I need some good examples to help distinguish between the law of noncontradiction and the law of the excluded middle.
Can anyone help?
They complement each other. The law of noncontradiction says that something cannot be both x and ~x simultaneously. BUT is there something that is neither x or ~x? The law of the excluded middle says no, something is always either x or ~x.
Let me give you an example that gives the two concept a workout. What is the opposite of red? Artist would say it’s green. Can something (monochromatic to make it easy) be red and green at the same time? No - but it is possible to be not red and not green athe the same time. Like for example yellow. So law of noncontradiction holds but the excluded middle doesn’t.
Of course the logician says that the opposite of red is not red so blue, green yellow, brown, black, white, etc. all fall in that not-red arena. But where does that put red-orange?
That really helps. Thanks.
I can help you with your question and I can’t help you, but I am going to do neither.
If visualizing helps you, imagine a bar divided into two different colored sections.
//////////\\\\\
The law of non-contradiction says they can’t overlap:
/////XXXXX\\\
The law of the excluded middle says there can’t be a gap between:
//////////_____\\\\\
Another example: If you flip a coin, it has to come up either heads or tails.
If it’s heads, the law of non-contradiction tells you that it can’t be tails. It can’t be both heads and not-heads.
If it isn’t heads, the law of the excluded middle tells you it must be tails. It has to be either heads or not-heads; there is no “middle” alternative.
Take two classics statements:
“This statement is false.”
“This statement is true.”
The first is neither true nor false. The second is both true and false. In some sense. Ideally, we want a formal logic to be unable to express either of them. (But if your formal logic is powerful enough to express significant things, you can’t avoid the problem entirely.)