*Tracy Bowell and Gary Kemp Critical Thinking: A Concise Guide, 3rd Ed. Routledge 2010
4 logic: inductive force
… An inductive inference is a deductively valid inference whose premise is a generalisation about some sample of a given population, and whose conclusion is a generalisation about the population as a whole, or about a particular member or set of members of the population outside the sample. In general, the force of an inductive inference depends on the degree to which the sample is representative of the population [i.e., ‘relevantly similar’].
5 the practice of argu8 ment reconstruction
Balancing costs, benefits and probabilities
What we are leading up to is the concept of expected value
Let o1 , o2 … on be the possible outcomes of an action A; let V(o) be the value (cost or benefit) of each outcome o, and let P(o) be the probability of each outcome (given that action A was performed). Then the expected value of an action A is:
[P(o1)×V(o1)]+[P(o2)×V(o2)]+ … [P(on)×V(on)].
It is somewhat controversial to suppose that the rationality of all action depends on its expected value. … [There] is a certain limit to the application of expected value calculations: the expected value of a proposed action tells us whether or not it would be rational to do something, unless it is overridden buy the existence of rights or moral rules. …
… Almost all fallacies fall under one of the following two types:
- Formal fallacies. … simply a logical mistake.
- Substantive fallacies. [These] involve reliance on some every general unjustified assumptions or inferences. We need only make these premises explicit in order to see that they are false and unjustified. What distinguishes a fallacious argument of this kind from an ordinary unsound argument is that the implicit, false, or dubious premise will be of a very general nature, having nothing specifically to do with the subject-matter of the argument.
8 truth, knowledge and belief
… It remains one of the defining endeavours of philosophy to formulate a consistent theory of knowledge that does not result in scepticism.
Much of my life has been spent trying to unmask substantive fallacies, using mathematics to demonstrate in the clearest possible way that certain common arguments in their favour have some questionable premises, which the mathematics makes starkly explicit. Yet on the whole people with these beliefs either resist the idea the premises are false and unjustified or shift their ground to different premises or, more common still, become vague about the justification except by appeals to a majority of their peers or an authority of their peers. Sometimes the claimed authorities do not actually claim the supposed generality and may even explicitly rule out the type of case at hand. Yet people may still believe.
Inductive inference is at the root of all practical reasoning. As described above it relies on the case at hand being ‘relevantly similar’ to the samples from which inferences are to be made. For example, when the anyone makes financial or economic forecasts there is an implicit assumption that the future will be relevantly similar to the past. So is no good looking at the forecasts and concluding, for example, that nothing very bad will happen. In the first place, as Taleb [*] notes a sample of the last 10 years’ of data is not worth much if one is interested in ‘one in a hundred year events’. In the second, if we think of an economy as being characterised by stochastic equations, these equations have changed from time to time, and may do so again. Sampling the current economy only tells you about the current economy, not the next.
Inductive inference (and in practice, all inference is inductive) only tells you what will probably just so long as the situation remains relevantly similar. If you want to understand the likelihood of a transition, and what you may transition to, then you need a representative sample of a population with such transitions. It is by no means obvious that this is at all possible for situations that are influenced by sentient beings.
The method of balancing costs, benefits and probabilities by maximising ‘expected value’ (or ‘utility’) is identical to the usual mathematical procedure with the sole exception that the scope of the approach is very much broader than mathematics can justify.
Precision, degrees of belief
Mathematics can often only justify imprecise constraints on probability, degrees of belief, etc. This book claims that one can always be precise, without providing any justification. I am sceptical.
Russell’s Human Knowledge, which covers some overlapping ground, is more logically grounded and links to relevant mathematics.