Have a Migraine

The conditional sentence is highly maligned in logic, but it is actually very common in our thoughts. The conditional sentence is anything like this: "If (this thing is true) then (this other thing is true)." "If I see a cat, I see a mammal," for example, or "If you sneeze on me, I will soon be sneezing too." A conditional sentence need not be true, or even be possible, for us to be able to know how things would be if we knew whether it were true or false. "If it is rainy, then it is windy," or "if my dog is blue, then I am orange," or even "If plipts can yell, than dundledorfs can speak" are all perfectly usable examples of conditional sentences.

A conditional sentence (for now let's take one that's totally sound, "If I see a cat, then I see a mammal") can be rendered like this, where "I see a cat" is P, and "I see a mammal" is Q:

P -> Q

The "->" symbol refers to the relationship between the two basic sentences, ("I see a cat," and "I see a mammal") and forms a non-basic sentence, "If I see a cat, then I see a mammal." "->" very roughly translates to "If... then..."

Assuming that one either sees a cat or does not see cat, and that one either sees a mammal or does not see a mammal, we can give the sentences one of two truth values: true or false. The table for all possible combinations of these values is this:

P Q P -> Q
True True True
True False False
False True True
False False True

Which is to say, if P is true and Q true, then P -> Q is true.
And if P is true and Q false, then P -> Q is false.
And if P is false and Q is true, then P -> Q is true.
And if P is false and Q is false, then P -> Q is true.

To clarify, if it is true that I see a cat, and it is true that I see a mammal, then "If I see a cat, then I see a mammal" is true too.
If I see a cat, but I do not see a mammal, then "If I see a cat, then I see a mammal" is false.
If I do not see a cat, but I see a mammal, then "If I see a cat, I see a mammal" is true. (I could see a wild boar)
If I do not see a cat, and I do not see a mammal, then "If I see a cat, I see a mammal" is true. (Does seeing a reptile make this sentence false?)

Notice that the only possibility that would never happen (seeing a cat, but not a mammal) turns out to be the only possibility that is false.

It works with nonsense as well, such as, "if my dog is blue, then I am orange." The same results would occur if this sentence were analyzed with a truth table, that is, from top to bottom, True, False, True, True.

If you don't understand, I read an illustration somewheres that goes like this: imagine your friend tells you, "If I see a frog jump, I start to sing." Under what circumstances could you conclude that your friend's statement was false?

1. When he saw a frog jump, and started to sing?
2. When he saw a frog jump, and did not start to sing?
3. When he did not see a frog jump, and started to sing?
4. When he did not see a frog jump, and did not start to sing?

The only time you could say that your friend was not telling the truth is #2. If your friend sees a frog jump, and does not start to sing, his statement, "if I see a frog jump, I start to sing," is false.

Here is a foursome of truth tables to puzzle over, about my friend Doug:

If Doug loves when people give to charity, then Doug is good:
Doug loves when people give to charity Doug is good If Doug loves when people give to charity, then Doug is good
True True True
True False False
False True True
False False True

Thus, the sentence "If Doug loves when people give to charity, then Doug is good" is true in all cases except if Doug loves when people give to charity, and is not good. In italics are the truth values of the basic sentences that most of Doug's friends believe, as well as the resulting truth value of the non-basic sentence they form.

If Doug loves when people give to charity, then Doug is not good:
Doug loves when people give to charity Doug is not good If Doug loves when people give to charity, then Doug is not good
True True True
True False False
False True True
False False True

Thus, the sentence "If Doug loves when people give to charity, then Doug is not good" is true in all cases except if Doug loves when people give to charity, and is good. In italics are the truth values of the basic sentences that most of Doug's friends believe, as well as the resulting truth value of the non-basic sentence they form.

If Doug does not love when people give to charity, then Doug is good:
Doug does not love when people give to charity Doug is good If Doug does not love when people give to charity, then Doug is good
True True True
True False False
False True True
False False True

Thus, the sentence "If Doug does not love when people give to charity, then Doug is good" is true in all cases except if Doug does not love when people give to charity, and is not good. In italics are the truth values of the basic sentences that most of Doug's friends believe, as well as the resulting truth value of the non-basic sentence they form.

If Doug does not love when people give to charity, then Doug is not good:
Doug does not love when people give to charity Doug is not good If Doug does not love when people give to charity, then Doug is not good
True True True
True False False
False True True
False False True

Thus, the sentence "If Doug does not love when people give to charity, then Doug is not good" is true in all cases except if Doug does not love when people give to charity, and is good. In italics are the truth values of the basic sentences that most of Doug's friends believe, as well as the resulting truth value of the non-basic sentence they form.

The truth value of the non-basic sentences of the past four truth tables may or may not be the common beliefs of Doug's friends, but nevertheless they follow from the truth values of the two basic sentences to their left.

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