Part 7: Representing Statements Symbolically
Why Symbolic Representation Matters
Imagine you have multiple puzzle pieces—negation, conjunction, disjunction, conditional, and biconditional—and you're trying to arrange them to see the full picture of an argument. Symbolic representation is like having a clear blueprint for how those puzzle pieces snap together. When statements grow more complex, carefully chosen symbols and rules of precedence help us avoid confusion and spot logical relationships more easily.
Our Five Propositional Connectives
So far, we've introduced five connectives. Here's a concise summary:
| Symbol | Meaning | Common Name |
|---|---|---|
| \(\neg\) | not | negation |
| \(\land\) | and | conjunction |
| \(\lor\) | or | disjunction |
| \(\to\) | implies | conditional |
| \(\leftrightarrow\) | if and only if | biconditional |
(You may sometimes see "&" instead of "\(\land\)," or "∨" in place of "\(\lor\)." We'll stay consistent with these current symbols.)
Rules of Precedence and Parentheses
When writing logical expressions, parentheses clarify which operations apply first. Just as in arithmetic—where multiplication typically takes precedence over addition—certain connectives bind more tightly than others. To streamline our writing, we adopt these conventions:
-
Negation Precedence
\(\neg\) (not) takes highest precedence over \(\land\), \(\lor\), \(\to\), and \(\leftrightarrow\).
Example: \(\neg p \land q\) means \((\neg p) \land q\), not \(\neg(p \land q)\). -
\(\land\) and \(\lor\) Precedence
Both \(\land\) (and) and \(\lor\) (or) come next in precedence, above \(\to\) (implies) and \(\leftrightarrow\) (if and only if).
Example: \(p \lor q \to r\) means \((p \lor q) \to r\), not \(p \lor (q \to r)\). -
Associativity of \(\land\)
\((p \land q) \land r\) is logically equivalent to \(p \land (q \land r)\).
So we can write \(p \land q \land r\) without ambiguity. -
Associativity of \(\lor\)
Similarly, \((p \lor q) \lor r\) is equivalent to \(p \lor (q \lor r)\).
We can write \(p \lor q \lor r\). -
\(\to\) and \(\leftrightarrow\) Are NOT Associative
Parentheses are always needed to distinguish \((p \to q) \to r\) from \(p \to (q \to r)\), which can have different truth values.
Likewise, \((p \leftrightarrow q) \leftrightarrow r\) might not mean the same as \(p \leftrightarrow (q \leftrightarrow r)\).
Similar to Arithmetic
Recall in math:
* Multiplication binds tighter than addition.
\(-x\) has higher precedence than multiplication or addition.
You can drop parentheses in \((x \cdot y) \cdot z\), but not in \((x + y) \cdot z\).
These parallels help us remember the rules of logical precedence and associativity.
Examples of Omitting and Requiring Parentheses
-
\(\neg\) Precedence
\(\neg p \land q\) means \((\neg p) \land q\).
If you meant \(\neg(p \land q)\), you must include parentheses. -
\(\land\) Over \(\to\)
\(p \to q \land \neg r\) means \(p \to (q \land (\neg r))\).
Parentheses show that you first combine \(q\) and \(\neg r\), then connect with \(p\) via \(\to\). -
\(\lor\) Over \(\to\)
\((p \lor q) \to r\) can be shortened to \(p \lor q \to r\), acknowledging that \(\lor\) has higher precedence. -
Non-Associativity of \(\to\)
\((p \to q) \to r\) is not the same as \(p \to (q \to r)\).
Example: If \(p\) is true and \(q\) is false, \((p \to q)\) is false, which might make the entire \((p \to q) \to r\) statement true or false depending on \(r\). Meanwhile, \(p \to (q \to r)\) checks a different relationship.
Common Mistake
People often drop parentheses incorrectly around \(\to\) or \(\leftrightarrow\). That can completely change the meaning of a statement, so be careful when rewriting or simplifying expressions.
Real-Life Angle: Why Bother With All These Rules?
- Streamlined Communication: Just as you wouldn't want a math expression cluttered with unnecessary parentheses, logical statements are easier to read when you apply consistent conventions.
- Reduced Ambiguity: Adhering to precedence rules ensures that everyone interprets the statement in exactly the same way.
- Avoiding Logical Errors: In coding (like in Python or C), an operator's precedence can change program behavior. In logic, forgetting parentheses around an implication can lead to a completely different conclusion.
Common Misconceptions
"All parentheses can be dropped, because it's all just logic."
Reality: Only some parentheses can be safely dropped under specific precedence or associativity rules. Others are essential.
"\(\land\) and \(\lor\) also work like \(\to\)—we can't drop parentheses."
Reality: \(\land\) and \(\lor\) are associative, so parentheses can be dropped. Conditionals (\(\to\)) and biconditionals (\(\leftrightarrow\)) are not.
"\((p \lor q) \to r\) and \(p \lor (q \to r)\) must have the same meaning."
Reality: They can have completely different truth conditions.
Exercises
Symbolic Representation
For each of the following, translate the English statement into symbolic logic. Introduce your own letters (e.g., \(p, q, r, s\), etc.) for the component propositions, and insert logical connectives (\(\neg, \land, \lor, \to, \leftrightarrow\)) where 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 television show."
7.7. "I will win the chess game if I checkmate my opponent, if she resigns, or if she runs out of time."
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 to preserve the original meaning.
7.10. \((a \land (\neg b)) \leftrightarrow (c \lor d)\)
7.11. \((\neg(a \lor b)) \leftrightarrow ((\neg a) \land (\neg b))\)
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. \((a \lor b) \land (c \land d) \land (b \lor c) \land a\)
Showing Logical Distinctions
7.16. Show that \((p \lor q) \land r\) and \(p \lor (q \land r)\) are logically distinct by assigning specific truth values for \(p\), \(q\), and \(r\) that make one statement true and the other false.
7.17. Similarly, demonstrate that \((p \to q) \to r\) and \(p \to (q \to r)\) are logically distinct by providing truth values for \(p\), \(q\), and \(r\) such that one expression is true while the other is false.
Extra Practice
7.18. Invent a complex statement (using at least four propositions and multiple connectives) and write it with unnecessary parentheses. Then remove as many parentheses as your rules allow without changing the meaning.
7.19. Explain (in words) why \((p \land q) \land r\) is simpler to read than \(((p \land q) \land r)\), but why something like \((p \to (q \to r))\) can't lose parentheses.
Final Thoughts
Symbols and rules of precedence might seem technical, but they're essential for precision in logic—much like standardized notations in math, engineering, or programming. By becoming comfortable with these conventions, you avoid misunderstandings and communicate complex logical ideas with clarity and confidence.
In the next part, we'll continue exploring more about how to use these symbols to form truth tables, detect equivalences, and eventually build up to formal proofs. Each step is about refining your logical "toolkit" so you can apply it in mathematics, daily reasoning, and beyond.