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Some have thought that logic will one day be completed and all its problems solved. Now we know it is an endless task
Spotting inconsistencies in what is said can enable us to work out that a relative is confused, or that a public figure is lying. Logic is one basic check on what politicians say.
Except for a few special cases, logic can’t tell you whether the premises or conclusion of an argument are true. It can’t tell you whether Maria is at home, or whether she’s in the office, or whether she’s in neither of those places. What it tells you about is the connection between them; in a valid argument, logic rules out the combination where the premises are all true while the conclusion is false.
Boolean logic has its limits. In particular, it doesn’t cover the logic of ‘some’ and ‘all’.
Within this formal system, Frege showed how to understand natural numbers as abstractions from sets with equally many members. For example, the number 2 is what all sets with exactly two members have in common. Two sets have equally many members just when there is a one-one correspondence between their members.
Contrary to what some philosophers claim, these patterns are not just linguistic conventions. We cannot make something not self-identical, however hard we try. We could mean something else by the word ‘identity’, but that would be like trying to defeat gravity by using the word ‘gravity’ to mean something else. Laws of logic are no more up to us than laws of physics.