Why we symbolize logical connectives? What is the advantages of symbolizing connectives like (¬) for 'not', (∧) for 'and', (→) for if...then' etc., why don't we just write these connectives in english? Please show me some examples if possible for disadvantages of not using symbols and rather using english letters/words.
 A: One advantage that logic has over natural language is the use of parentheses. Try expressing $P \land ((Q \to (R \lor S) \land T) \lor U)$ in English without it becoming ambiguous. I suppose you could do it but it would be a very awkward sentence and it would be hard for us to work with to see what would follow from it, how it can be combined with other sentences, or whatever you want to do with it.
All languages are tools to enhance our cognitive capabilities, but in certain circumstances and for certain tasks, certain languages are just much more effective than other languages.
A: There are numerous reasons why logical symbols are preferable:

*

*They take up less space than the English words, and the notation is independent of whatever language the reader speaks. It's the same reason as why we write the Pythagorean Theorem as $a^2 + b^2 = c^2$, instead of "a squared plus b squared is equal to c squared" or "a quadrato più b quadrato è uguale a c quadrato".


*Natural language words "and", "or", etc can have unintended meanings depending on context. For example, plain English "or" is sometimes inclusive, but other times exclusive. Sometimes "or" doesn't act as a truth connective at all, but means "in other words".


*We can use parentheses to group specific parts of an expression together, which further eliminates ambiguity. In English, the statement "A or B and C" could be interpreted as either of the following:

*

*"A or B is true and C is also true".

*"Either A is true, or it's B and C that are both true".

Formal logic allows us to solve this problem by writing the first as $(A \lor B) \land C$, and the second as $A \lor (B \land C)$.
A: In my opinion using symbols to represent logical connectives allows for a more compact and precise way of expressing logical statements. It also allows for easier manipulation of logical expressions and can help prevent ambiguity.
The disadvantage of writing the symbol in English format like "If A then B" instead of using arrow symbol "A → B" is that it can be less clear and more prone to misinterpretation. Additionally, it can take up more space and make it harder to perform formal manipulation of logical expressions.
An example of misinterpretation when using English format instead of symbols is the statement "If A then B" versus "A only if B",
The statement "If A then B" is written in English format and is represented symbolically as A → B.
The statement "A only if B" is also written in English format, but it is represented symbolically as B → A.
As we can see, these two statements have a different meaning but they are written similarly in English format and it can be misinterpreted.
Another example of less clear statement in English format is "A and B" versus "A or B".
The statement "A and B" is written in English format, and it is represented symbolically as A Ʌ B.
The statement "A or B is also written in English format, but it is represented symbolically as A V B.
As you can see, these two statements have a different meaning but they are written similarly in English format and it can be less clear.
