Intro to Logic: Conclusion
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| Type: | Course Related Materials |
| Grade Level: | Post-secondary |
Abstract: We saw three stages of logics: Propositional logic, with formulas like ? Predicate logic, where the same statement was phrased as ?. This introduced the idea of variables, which ranged over atomic propositions. First-order logic, which included quantifiers; the above statements were superceded with the likes of ?. So why, you might ask, didn't we just start out with first-order logic in the first lecture? One reason, clearly, is to introduce concepts one at a time: everything you needed to know about one level was needed in the next, and then some. But there's more: by restricting our formalisms, we can 't express all the concepts of the bigger formalism, but we can have automated ways of checking statements or finding proofs. In general, this is a common theme in the theory of any subject: determining when and where you can (or, need to) trade off expressibility for predictive value. For example, ? Linguistics: Having a set of precise rules for (say) Tagalog grammar allows you to determine what is and isn't a valid sentence; details of the formal grammar can reveal relations to other languages which aren't otherwise so apparent. On the other hand, a grammar for any natural language is unlikely to exactly capture all things which native speakers say and understand. If working with a formal grammar, one needs to know what is being lost and what is being gained. Dismissing a grammar as irrelevant because it doesn't entirely reflect usage is missing the point of the grammar; Conversely, condemning some real-life utterances as ungrammatical (and ignoring them) forgets that the grammar is a model which captures many (if not all) important properties. Of course, any reasonable debate on this topic respects these two poles and is actually about where the best trade-off between them lies. Psychology: Say, Piaget might propose four stages of learning in children. It may not trade off total accuracy, for (say) clues of what to look for in brain development. Physics: Modern pedagogy must trade off quantum accuracy for Newtonian approximations. Researchers exploring fields like particle physics must trade off exact simulations for statistical ("stochastic") approximations. Understanding the theoretical foundations of a field is often critical for knowing how to apply various techniques in practice.
