'When Are Relations Neither True Nor False?
in [Math]
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From: Michael Stemper on 19 Feb 2010 13:39 In article <taofn.894$OJ6.99(a)newsfe22.iad>, Nam Nguyen <namducnguyen(a)shaw.ca> writes: >You summarily dismissed Calvin's appropriately mentioning of CH in this >thread topic, on the ground that it's philosophical, which I don't see >as justified at all. > >Look, there's a false relation, e.g. the empty set {} for a 1-ary relation >symbol, right? I don't see any meaning to the word "false" here. {} is a relation, period. It has no members, but it's a relation. > There's also a true relation, a non-empty Universe U for >a 1-ary relation symbol, right? I'm going to try to decode what you're saying here. My guess is that you mean "UxU" is a relation from U to U. Is my guess correct? If so, it's a relation, just like {} is. These are the trivial cases, but they're both valid relations. The statement "(x,y) ∈ R" can be true or false. "R" is neither true nor false. It's just a set (of ordered pairs). -- Michael F. Stemper #include <Standard_Disclaimer> Time flies like an arrow. Fruit flies like a banana.
From: Marshall on 19 Feb 2010 18:57 On Feb 19, 10:39 am, mstem...(a)walkabout.empros.com (Michael Stemper) wrote: > In article <taofn.894$OJ6...(a)newsfe22.iad>, Nam Nguyen <namducngu...(a)shaw..ca> writes: > >You summarily dismissed Calvin's appropriately mentioning of CH in this > >thread topic, on the ground that it's philosophical, which I don't see > >as justified at all. > > >Look, there's a false relation, e.g. the empty set {} for a 1-ary relation > >symbol, right? > > I don't see any meaning to the word "false" here. {} is a relation, > period. It has no members, but it's a relation. In some relational theories, relations of any arity >= 0 are allowed. In such systems, sometime the 0-arity, empty relation is identified with false and the 0-arity, non-empty relation is identified with true. These relations are the identity and fixpoint values for certain operators. Marshall
From: Newberry on 19 Feb 2010 23:50 On Feb 19, 7:30 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Newberry <newberr...(a)gmail.com> writes: > > You are asking me way too many questions: What gave me this peculiar > > idea? Why it is an objection? What is the objection against? I do not > > have a clue. > > Why then go on about such matters? I do not have a clue. > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: OP on 20 Feb 2010 00:15 Newberry wrote: > On Feb 19, 7:30 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >> >> Why then go on about such matters? > > I do not have a clue. Well it's settled then.
From: Nam Nguyen on 20 Feb 2010 01:59
Michael Stemper wrote: > In article <taofn.894$OJ6.99(a)newsfe22.iad>, Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> You summarily dismissed Calvin's appropriately mentioning of CH in this >> thread topic, on the ground that it's philosophical, which I don't see >> as justified at all. >> >> Look, there's a false relation, e.g. the empty set {} for a 1-ary relation >> symbol, right? > > I don't see any meaning to the word "false" here. {} is a relation, > period. It has no members, but it's a relation. Let's remember that in mathematical reasoning we're allowed to have definitions, as long as they're technically correct. (If they reasonably make sense then they'd be "sound" definitions, but they in the first place must be technically correct). In model-definition, a relation R is a set of n-tuples in which a n-ary predicate P <-> P(x1, x2, ..., xn) [about individuals x1, x2, .., xn] will be determined as true or false. And the way we determine the truth or falsehood of P is by looking the set R: if the n-tuple (x1, x2, ..., xn) is _in R_ then P is true, otherwise P is false. But since {} contains *no* elements, if R = {} then *all* predicates (formulas) of the form P(x1, x2, ..., xn) are _false_. On the other hand, if R = the set of all n-tuples, then *all* predicates P(x1, x2, ..., xn)'s are _true_. It just so happens that in the case of 1-ary relation, the relation R is just a subset of U, the universe of individuals of the model in question. Again, in this case, if R is an empty subset then it's a false relation in the sense that all 1-ary predicate of the form P(x) are _false_, while if R = U, then all predicates of the form P(x) are true, hence R is a true relation in this sense. We could generalize the definitions as: - R is a true relation if the following formula is true in R: Ax1...xn[P(x1, ..., xn)] - R is a false relation if the following formula is true in R: Ax1...xn[~P(x1, ..., xn)] >> There's also a true relation, a non-empty Universe U for >> a 1-ary relation symbol, right? > > I'm going to try to decode what you're saying here. My guess is that > you mean "UxU" is a relation from U to U. Is my guess correct? If so, > it's a relation, just like {} is. These are the trivial cases, but > they're both valid relations. Again I've explained the case of 1-ary where R would be just a subset of U. > > The statement "(x,y) ∈ R" can be true or false. "R" is neither > true nor false. It's just a set (of ordered pairs). ***** Given what we've defined as true and false relations above, then the answer to the (thread-title) question is extremely trivial: *in general*, a relation is neither true or false, simply because in a relation there might be true predicates as well as false ones in the relation, say, R. For example P(x1,x2) might be true [i.e. (x1,x2) is in R], while P(x2,x1) might be false [i.e. (x2,x1) isn't in R]. That's to say if we take the title-question "When Are Relations Neither True Nor False?" rather _too literally_! If we take the liberty to give the question a benefit of a doubt and grant it a slightly different intention: (1) When is a relation R such that a particular formula F would be neither true nor false in it? then that's a different question but it's still a technically valid question the answer of which is either "never", or "yes, there are such cases". I already explained to AK the "yes" answer as the correct one to question (1). Basically it's the cases when in defining a model/relation instance we don't have complete one. But based on his terse/short 10-commandment- like dismissing-assertions or questions throughout the thread, I don't believe he has a desire to technically explore that answer. (He still could prove me wrong in my belief here though). |