2.2. Logical Operators and Propositions

In the last lecture we discussed about propositional logic, various logical operators and how do we form compound propositions from simple propositions using logical operators.

In propositional logic, we use logical operators to create compound propositions from simpler ones. Logical operators include 'and', 'or', 'not', 'if...then', and 'if and only if'. For example, if we have two simple propositions, A and B, we can make new statements like 'A and B', 'A or B', and 'not A'. These operators allow us to explore complex logical relationships.

Think about following rules in a game. If the rule says 'if it rains, then we will stay inside', then you could think of 'raining' as one proposition (A) and 'staying inside' as another proposition (B). Using logical operators allows you to create simple rules and combine them for complex game scenarios.

So, remember if p then q is represented by p → q and truth table of p → q is this.

The statement 'if p then q' is a conditional statement. In logical terms, this is represented as 'p → q'. The truth table for this expression shows all possible truth values for p and q, demonstrating when the conditional is true or false. It's important to understand that a conditional statement is only false when p is true and q is false.

Imagine a traffic signal: 'If the light is green (p), then you can go (q)'. This is true in all cases except when the light is green, but the car still stops (which would be an unexpected situation). This helps illustrate how the truth values work.

The converse of p → q is denoted by q → p and the inverse of p → q is denoted by ¬p → ¬q. The contrapositive is ¬q → ¬p.

The converse of a statement flips the original condition and conclusion. For example, if the original statement is 'if it rains, then the ground is wet', the converse is 'if the ground is wet, then it rained'. The inverse negates both the condition and conclusion, while the contrapositive negates and flips them. It's vital to note that only the contrapositive is logically equivalent to the original statement.

Think of a light bulb: "If it is turned on, then it will light up" (p → q). The converse is, "If it lights up, then it is turned on" (q → p), which may not always be true (maybe it is malfunctioning). The contrapositive, however, "If it is not lighting up, then it is not turned on," is always true as it holds the logical integrity.

Let me first define a bi conditional operator or a bi conditional statement which for which we use this notation ↔.

A biconditional operator is used to denote statements that are true in both directions, such as 'p if and only if q', noted as p ↔ q. This means that both statements are equivalent; both must be true or both must be false for the biconditional statement to hold true.

Consider a situation where 'you can enter the club if you have a membership'. This means 'you have a membership if and only if you can enter'. If you lack a membership (one condition), you cannot enter (the other condition) – both conditions are interdependent.

Now let us next define tautology, contradiction and contingency.

A tautology is a proposition that is always true regardless of the truth values of its variables (e.g., p ∨ ¬p). A contradiction is always false regardless of the truth values (e.g., p ∧ ¬p). Contingency refers to a proposition that can be either true or false, based on the truth values assigned (e.g., p ∧ q). Understanding these concepts is important in evaluating logical statements.

Imagine a scenario where a device can either be on or off: A tautology would be a statement like 'the device is on or not on', which is always true. A contradiction could be 'the device is on and not on simultaneously', which is always false. A contingency might be 'the device is on and the light is green', which can either be true or false depending on the situation.

So before trying to understand what are logical equivalent statements?...two expressions are the same expression.

Two statements or propositions are said to be logically equivalent if they yield the same truth values for all possible scenarios. This is often denoted as X ≡ Y. A practical way to check for logical equivalence is through the biconditional—if X ↔ Y is a tautology, the two statements are equivalent.

Consider two math equations, like '2 + 3 = 5' and '5 = 2 + 3'. Both statements express the same idea despite being written differently, showing logical equivalence. In the context of logic, just like in algebra, different expressions can represent the same truth.

So there are various standard logical equivalent statements which are available which are very commonly used in mathematical logic.

Certain logical identities, like the identity law (p ∧ true = p), double negation (¬(¬p) = p), and De Morgan's laws (¬(p ∧ q) = ¬p ∨ ¬q) are fundamental in logical reasoning. These laws can help simplify complex propositions and articulate logical relationships efficiently.

It's like knowing the rules in cooking. Just as 'if you have salt, and add more salt, it still tastes of salt' (identity law) and 'not not salt is just salt' (double negation), logical identities help simplify statements to their core meanings directly.

So how do we verify whether these logical identities are correct?

To verify logical identities, one can construct truth tables for both sides of an expression. If the truth values match across all scenarios, the identity holds true. This method, however, is limited to a small number of propositional variables due to the exponential growth of combinations as variables increase.

If you were to check if two recipes yield the same dish (e.g., cake A and cake B), comparing one ingredient at a time with a checklist (truth table) helps verify if both are indeed cakes. But with many more ingredients, it becomes complex, similar to many variables in logical expressions.

So we are trying to do the same thing even in the mathematical logic...while doing the simplification.

When faced with complex logical expressions, one can simplify them using known logical identities to show equivalence to another expression. The aim is to use established rules, much like solving equations step by step until you arrive at the desired form efficiently.

Imagine restructuring a large piece of furniture. Instead of tackling it from scratch, you recognize and use known techniques for assembly (like logical identities) to simplify construction and eventually fit the pieces together seamlessly into the final furniture form.

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