Modeling: Option to cross-validate and predict afterwards This is beyond a R question,
But why doesn't it make sense to fit a model, say, a linear discriminant (LDA) model, with leave-out-one cross validation, and afterwards to use this model to predict a new data set?
Is this reasoning right or not: using such cross validation is a way to measure the performance of the model (instead of testing the model on new test data). If the model performs well, you model it again on the complete training data set and this final model you then use to predict new values.
 A: 
But why doesn't it make sense to fit a model, say, a linear discriminant (LDA) model, with leave-out-one cross validation, and afterwards to use this model to predict a new data set?



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*Because resampling validation (e.g. loo) fits lots of surrogate models. Thus, you do not have one model to use for prediction 

*Instead, for prediction the one model fit on all data is used. As it has a (slightly) larger training data base, you hope that it may be better than the surrogate models.

Is this reasoning right or not: using such cross validation is a way to measure the performance of the model (instead of testing the model on new test data). If the model performs well, you model it again on the complete training data set and this final model you then use to predict new values.

This reasoning is first of all very muddled: validation measures the performance. Testing is just another word for this. 
There are different methods to measure performance, e.g. resampling or testing with an independent/held out test set. 
Many terms in this context, in particular validation and testing are used with different meaning in different fields. You need to explicitly state which meaning you use.
The point that is correct in your statement is that you use a model built on all cases for prediction. However, you don't just do that if the cross validation yielded nice results: 


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*If the cross validation was done to estimate/measure the "final" model's performance, you are not allowed to choose on the basis of the cross validation results. That would be data-driven model selection/optimization.

*If you want to do data-driven model selection/optimization, you need to validate (= measure performace) the chosen model. That requires independent test data again, which can come from an outer cross validation loop or an outer independent test set. 
Finally, yes resampling validation is used when the data set is not large enough to allow a one-time splitting. However, loo is not the preferred choice, other resampling techniques like out-of-bootstrap or iterated $k$-fold/leave-$n$-out cross validation is better.
