lingual-logical-equivalence.md

Lingually Understanding Logical Equivalence (IFF)

Your brain is a powerful language processor. When you encounter mathematical logic, you can leverage your natural language understanding instead of fighting against it.

Consider the confusingly worded "if and only if" used in sentences to represent logical equivalence. When your encounter "if and only if," your language processing actually halts - it has to switch to logical reasoning to understand the unnatural construct. This switching between language and logic interrupts and thus slows your ability to understand a statement.

But your brain already understands equality through the simple word "is":

• Water is H2O
• A square is a rectangle with equal sides
• The derivative is the rate of change

When you see logical equivalence represented as "P if and only if Q", $P \leftrightarrow Q$, or "P IFF Q", interpret it as "P is Q". You naturally grasp that this means P and Q are the same.

For more complex statements where just using is becomes confusing, you can use these natural variations:

• is the same as
• is identical to
• when it is
• when it is the same as

Instead of parsing "A if and only if B if and only if C", think "A is the same as B, which is identical to C". This matches how you naturally express equivalence.

You can use these natural equivalence expressions constantly:

• The morning star is the same as the evening star
• Thunder is the sound of lightning
• Being prepared is identical to having a plan
• Success is the same as achieving your goals

Each of these statements expresses equivalence - the terms on either side are equivalent. You understand these effortlessly because they use your natural language equivalence patterns.

Even in technical contexts, natural equivalence flows better:

• A function is continuous when its graph has no breaks
• A matrix is singular is the same as it having no inverse
• An argument is valid when it is impossible for the premises to be true and the conclusion false

The key insight: don't fight you natural language processing. Use your understanding of equivalence through is and its variations. Mathematical logic becomes clearer when it aligns with how you naturally process meaning.

What are some other more natural language translations that you use for better understanding logical expressions?

#Logic, #MathEducation, #LanguageProcessing