After building truth tables for basic connectives, we need to translate between ordinary language and symbolic form. Symbolic translation of compound statements is a skill used in proofs, argument testing, programming conditions, and mathematical modelling. The goal is not to replace language, but to remove ambiguity from it. In this lesson, symbolic translation will be practised in both directions. :::definition[Symbolic Translation] Symbolic translation is the process of converting English statements into logical notation, or converting logical notation into English statements. Primitive statements are first represented by letters, and logical connectives are then used to build compound statements. ::: A useful method is to identify the primitive statements first, assign a letter to each, and then translate the connective words. The word not corresponds to $\neg$, and corresponds to $\land$, or usually corresponds to $\lor$, if-then corresponds to $\to$, and if and only if corresponds to $\leftrightarrow$. The word but has the same truth-functional meaning as and, although it may carry a different tone in ordinary language. Translation must preserve logical meaning rather than emotional emphasis. Use the translator to practise the first step of symbolic translation. Choose a common English phrase and decide which logical symbol or structure represents it. The feedback focuses on the connective word rather than the full sentence, so it helps you build reliable translation habits. Keep necessary and sufficient language separate from this basic connective practice because implication direction needs special care. :::calculator[Logical Phrase Translator] ::: :::solved-problem Let $s:$ The student attends the seminar, $t:$ The registration form is complete, and $u:$ The fee is unpaid. Translate $(t\land\neg u)\to s$, $t\to(\neg u\to s)$, and $\neg(s\leftrightarrow(u\lor t))$ into English. ::: :::solution Let $s,t,u$ have the stated meanings. [1] The statement $(t\land\neg u)\to s$ means: If the registration form is complete and the fee is not unpaid, then the student attends the seminar. [2] The statement $t\to(\neg u\to s)$ means: If the registration form is complete, then if the fee is not unpaid, the student attends the seminar. [3] The statement $\neg(s\leftrightarrow(u\lor t))$ means: It is not the case that the student attends the seminar if and only if the fee is unpaid or the registration form is complete. ::: :::solved-problem Let $s:$ The student attends the seminar, $t:$ The registration form is complete, and $u:$ The fee is unpaid. Write each statement in symbolic form: (1) The student attends the seminar if and only if the registration form is complete. (2) If the fee is unpaid and the registration form is not complete, then the student does not attend the seminar. (3) The fee is unpaid, but the student attends the seminar. ::: :::solution [1] The phrase if and only if corresponds to biconditional. Hence, the symbolic form is $$\begin{equation*} s\leftrightarrow t. \end{equation*}$$ [2] The hypothesis is the fee is unpaid and the registration form is not complete, namely $u\land\neg t$. The conclusion is the student does not attend the seminar, namely $\neg s$. Hence, the symbolic form is $$\begin{equation*} (u\land\neg t)\to\neg s. \end{equation*}$$ [3] The word but has the same truth-functional meaning as and. Hence, the symbolic form is $$\begin{equation*} u\land s. \end{equation*}$$ ::: :::solved-problem Let $p:$ I complete the project before Friday, $q:$ I present the project in class, $r:$ The instructor approves the topic, and $s:$ The report is typed. Write each statement in symbolic form: (1) If the instructor approves the topic, then I present the project in class. (2) Completing the project before Friday is necessary for presenting the project in class. (3) Having an approved topic and a typed report is sufficient for presenting the project in class. ::: :::solution [1] The hypothesis is $r$ and the conclusion is $q$. Hence, the symbolic form is $$\begin{equation*} r\to q. \end{equation*}$$ [2] Since $p$ is necessary for $q$, the statement $q$ cannot be true unless $p$ is true. Hence, the symbolic form is $$\begin{equation*} q\to p. \end{equation*}$$ [3] Since $r\land s$ is sufficient for $q$, the symbolic form is $$\begin{equation*} (r\land s)\to q. \end{equation*}$$ ::: :::solved-problem Let $p:$ Quadrilateral $ABCD$ is a rectangle, $q:$ Quadrilateral $ABCD$ is a square, and $r:$ Quadrilateral $ABCD$ has four right angles. Translate $q\to p$, $\neg p\to\neg q$, $p\leftrightarrow r$, $p\land\neg q$, and $q\to r$ into English. ::: :::solution [1] The statement $q\to p$ means: If quadrilateral $ABCD$ is a square, then quadrilateral $ABCD$ is a rectangle. [2] The statement $\neg p\to\neg q$ means: If quadrilateral $ABCD$ is not a rectangle, then quadrilateral $ABCD$ is not a square. [3] The statement $p\leftrightarrow r$ means: Quadrilateral $ABCD$ is a rectangle if and only if quadrilateral $ABCD$ has four right angles. [4] The statement $p\land\neg q$ means: Quadrilateral $ABCD$ is a rectangle and quadrilateral $ABCD$ is not a square. [5] The statement $q\to r$ means: If quadrilateral $ABCD$ is a square, then quadrilateral $ABCD$ has four right angles. ::: :::tip When translating necessary and sufficient conditions, first decide which statement forces the other. If $p$ is sufficient for $q$, write $p\to q$. If $p$ is necessary for $q$, write $q\to p$. ::: :::exercise Let $a:$ The alarm is active, $b:$ The door is locked, and $c:$ The sensor reports motion. Translate into symbols: (1) The alarm is active and the door is locked. (2) If the sensor reports motion, then the alarm is active. (3) The door is locked only if the alarm is active. (4) It is not the case that the alarm is active or the sensor reports motion. ::: :::answer (1) $a\land b$. (2) $c\to a$. (3) $b\to a$. (4) $\neg(a\lor c)$. ::: :::faq[Frequently Asked Questions] Q: Does but translate differently from and? A: In truth-functional propositional logic, but is represented by $\land$. Q: How should only if be translated? A: The statement $p$ only if $q$ is translated as $p\to q$. Q: Why assign letters first? A: Letters keep the logical structure visible while preventing long English sentences from hiding the connectives. ::: :::call-to-action[Practise Symbolic Translation] subtitle: Translate short English statements before attempting longer compound statements with nested connectives. button: Next Lesson | /discrete-mathematics/mathematical-logic/basic-logical-connectives-and-truth-tables/conditional-statements-and-programming-decisions button-ghost: Previous Lesson | /discrete-mathematics/mathematical-logic/basic-logical-connectives-and-truth-tables/truth-tables-for-basic-connectives :::