laws of contradiction in the mathematical language
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We say that a statement, or set of statements is logically consistent when it involves no logical contradiction. A logical contradiction is the conjunction of a statement S and its denial not-S. In logic, it is a fundamental law- the law of non contradiction- that a statement and its denial cannot both be true at the same time. Here are some simple examples of contradictions.
1. I love you and I don't love you.
2. Butch is married to Barb but Barb is not married to Butch.
3. I know I promised to show up today, but I don't see why I should come if I don't feel like it.
4. The restaurant opens at five o'clock and it begins serving between four and nine.
5. John Lasagna will be a little late for the party. He died yesterday.
These all seem to be contradictions because they seem either explicitly to state or logically imply a certain statement and its denial. (1) is an explicit contradiction. You can't love someone and not love someone at the same time. (2) is an implicit contradiction. It depends on the unstated but well known principle: if x is married to y, then y is married to x. (3) is also an implicit contradiction. It depends on the unstated meaning of promising, namely, that whenever you promise to do something you thereby acquire a moral obligation to do it.
Very often contradictions are only apparent. For example someone in a love-hate relationship might utter something like (1), meaning "I love you in some ways, but I hate you in others". This, of course, is not a contradiction at all. (4) also can look like a contradiction, but this may just be the result of unclarity. Perhaps the restaurant opens at 5:00 in the morning. (5) is not literally a contradiction, since a dead person could show up at a party. We call it a contradiction just because the statement "John will be a little late for the party," strongly suggests that John will be alive when he shows up.
When we tell people that they aren't making any sense, it is often because we think that they are saying something contradictory. In a Dilbert cartoon one of Dilbert's office mates is complaining that she hasn't been trained how to use the new computer. The conversation proceeds as follows:
Dilbert: Why don't you just read the manual?
Office mate: Right. Who has time to do that?
Dilbert: You mean you have time to go to a training session, but you don't have time to read the manual? That doesn't add up.
Here Dilbert's point is obviously that she is contradicting herself : She is saying that she has time to learn and she doesn't. Of course she might not really be contradicting herself at all. It may be that she finds computer manuals very hard to understand, so that the time it takes to be trained really is far less than the time it would take to learn from the manual.
This example shows that while it is very important to be logically consistent, it is also important to permit people to be so. When people speak in a way that seems logically contradictory it is often just because they are not speaking completely or clearly. So the point of exposing apparent contradictions is not, ultimately, to criticize peoples views as nonsensical, but rather to make them be clearer about what they are saying.
The logical law stating that no proposition can be true simultaneously with its negation. In the language of propositional calculus the law of contradiction is expressed by
The logical law stating that no proposition can be true simultaneously with its negation. In the language of propositional calculus the law of contradiction is expressed by¬(A&¬A)
The logical law stating that no proposition can be true simultaneously with its negation. In the language of propositional calculus the law of contradiction is expressed by¬(A&¬A)This formula is derivable in classical as well as in intuitionistic constructive propositional calculus (cf. also Propositional calculus).
A contradiction is a sentence together with its negation, and a theory is inconsistent if it includes a contradiction. Inconsistent mathematics considers inconsistent theories. As a result, inconsistent mathematics requires careful attention to logic.
Proof by contradiction in logic and mathematics is a proof that determines the truth of a statement by assuming the proposition is false, then working to show its falsity until the result of that assumption is a contradiction.
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