averaging above 60% is a sufficient condition for passing.

Translation: A → P

Buying a ticket is a necessary condition for winning the jackpot.

Translation: W → B or ~B → ~W

Knowing Björk is necessary and sufficient for loving Björk.

Translation: K ↔ L

Translating Complex Statements

For these we need to combine approaches.

You’ll ENJOY Scary Movie provided you have a SENSE of humor and you LIKE horror movies.

Translation: (S & L) → E

Unless ALLIE and ERIN go to the party, I don’t WANT to go.

Translation: ~(A & E) → ~W

To properly place parentheses, make use of:

• context
• common sense
• punctuation

More examples:

It’s not true that I am BALD, and I RESENT the insult.

Translation: ~B & R

It’s not true that I’m BALD and I’m LAZY.

Translation: ~(B & L)

If they go to the STORE at 7pm then they’ll arrive HOME at 9pm, but I’ll be GROWING hungry by 8pm.

Translation: (S → H) & G

If they go to the STORE at 7pm, then they’ll arrive HOME at 9pm and they will MISS the start of the movie.

Translation: S → (H & M)

Practice problems