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.
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7$\begingroup$ Why do we use arithmetic symbols like $+$ or $-$? Why not just talk about sums and differences? $\endgroup$– John DoumaJan 27 at 17:57
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$\begingroup$ Why do we write algebraic expressions symbolically instead of writing them out in English, e.g. "the roots of a quadratic function are the negative of the linear coefficient, plus or minus the square root of the difference between the square of the linear coefficient and product of 4, the square coefficient and the constant coefficient, all divided by double the linear coefficient..."? $\endgroup$– Daniel ScheplerJan 27 at 17:58
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2$\begingroup$ The primary disadvantage is loss of compactness of notation. But still, I think it's a bit ridiculous and far less readable to throw in those symbols (for connectives) instead of using English, especially in the middle of a normal English sentence. $\endgroup$– MPWJan 27 at 18:01
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1$\begingroup$ For example: I could have written the ludicrous "I think it's a bit ridiculous $\wedge$ far less readable...", which is just plain pretentious and pseudo-scientific. $\endgroup$– MPWJan 27 at 18:10
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$\begingroup$ There are reasons, as you find in the answers. However: sometimes we see symbolic logical connectives overused, where using words would be better. $\endgroup$– GEdgarJan 28 at 0:16
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3 Answers
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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.
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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 $A \lor (B \land C)$.
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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.
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$\begingroup$ "A only if B" is still actually $A \to B$. However, "Only if A, B" and "A, if B" would both be $B \to A$. $\endgroup$– NicolinoFeb 28 at 17:42