2/9 2. Logical Connectives and Symbolic Translation

Identify propositions and statement structure

A proposition is a complete declarative sentence with exactly one truth value: true or false. For example, “The library is open” can be represented by $p$, while “The temperature is below freezing” can be represented by $q$.

A compound statement combines propositions using logical connectives. When translating, first separate the sentence into its basic propositions. Then identify the connective that describes how those propositions are related.

A useful first question is: “What is the main structure of the sentence?” It may be a negation, conjunction, disjunction, conditional statement, or biconditional.

Takeaway: Identify complete declarative statements before translating the words that connect them.

Translate not, and, and or

A negation means “not.” The symbol for negation is $\lnot$, so $\lnot p$ means “not $p$” or “it is not the case that $p$.” Negation reverses truth value:

• If $p$ is true, then $\lnot p$ is false.

• If $p$ is false, then $\lnot p$ is true.

A conjunction means “and” and uses $\land$. The statement $p \land q$ is true only when both component propositions are true. Words such as “but” and sometimes “although” can also express conjunction in a logical translation.

A disjunction means “or” and uses $\lor$. Standard logical “or” is inclusive, so $p \lor q$ is true when at least one of $p$ and $q$ is true, including the case in which both are true.

For example, if $p$ means “Maya submitted the form” and $q$ means “Maya paid the fee,” then $p \land q$ means that both actions occurred. If $p$ means “The package will arrive on Monday” and $q$ means “The package will arrive on Tuesday,” then $p \lor q$ allows either day or both days.

Takeaway: Negation reverses truth value, conjunction requires both parts, and inclusive disjunction requires at least one part.

Translate conditional relationships

A conditional statement has the form $p \to q$, read “if $p$, then $q$.” The antecedent is $p$, and the consequent is $q$. In standard propositional logic, the conditional is false only in one situation: the antecedent is true while the consequent is false. It is true in the other three truth-value combinations.

The phrase “$p$ only if $q$” translates as $p \to q$. The statement after “only if” gives the necessary condition. For example:

• $p$: You may enter.

• $q$: You have a ticket.

• “You may enter only if you have a ticket”: $p \to q$.

Do not reverse this translation. “If you have a ticket, then you may enter” is $q \to p$, which is a different conditional.

For $p \to q$, the related forms are:

• Converse: $q \to p$

• Inverse: $\lnot p \to \lnot q$

• Contrapositive: $\lnot q \to \lnot p$

A conditional is logically equivalent to its contrapositive, but it is not generally equivalent to its converse or inverse.

Takeaway: Pay special attention to “only if,” and remember that a conditional and its contrapositive have the same logical meaning.

Understand if and only if

A biconditional has the form $p \leftrightarrow q$ and is read “$p$ if and only if $q$.” It asserts both directions at once:

A biconditional is true when $p$ and $q$ have the same truth value: both are true or both are false. It is false when exactly one of them is true.

For example, let $p$ mean “An integer is even” and let $q$ mean “The integer is divisible by 2.” The statement $p \leftrightarrow q$ says that each condition implies the other. By contrast, a statement such as “If an integer is divisible by 2, then it is even” gives only the direction $q \to p$.

The abbreviation “iff” is commonly used for “if and only if.”

Takeaway: A biconditional requires two valid directions, not merely one.

Combine connectives and check translations

Parentheses make the structure of a compound statement explicit. They are especially important when a sentence contains more than one connective.

For example, let:

• $p$: The store is open.

• $q$: I have enough money.

• $r$: I will buy the item.

The sentence “If the store is open and I have enough money, then I will buy the item” translates as:

The parentheses show that the entire conjunction $p \land q$ is the antecedent of the conditional.

A commonly used precedence order is:

1. Parentheses

2. Negation $\lnot$

3. Conjunction $\land$ and disjunction $\lor$

4. Conditional $\to$

5. Biconditional $\leftrightarrow$

When translating an English sentence, use this procedure:

1. Assign a letter to each basic proposition.

2. Identify the main connective.

3. Replace connective phrases with their symbols.

4. Add parentheses to show the intended grouping.

5. Read the symbolic expression back into English to check its meaning.

For example, if $p$ means “The application is complete,” $q$ means “The fee is paid,” and $r$ means “The request will be processed,” then “If the application is complete and the fee is paid, then the request will be processed” becomes:

Takeaway: Translate the main structure first, group related parts with parentheses, and verify the result by reading it back in English.