Introduction to Symbolic Logic and Truth Values

1. Negation (NOT)

The negation symbol, often represented by a squiggly line ($\neg$), reverses the truth value of a statement. If statement $P$ is "The dog is sleeping," then $\neg P$ is "The dog is not sleeping".

• If $P$ is True, $\neg P$ is False.
• If $P$ is False, $\neg P$ is True.

2. Conjunction (AND)

Represented by the symbol $\land$, a conjunction connects two statements. In natural language, words like "but," "however," or "nevertheless" function logically as an "and". For a conjunction to be true, both individual statements must be true.

3. Disjunction (OR)

Represented by the symbol $\lor$, a disjunction connects two statements using "or". In symbolic logic, we use the inclusive or, meaning the statement is true if the first part is true, the second part is true, or both are true.

4. Conditional (IF... THEN)

The conditional uses an arrow ($\rightarrow$) to show implication. It is structured as "If [antecedent], then [consequent]". For example, "If the dog runs away, then the child cries" is symbolized as $P \rightarrow Q$.

5. Biconditional (IF AND ONLY IF)

The biconditional uses a double-headed arrow ($\leftrightarrow$). It represents a relationship where $P$ and $Q$ are tied together such that they must have the same truth value to be considered true.

Determining Truth Values

The ultimate goal of symbolic logic is to find the truth value of a complex expression. You can do this using a truth table, which maps out every possible combination of True and False for the variables involved.

For a compound statement with two variables ($P$ and $Q$), there are always four possible cases to consider:

1. $P$ is True, $Q$ is True
2. $P$ is True, $Q$ is False
3. $P$ is False, $Q$ is True
4. $P$ is False, $Q$ is False

Exam Tip: When translating English to symbols, be careful with commas! A comma in a sentence often indicates where parentheses should be placed in your symbolic expression to group statements together.

Worked Example: Evaluating without a full table

Sometimes, you don't need a full table if you are given specific values for your variables. Let's evaluate the following: Given: $P$ is True, $Q$ is False, $R$ is True. Expression: $(P \lor Q) \land \neg R$

To solve this, follow the logical order of operations:

1. Handle Negation: Since $R$ is True, $\neg R$ becomes False.
2. Solve Parentheses: Evaluate $(P \lor Q)$. Since $P$ is True and $Q$ is False, the "inclusive or" makes this part True (because at least one is true).
3. Final Conjunction: Now we have $(\text{True}) \land (\text{False})$. Since a conjunction requires both sides to be true, the final result is False.

By breaking complex sentences into these small, manageable pieces, we can navigate even the most complicated logical arguments with mathematical precision.