Part 7: Representing Statements Symbolically

Why Symbolic Representation Matters

Think of complex logic like a puzzle: you have pieces (negation, conjunction, disjunction, conditional, biconditional) and you need to arrange them to see the full picture of an argument. Symbolic representation is like having a clear blueprint for how those puzzle pieces fit together. When statements get complicated, using well-defined symbols and following rules of precedence helps us avoid confusion and spot logical relationships easily.

Architect's Blueprint

Building an argument is like constructing a house:
- You have bricks (basic propositions) and different ways to join them (connectives).
- A detailed blueprint ensures you build in the correct order and structure.
- Without a clear diagram, you risk confusion about which part connects where. Symbolic logic provides that clear diagram for reasoning.

Our Five Propositional Connectives

So far, we have five logical connectives. Let's summarize them with their symbols:

Symbol	Meaning	Name
$\neg$	not	negation
$\land$	and	conjunction
$\lor$	or	disjunction
$\to$	implies	conditional
$\leftrightarrow$	if and only if	biconditional

Note: In different texts you might see "&" instead of $\land$, or "∨" instead of $\lor$. In this course we'll stick to the symbols above for consistency.

Rules of Precedence and Parentheses

When writing logical expressions, parentheses clarify which operations apply first. Similar to arithmetic (where multiplication happens before addition unless parentheses dictate otherwise), certain logical connectives bind more tightly.

To avoid ambiguity, we use these precedence conventions (from highest precedence to lowest):

Negation ($\neg$) – evaluate this first.
Example: $\neg p \land q$ means $(\neg p) \land q$, not $\neg(p \land q)$. If you meant the latter, you'd need parentheses.
Conjunction ($\land$) and Disjunction ($\lor$) – evaluate these next, before implications. They share the same precedence level.
Example: $p \lor q \to r$ is interpreted as $(p \lor q) \to r$, not $p \lor (q \to r)$. If your intent was the latter, parentheses are required.
Conditional ($\to$) and Biconditional ($\leftrightarrow$) – these have the lowest precedence and are evaluated last (unless parentheses dictate otherwise). They are not associative, meaning you can't just drop parentheses if you chain them.

Additionally, some connectives are associative, meaning the grouping doesn't matter for consecutive occurrences: - Associative: $\land$ and $\lor$ are associative. $(p \land q) \land r$ is equivalent to $p \land (q \land r)$, so we can write $p \land q \land r$ unambiguously. Likewise for $p \lor q \lor r$.
- Non-Associative: $\to$ and $\leftrightarrow$ are not associative. $(p \to q) \to r$ is generally not the same as $p \to (q \to r)$. We always need parentheses when chaining multiple $\to$ or $\leftrightarrow$ in one expression to avoid confusion.

Analogy with Arithmetic

Multiplication binds tighter than addition (just like $\land$ or $\lor$ bind tighter than $\to$).
A unary minus (like $-x$) has high precedence (like $\neg$ does in logic).
You can drop parentheses for $(x + y) + z$, but not for $(x + y) \cdot z$ without changing meaning. Similarly, we can drop some parentheses with $\land$ or $\lor$, but not around $\to$ or $\leftrightarrow$.

Examples: When to Omit or Use Parentheses

Let's practice applying these rules:

Negation Precedence
$\neg p \land q$ means $(\neg p) \land q$.
If you wanted $\neg(p \land q)$, you must write the parentheses.
$\land$ and $\lor$ over $\to$
$p \to q \land \neg r$ by precedence means $p \to (q \land (\neg r))$.
If you wrote $p \to q \land \neg r$ without thinking, it's read as $p \to (q \land \neg r)$. To express $ (p \to q) \land \neg r$, add parentheses: $(p \to q) \land \neg r$.
Dropping parentheses for consecutive $\land$ or $\lor$
$(p \lor q) \lor r$ can be simply written $p \lor q \lor r$ (since $\lor$ is associative).
But $(p \to q) \to r$ cannot drop parentheses because it would mean something different if grouped differently.
Non-Associativity of $\to$
$(p \to q) \to r$ vs $p \to (q \to r)$: these can have different truth values (Exercise 7.17 will have you examine this difference). Always use parentheses to clarify in such cases.

Common Mistake

Dropping parentheses around implications ($\to$) or biconditionals ($\leftrightarrow$) is a frequent error. It can completely change the meaning of an expression. When in doubt, put parentheses to enforce the intended grouping.

Real-Life Angle: Why Bother With All These Rules?

Streamlined Communication: Just as you wouldn't write a math expression with every possible parenthesis (that would be cluttered), you don't want to clutter logical expressions either. Precedence rules let us omit unnecessary parentheses, making statements easier to read while still being clear.
Reduced Ambiguity: By everyone following the same conventions, anyone reading the expression interprets it the same way. This avoids miscommunication in proofs or program code.
Avoiding Logical Errors: In programming, the precedence of operators can change how code runs. Similarly in logic, forgetting a needed parenthesis around an implication can lead you to interpret a statement incorrectly and draw wrong conclusions.

Common Misconceptions

"All parentheses are optional in logic; it's all about context."
Reality: Only some parentheses can be safely dropped under specific precedence or associativity rules. Others are essential to preserve meaning.
"$\land$ and $\lor$ work like $\to$, so we can't drop any parentheses with them."
Reality: $\land$ and $\lor$ are associative, so you can drop parentheses in a chain of all ANDs or all ORs. Conditionals ($\to$) and biconditionals ($\leftrightarrow$) are not associative, so you cannot drop those parentheses.
"$(p \lor q) \to r$ means the same as $p \lor (q \to r)$."
Reality: They are generally not the same. For instance, if $p$ is true, $(p \lor q) \to r$ only cares about $r$ for truth value, whereas $p \lor (q \to r)$ will be true regardless of $r$ because $p$ is true (no conditional in that form). They differ.

Exercises

Symbolic Representation

For each of the following English statements, translate into symbolic logic using the proper symbols and keeping the intended meaning clear. Introduce letters (e.g., $p, q, r, s$) for the component propositions, and use logical connectives ($\neg, \land, \lor, \to, \leftrightarrow$) as appropriate:

7.1. "If it rains in the forest, the trees get wet."
7.2. "If you pass the test, then you won't fail the course, and if you fail the course, you will have to go home."
7.3. "Bob will sit next to Alice if and only if Jose does not sit next to Darren."
7.4. "If a tree falls in the forest and no one is there to hear it, then it makes no sound."
7.5. "If the slithy toves did gyre and gimble in the wabe, then either the mome raths were not outgrabe or it was brillig."
7.6. "If I don't finish my homework tonight, then either I will have to finish it tomorrow or I will get a bad grade; but if I do finish my homework tonight, then I will miss my favorite TV show."
7.7. "I will win the chess game if I checkmate my opponent, if she resigns, or if she runs out of time." (Hint: restructure as a logical OR of conditions implying your win.)
7.8. "I will declare a major in astronomy, and I will declare a minor in either philosophy or anthropology."
7.9. "If it is sunny outside, then I will go to the beach if I have some free time."

Removing Parentheses

Apply the precedence and associativity rules to eliminate parentheses where it's valid. Do not remove parentheses if they're necessary for the original meaning:

7.10. $(a \land (\neg b)) \leftrightarrow (c \lor d)$ (Hint: Can any outer parentheses be dropped or re-grouped?)
7.11. $(\neg(a \lor b)) \leftrightarrow ((\neg a) \land (\neg b))$ (This is a classic equivalence — see if you can simplify the notation.)
7.12. $((a \to b) \land (b \to a)) \leftrightarrow (a \leftrightarrow b)$
7.13. $((a \to b) \lor (b \to a)) \lor (a \land (\neg a))$
7.14. $\neg(a \leftrightarrow c) \land \neg(b \leftrightarrow c)$
7.15. $(a \lor b) \land (c \land d) \land (b \lor c) \land a$ (Use associativity and commutativity of $\land$ and $\lor$ as needed.)

Showing Logical Distinctions

7.16. Show with specific truth values for $p$, $q$, and $r$ that $(p \lor q) \land r$ and $p \lor (q \land r)$ are not logically equivalent (i.e. find a counterexample where one is true and the other is false).

7.17. Similarly, find truth values for $p$, $q$, and $r$ such that $(p \to q) \to r$ and $p \to (q \to r)$ have different truth values, demonstrating they are not equivalent expressions.

Extra Practice

7.18. Invent a complex statement (using at least four distinct propositions and multiple connectives). Write it out with full parentheses showing every grouping. Then rewrite it in a cleaner way by removing any unnecessary parentheses using the rules learned. Ensure the meaning stayed the same.

Next: In Part 8: More on Truth Tables, we'll deepen our understanding by analyzing complex statements via truth tables and exploring concepts like tautologies, contradictions, and logical equivalences.

Symbol	Meaning	Name
\(\neg\)	not	negation
\(\land\)	and	conjunction
\(\lor\)	or	disjunction
\(\to\)	implies	conditional
\(\leftrightarrow\)	if and only if	biconditional