Understanding the Tautology and Logical Equivalence in Propositional Logic

Lets start by demonstrating this very disturbing tautology:

Prightarrow Rvee Qrightarrow Rquad RHSq

This tautology is simplified through logical transformations:

By the definition of implication, Prightarrow Rvee Qequiv neg pensevere Rvee Q. Using the associative and commutative properties of disjunction (vee), neg deste le R vee Q vee R. The idempotence of disjunction simplifies to neg P vee Q vee R. Applying De Morgan's laws, neg P vee (neg Q vee R). Finally, it is rephrased as P wedge Qrightarrow R, which is logically equivalent to the original statement.

Why Does This Tautology Seem Intuitively Wrong?

This question is relevant and legitimate, as it highlights a common intuition that can mislead. The confusion arises from the way our brain processes logical statements and the fact that vee and rightarrow represent different logical operations.

Logical Equivalence: P wedge Q to R

Consider the following formula: P wedge Q to R. By definition of implication, this is equivalent to:

neg{P wedge Q} vee R Using De Morgan's laws, this becomes: neg{P} vee neg{Q} vee R By the commutativity and idempotence of disjunction, we simplify to: neg{P} vee R vee neg{Q} vee R Finally, this is rephrased as P to R vee Q to R

The "truth table" confirms that these two things are equivalent, but the intuition of such an equivalence can often be counterintuitive in propositional logic.

Step-by-Step Guide to Understanding Propositonal Logic

This conceptual understanding can be further solidified with a step-by-step approach. Here are two easy steps:

Step 1: Understand Logical Sentences and Formulas

Consider expressions like 3 le 5 or 3ge 5, which are logical sentences. While 3 le 5 is true, 3ge 5 is false. An expression like xle y isn't true or false on its own; it becomes a true or false statement only when specific values are assigned to x and y. For example, when x3 and y5, the expression 3le 5 becomes true.

You might write Let Pxy be the formula xle y instead of Let P be the sentence xle y for clearer understanding and consistency. This approach will help you see that both P35 wedge Q35 Rightarrow R35 and P35Rightarrow R35vee Q35Rightarrow R35 are true sentences. Meanwhile, Pxy wedge Qxy Rightarrow Rxy and PxyRightarrow Rxyvee QxyRightarrow Rxy are formulas, which aren't true or false until variables are fixed.

Step 2: Analyze the Logical Implication of Universal Statements

The sentences For all xy we have xle y Rightarrow xy and For all xy we have xge y Rightarrow xy are both false. Since these statements are false, their disjunction For all xy xle y Rightarrow xy vee xge y Rightarrow xy is also false. However, when you consider a statement like:

For all xy we have (xle y Rightarrow xy vee xge y Rightarrow xy)

The key difference is that here, you first consider the disjunction of logical implications for all x and y, and then assert that this disjunction holds true for all x and y. Since the alternative is true for any fixed x and y, the whole statement becomes true.

Conclusion

Understanding and applying propositional logic, particularly the use of implication, disjunction, and universal quantification, can be challenging but rewarding. By following these steps and carefully analyzing logical statements, you can better grasp the nuances of propositional logic and its implications.