Contexts where one does not measure uncertainty via numerical probabilities

Clearly one can go through much of ordinary life with only common sense notions of likely/unlikely, illustrated by the saying "When you hear hoofbeats, think of horses, not zebras". A noteworthy context in which non-quantitative assessment is mandated is the famous "beyond reasonable doubt" criterion for criminal conviction. The legal profession explicitly refuses to quantify this; if you as a juror were to ask the judge whether a 97% probability was sufficient, the judge would not give you a straight answer!

As a more substantial example, the Intergovernmental Panel on Climate Change (IPCC) issues periodic reports, widely regarded as the most authoritative analysis of scientific understanding of climate change caused by human activity. Future predictions involve uncertainty, and they want their many authors to be consistent in how they write about uncertainty, so provide technical documents such as Guidance Notes for Lead Authors of the IPCC Fourth Assessment Report on Addressing Uncertainties from which I have extracted the table below, there labelled "A simple typology of uncertainties".

Type Indicative examples of sources Typical approaches or considerations
Unpredictability Projections of human behaviour not easily amenable to prediction (e.g. evolution of political systems). Chaotic components of complex systems. Use of scenarios spanning a plausible range, clearly stating assumptions, limits considered, and subjective judgments. Ranges from ensembles of model runs.
Structural uncertainty Inadequate models, incomplete or competing conceptual frameworks, lack of agreement on model structure, ambiguous system boundaries or definitions, significant processes or relationships wrongly specified or not considered. Specify assumptions and system definitions clearly, compare models with observations for a range of conditions, assess maturity of the underlying science and degree to which understanding is based on fundamental concepts tested in other areas.
Value uncertainty Missing, inaccurate or non-representative data, inappropriate spatial or temporal resolution, poorly known or changing model parameters. Analysis of statistical properties of sets of values (observations, model ensemble results, etc); bootstrap and hierarchical statistical tests; comparison of models with observations.

This table is addressing the issue of uncertainty and mathematical modeling. It makes the point that, within a complex setting (such as future climate change), any asserted numerical probability is (at best) an output from some complicated model in which all these different kinds of uncertainty are present. This point is obvious once you think about it; but it's just different from what's said in textbooks on the mathematics or philosophy of probability.