The phrase "a rational preference" is correct and usable in written English.
It can be used when discussing choices or decisions that are based on logical reasoning or sound judgment.
Example: "In making a decision about which product to buy, I always try to have a rational preference rather than being swayed by marketing tactics."
Alternatives: "a logical choice" or "a reasoned preference".
A very common assumption is that a rational preference ordering must be transitive (i.e., if A is preferred to B, and B is preferred to C, then A must be preferred to C. See the entry on preferences (Hansson and Grüne-Yanoff 2009) for an extended discussion of the rationale behind this assumption).
"But if people accustom their taste so that they enjoy hamburgers," Mr. Murphy said, "then trading health for taste is a rational preference".
One interpretation is that, although each of the individual voters has a rational preference ordering, the group's preference ordering (defined as the majority ordering) is not rational.
The theorem is constructed to resolve the question of whether there is any mathematical procedure for amalgamating individual preferences that results in a collectively rational preference ordering of all the possible outcomes.
The incommensurability of certain dimensions of value, say, freedom versus welfare, or average versus total welfare, may even be a feature of some ethical theories; this puts a lot of pressure on completeness being a reasonable idealisation, let alone a requirement, for rational preference.
In short, Premise 1 is in tension with Premise 3. The issue then becomes whether continuity is a requirement on rational preference.
No general rule satisfying reasonable constraints can be devised that can transform any set of individual preferences into a rational social preference.
For instance, social choice theory questions the idea that there can be a fair decision making function that transforms a set of individual preferences into a rational collective preference.
The above can be taken as a preliminary characterisation of rational preference over options.
Some people find the Continuity axiom an unreasonable constraint on rational preference.
Others (e.g., Broome 1991a) argue that Transitivity is part of the very meaning of the betterness relation (or objective comparative desirability); if rational preference is a judgment of betterness or desirability, then Transitivity is non-negotiable.
