4.4 Lost in Translation
Translate these two sentences into BOOL.
"And" and "but" don't have identical meanings in English. The word "but" suggests a contrast, but "and" doesn't.
Their translations into BOOL, though, are the same. The point is that BOOL doesn't capture every nuance of English: some bits are lost in translation.
You might think that means BOOL is a bad tool, but that's not so.
It just means that symbolic languages and normal languages have different purposes, different advantages and disadvantages.
If you want to write poetry, then English is a great tool. If you want to eliminate all nuance and unclarity, so that we know exactly what a claim is committed to and what it logically entails, then BOOL is a great tool.
Remember, logical systems like BOOL are models for helping us study how we reason. Models often involve simplification, which gives them their power.
Compare it with Newtonian mechanics in physics: Newtonian mechanics is a great tool for understanding how objects behave as we typically encounter them, but it's usefulness depends on the simplifications that it makes.
If BOOL were as messy and nuanced as English, we wouldn't need it--we could just stick with English.
When we translate a complex sentence into BOOL, we want to capture the logical commitments of the sentence. The logical commitments are what is required to make the sentence true.
For example, the commitment of ~P is that P is false. That is what is required in order to make the complex sentence ~P to be true.
When someone uses "but", "however" or "although", they are committed to the truth of the two things they are contrasting. We capture that commitment with "&", and we accept that other parts of the meanings of those words get ignored.
Notice how powerful this is: we can capture the logical commitments of many different expressions in English with just three little symbols.