If an axiomatic system is consistent it means that if x logically follows, then “not x” does not logically follow.
While this might seem overly pedantic (hardly something to complain about in this discussion), I’d like to point out that this definition only matches the usual one if you also accept the law of non-contradiction. More precisely, a system is consistent if it does not contain a contradiction.
Also, your definitions of Mathematics and “logically follows” don’t seem very… good. Did you make them up?
EDIT: changed “excluded middle” to “non-contradiction”. duh.
More precisely, a system is consistent if it does not contain a contradiction.
What we mean by a contradiction is the truth of a statement of the form “x and not x”, or, in Bynerma’s terms, that there is a statement x such that both x and “not x” logically follow (from the axiomatic system). (Things that we (correctly) intuitively recognize as contradictions allow us to derive a statement of that form.) So, although Bynerma used slightly unusual language, she correctly expressed the content of the usual definition.
The Law of the Excluded Middle is not needed here. It says that statements of the form “x or not x” are true (provable even, according to Wikipedia).
I see what I was doing here. The law of excluded middle is equivalent to the law of non-contradiction. Probably why I had the two confused. Example in sentential logic:
While this might seem overly pedantic (hardly something to complain about in this discussion), I’d like to point out that this definition only matches the usual one if you also accept the law of non-contradiction. More precisely, a system is consistent if it does not contain a contradiction.
Also, your definitions of Mathematics and “logically follows” don’t seem very… good. Did you make them up?
EDIT: changed “excluded middle” to “non-contradiction”. duh.
ETA: Yeah, my point was stupid. Nevermind.
What we mean by a contradiction is the truth of a statement of the form “x and not x”, or, in Bynerma’s terms, that there is a statement x such that both x and “not x” logically follow (from the axiomatic system). (Things that we (correctly) intuitively recognize as contradictions allow us to derive a statement of that form.) So, although Bynerma used slightly unusual language, she correctly expressed the content of the usual definition.
The Law of the Excluded Middle is not needed here. It says that statements of the form “x or not x” are true (provable even, according to Wikipedia).
Absolutely correct—note edit
I see what I was doing here. The law of excluded middle is equivalent to the law of non-contradiction. Probably why I had the two confused. Example in sentential logic:
¬(p∧¬p) (law of non-contradiction)
¬p∨¬¬p | 1, de morgan’s
¬p∨p | 2, double negation
p∨¬p | 3, commutation (law of excluded middle)