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There is quite a bit of conflict between the way certain words are used in English, and how the same words are used in Logic. Essentially, Logic uses precisely defined words, while English is much more vague and ambiguous. This ambiguity can be a problem, but it is also the main reason for the flexibility and power of a natural language like English. This article is about the difference in meaning, of some simple words.
Simple Statements
Statements in Logic are either true or false. In English, we often assume them to be true, and ignore the possibility of their being false.
Also, "He's a happy camper," has hidden meaning. It does not mean that this person is actually a camper. This is an idiomatic expression, or an idiom. Some of these take some thought to express in Logic. This one is fairly easy: "He is a happy person." In logic, statements have explicit meaning. In English, much of the meaning may be implicit.
And
To start out, we sometimes use "and" to mean arithmetic addition (instead of a logical "and"): "three girls and two boys." In Logic, we often lengthen a statement to make it clearer: "My car is small and blue," becomes "My car is small, and my car is blue." The use of "and" can have hidden meaning: "I like strawberries and cream," does not mean "I like strawberries, and I like cream." Instead it means that "I like a dish called 'strawberries and cream.'" I guarantee that I would be unhappy if I were served cream, because the restaurant was fresh out of strawberries.
The word "but" is synonymous with "and" except that it packs hidden meaning: "The story was short but sweet," means "The story was short, and the story was sweet." But it also means that "short" and "sweet" do not normally go together, which is probably not true.
Can "and" ever mean "or?" Look at this: "We will decide between red and blue," which can become "We will choose red or we will choose blue." See the third paragraph concerning "or" below.
Or
There are two kinds of "or" in logic, inclusive and exclusive. In logic, the simple "or" is inclusive. "A or B" means "either A or B or both A and B". In other words, we include "A and B". An example of this might be: "I'll have cherry pie or vanilla ice cream." I guarantee that I will not object to having both.
But normally in English, "or" is exclusive: "I'll have lemon pie or chocolate pie." In this case, I do not want both, I am excluding both. If I want to use an inclusive "or", I would say, "I'll have lemon pie or chocolate pie or both." So, in English, "or" is usually exclusive, but is often ambiguous. You have to consider the context, sometimes in a subtle way.
Sometimes, "or" can even mean "and": "I like steak or cherry pie," which means "I like steak, and I like cherry pie," with the added meaning that I don't like them together. We use it this way, just because "I like steak and cherry pie," sounds like I do like them together.
Not
"Not" negates a statement. "I do not like broccoli," becomes "not (I like broccoli)." This is fairly simple, except for hidden meanings: "This elevator is not going up." Does that mean it is going "down"? Logically no, but in practice very likely.
Also, double negatives may not always cancel each other out: "He don't say nothing," which is poor English, but is a perfectly clear statement of fact.
If
"If" is used very differently in English and Logic. "If my shirt is red, then I will wear blue jeans," says nothing about what I will do when my shirt is not red. I may still wear blue jeans. So, "if" often is ambiguous or even meaningless when the condition (my shirt is red) is false. But, "If I have money, then I will eat a hamburger," actually implies a cause and effect: (money leads to hamburger) and (no money leads to no hamburger).
In Logic, "if" is similar to the above, when the condition (my shirt is red) is true. But, when the condition is false, then the entire statement is true. "If my shirt is red, then I am wearing blue jeans," is true when I wear a yellow shirt, regardless of whether I am wearing blue jeans or not. This is a big difference.
In logic an implication (as an "if" is called) is a statement which can be true or false (and is true when the condition is false). In English, an implication means intent or cause and effect.