Terrible answers. Here is the correct one.
Belief and Faith, because of our theological heritage have been conflated quite intentionally. So we have to deconflate (disambiguate) them before we can answer the question.
A belief or reported preference refers to that which you report (state) that you understand to be True, and honestly think you will or do, act as if is True. (whether or not you actually act as such is something different.)
A demonstrated preference refers to what we do regardless of what we believe, say we believe. This is why social sciences and psychology were pseudosciences and economics was necessary to stop them from spreading pseudoscience: people demonstrate preferences when they vote or purchase things, and they report, and say they believe very differently from how they vote or purchase. Hence we use only indirectly produced information to test people’s demonstrated preferences, and nearly all surveys are to large part meaningless on anything that someone would virtue signal (Google “Virtue Signaling”).
An article of Faith requires we preserve belief (act as if true) something that is contrary to the evidence in order to preserve the value of acting in accordance with Wisdom Literature in order to achieve desirable ends, even when we don’t understand the relationship between cause and effect. In economic terms faith allows us to buy cheap options on achieving a personal or collective good, and renders one’s plans and actions invulnerable to rational and scientific persuasion. That is their value. It turns out that faith in others is the optimum strategy for producing high trust cooperation. That was just a theory until we proved it in the past century.
An ideology functions, like literature, to inspire individuals to action under democracy. Ideologies need not be rational or consistent, and are less vulnerable to criticism if they are not. Ideology is the result of our change to (limited) democracy.
A philosophy provides methods of decidability in order to achieve a desired state of affairs. The domain of philosophy is individual preferences, and interpersonal good.
A logic provides a grammar (rules of continuous disambiguation) for the testing (criticism) of sets of constant relations for internal consistency between two or more states (falsification by competition).
All disciplinary languages (grammars) from math to logic, to programming, to contract language, to common language, to fiction (and even ficitnoalisms – meaning pseudoscience, and theology) consist of variations in the rules of grammar (rules of continuous disambiguation), including variations in permissible vocabulary (paradigms).
A science provides a formal process and makes use of instrumentation for the use of measurements for the elimination of ignorance, error, bias, and deceit, by the falsification (passage of testing) of categorical consistency, internal consistency, external correspondence, existential possibility, scope completeness, limits, and parsimony. If the science is a social science it must also include tests of rational choice given available knowledge and incentives (rationality), and if a matter of law, tests of voluntary reciprocity (morality)
As far as I know this is the ‘state of the art’ set of definitions.
The Propertarian Institute,
Andrew Heywood : Political Ideologies : An Introduction.
Emmanuel Todd: The Explanation of Ideology
Thomas Sowell: A Conflict of Visions: Ideological Origins of Political Struggles