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a single inconsistency in a formal logical system means you can prove anything. if both A and !A are true, then you can prove anything you wish via contradictio

a single inconsistency in a formal logical system means you can prove anything. if both A and !A are true, then you can prove anything you wish via contradiction. assume !B. A & !A - which is a contradiction. therefore, our assumption was wrong, and thus B is true. this means a single contradiction in a legal system renders the whole thing meaningless. everything becomes both legal and illegal at the same time. a single semicolon can break an entire program because the program _must_ be consistent. that doesn't happen in the law because you have a judge who, at some point, will get pissed off at you. it doesn't matter if two conflicting rules - one dealing with the importing of flowers, and the other dealing with navigating riverways - are in conflict with each other. the judge isn't going to accept that as an argument for why you should be allowed to jaywalk - even if a technically correct interpretation of the law would say you should. all of this is to say that the claim 'we have a nation of laws, not a nation of men' is entirely fatuous. it's just not true. we have a nation of men interpreting rules that are in a very convoluted language, designed to be 'unambiguous' - and yet still vague enough for an education person to find and exploit this ambiguity. anyone reasonable would agree with me. if you disagree, you probably think 'reasonable cause' is reasonably clear.

ArtB 12y ago
> a single inconsistency in a formal logical system means you can prove anything. if both A and !A are true, then you can prove anything you wish via contradiction. I believe this has been solved in intuitionistic logic (http://en.wikipedia.org/wiki/Intuitionistic_logic http://en.wikipedia.org/wiki/Intuitionistic_logic). If you don't except the law of the excluded middle and require constructive proofs you avoid many of these shenanigans.
- MereInterest 12y ago
  A question. Wouldn't this line of reasoning still work? It is entirely a constructive proof, as opposed to the GP's proof by contradiction. http://en.wikipedia.org/wiki/Principle_of_explosion#The_proof-theoretic_argument http://en.wikipedia.org/wiki/Principle_of_explosion#The_proo...
- quchen 12y ago
  Although intuitionistic logic does not have the axiom ------ (LEM) A ∨ ¬A that is, the law of the exluded middle (LEM) can not be derived "from nothing" for all propositions, it is still an inconsistency if you can prove A ∧ ¬A to be true, since ------ (Assumption) A ∧ ¬A ------ (Assumption) ------ (∧E2) A ∧ ¬A ¬A ------ (∧E1) ------ (Definition of ¬) A A ⇒ ⊥ ---------------------------- (⇒E) ⊥ --------------- (⇒I, discharge assumption) A ∧ ¬A ⇒ ⊥ --------------- (Definition of ¬) ¬(A ∧ ¬A) In other words, assuming A ∧ ¬A holds you can prove the false proposition ⊥ (from which you can prove anything). (Note that the De Morgan's law ¬(A ∧ ¬A) ⇒ A ∨ ¬A does not hold without the LEM, so the above proof cannot be simplified in terms of it!)
prutschman 12y ago
> this means a single contradiction in a legal system renders the whole thing meaningless. everything becomes both legal and illegal at the same time. The law is not a formal logical system.
- Tombone5 12y ago
  Yeah, have you heard of IP-law?
dragonwriter 12y ago
> a single inconsistency in a formal logical system means you can prove anything. First, the law is not a formal logical system (its a signalling mechanism between members of a community), and, second, there is no actual contradiction. The fact that the law provides a mechanism for changing itself that applies to the whole of the law, including the part specifying the mechanism for change, is not a "logical contradiction". It is -- even viewing the law mechanistically (which, as just explained, is a somewhat limited view) -- just means that the law in the future may not be the same as the law now in any of its particulars. But that's not a contradiction, its a choice (and its a choice made largely because such a guarantee would be empty, and providing a formal mechanism which is not impractical makes it more likely that change will be governed by the framework -- even if it changes the framework in the process -- rather than simply abandoning the framework [something that Framers were keenly aware of since they were in a very real sense engaged in exactly the process of abandoning the framework of the Articles of Confederation precisely because of its impractical unanimity requirement.]) > all of this is to say that the claim 'we have a nation of laws, not a nation of men' is entirely fatuous. Its only fatuous if you think law is meant to be mechanistic, rather than a communication mechanism; the "nation of laws, not a nation of men" is a relative description in reference to purely arbitrary authority. Yes, the distinction is not one of crisp binary categories, but then crisp binary categories usually are abstractions that fall apart when applied to the real world.
erehweb 12y ago
The philosopher Wittengstein has something to say about this, in part of a discussion of formal logic. He says contradictions are not as bad as you might think. "Or suppose that there is a contradiction in the statutes of a particular country. There might be a statute that on feast days the vice-president had to sit next to the president, and another statute that he had to sit between two ladies. This contradiction may remain unnoticed for some time, if he is constantly ill on feast-days. But one day a feast comes and he is not ill. Then what do we do? I may say 'We must get rid of this contradiction.' All right, but does that vitiate what we did before? Not at all. [...] When a contradiction appears, then there is time to eliminate it. We may even put a ring round the second rule and say 'This is obsolete.'"
- simonh 12y ago
  Clearly the sensible way to resolve this is to ensure that the president is always a lady.
- deleted 12y ago
  [deleted]
pjc50 12y ago
two conflicting rules - one dealing with the importing of flowers, and the other dealing with navigating riverways http://en.wikipedia.org/wiki/R_%28Factortame_Ltd%29_v_Secretary_of_State_for_Transport http://en.wikipedia.org/wiki/R_%28Factortame_Ltd%29_v_Secret... The 'normal' resolution of later legislation with earlier legislation would be that the later wins. However in this case, the House of Lords (at that time the UK's supreme court) ruled that Parliament had not intended that making fisheries rules should eject the UK from the EU and that the earlier European Communities Act had priority in this case and all future similar cases. (Law also spends a lot of time and effort working out what category a particular "thing" actually is - businesses,goods,property - so it can be determined which law applies)

ArtB 12y ago

> a single inconsistency in a formal logical system means you can prove anything. if both A and !A are true, then you can prove anything you wish via contradiction. I believe this has been solved in intuitionistic logic (http://en.wikipedia.org/wiki/Intuitionistic_logic http://en.wikipedia.org/wiki/Intuitionistic_logic). If you don't except the law of the excluded middle and require constructive proofs you avoid many of these shenanigans.

MereInterest 12y ago
A question. Wouldn't this line of reasoning still work? It is entirely a constructive proof, as opposed to the GP's proof by contradiction. http://en.wikipedia.org/wiki/Principle_of_explosion#The_proof-theoretic_argument http://en.wikipedia.org/wiki/Principle_of_explosion#The_proo...
quchen 12y ago
Although intuitionistic logic does not have the axiom ------ (LEM) A ∨ ¬A that is, the law of the exluded middle (LEM) can not be derived "from nothing" for all propositions, it is still an inconsistency if you can prove A ∧ ¬A to be true, since ------ (Assumption) A ∧ ¬A ------ (Assumption) ------ (∧E2) A ∧ ¬A ¬A ------ (∧E1) ------ (Definition of ¬) A A ⇒ ⊥ ---------------------------- (⇒E) ⊥ --------------- (⇒I, discharge assumption) A ∧ ¬A ⇒ ⊥ --------------- (Definition of ¬) ¬(A ∧ ¬A) In other words, assuming A ∧ ¬A holds you can prove the false proposition ⊥ (from which you can prove anything). (Note that the De Morgan's law ¬(A ∧ ¬A) ⇒ A ∨ ¬A does not hold without the LEM, so the above proof cannot be simplified in terms of it!)

prutschman 12y ago

> this means a single contradiction in a legal system renders the whole thing meaningless. everything becomes both legal and illegal at the same time. The law is not a formal logical system.

Tombone5 12y ago
Yeah, have you heard of IP-law?

dragonwriter 12y ago

> a single inconsistency in a formal logical system means you can prove anything. First, the law is not a formal logical system (its a signalling mechanism between members of a community), and, second, there is no actual contradiction. The fact that the law provides a mechanism for changing itself that applies to the whole of the law, including the part specifying the mechanism for change, is not a "logical contradiction". It is -- even viewing the law mechanistically (which, as just explained, is a somewhat limited view) -- just means that the law in the future may not be the same as the law now in any of its particulars. But that's not a contradiction, its a choice (and its a choice made largely because such a guarantee would be empty, and providing a formal mechanism which is not impractical makes it more likely that change will be governed by the framework -- even if it changes the framework in the process -- rather than simply abandoning the framework [something that Framers were keenly aware of since they were in a very real sense engaged in exactly the process of abandoning the framework of the Articles of Confederation precisely because of its impractical unanimity requirement.]) > all of this is to say that the claim 'we have a nation of laws, not a nation of men' is entirely fatuous. Its only fatuous if you think law is meant to be mechanistic, rather than a communication mechanism; the "nation of laws, not a nation of men" is a relative description in reference to purely arbitrary authority. Yes, the distinction is not one of crisp binary categories, but then crisp binary categories usually are abstractions that fall apart when applied to the real world.

erehweb 12y ago

The philosopher Wittengstein has something to say about this, in part of a discussion of formal logic. He says contradictions are not as bad as you might think. "Or suppose that there is a contradiction in the statutes of a particular country. There might be a statute that on feast days the vice-president had to sit next to the president, and another statute that he had to sit between two ladies. This contradiction may remain unnoticed for some time, if he is constantly ill on feast-days. But one day a feast comes and he is not ill. Then what do we do? I may say 'We must get rid of this contradiction.' All right, but does that vitiate what we did before? Not at all. [...] When a contradiction appears, then there is time to eliminate it. We may even put a ring round the second rule and say 'This is obsolete.'"

simonh 12y ago
Clearly the sensible way to resolve this is to ensure that the president is always a lady.
deleted 12y ago
[deleted]

pjc50 12y ago

two conflicting rules - one dealing with the importing of flowers, and the other dealing with navigating riverways http://en.wikipedia.org/wiki/R_%28Factortame_Ltd%29_v_Secretary_of_State_for_Transport http://en.wikipedia.org/wiki/R_%28Factortame_Ltd%29_v_Secret... The 'normal' resolution of later legislation with earlier legislation would be that the later wins. However in this case, the House of Lords (at that time the UK's supreme court) ruled that Parliament had not intended that making fisheries rules should eject the UK from the EU and that the earlier European Communities Act had priority in this case and all future similar cases. (Law also spends a lot of time and effort working out what category a particular "thing" actually is - businesses,goods,property - so it can be determined which law applies)