The idea of order is familiar.
Solid crystals have both short-range order and long-range order.
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First order logic and second-order logic are in a sense two opposite extremes.
By and large, results of many-sorted first order logic translate immediately to second-order logic.
This raises the question, how can we understand second-order validity from the second-order perspective?
Elementarism (Bergmann 1968) is the view that there are first-order properties but no properties of any higher order.
The ontological commitments of second-order theories will then depend on how the second-order variables are interpreted.
When second-order logic is used as a foundation for mathematics, a situation may emerge in which third order logic is needed.
Church’s notion of “order” is not quite one that is familiar from talk of “first order logic” and “second order logic”.
So the only way to get a well-order of, say, the real numbers, in second-order logic is to assume some form of second-order Axiom of Choice.
First-order sentences lack the expressive power of first-order theories in general, and first-order languages lack the expressive resources of higher-order languages.
The entry on second-order and higher-order logic provides more detail and gives some indication of the complexity of the set of valid second-order formulas in the standard model theory for second-order logic.
On Frankfurt’s account, there is no apparent reason not to regard both desires as authentic (the first-order desire because it is endorsed by the second-order desire, the second-order desire because it is a second-order desire).
To make ontological commitment transparent, Quine holds, second-order theories must be transformed into first-order theories by replacing second-order variables with first-order variables ranging over sets, and taking predication to be set membership.
In chapters 63 and 64, Kilwardby discusses different ways to consider the order of the sciences: (i) the order of discovery, (ii) the natural order, (iii) the ordering according to ends, (iv) the order of the degree of certainty, and (v) the order of teachability.
This argument cuts both ways, in the sense that it entails that the causal order is the temporal order (contra the arguments from time travel, simultaneous causation, and physics), but also entails that the causal order cannot be based on the temporal order on pain of circularity.
Eusocial behaviour is found in ants and bees (order Hymenoptera), some wasps in the family Vespidae, termites (order Isoptera; sometimes placed in the cockroach order, Blattodea), some thrips (order Thysanoptera), aphids (family Aphididae), and possibly some species of beetles (order Coleoptera).
That is, there are sentences of first-order arithmetic that can be deduced from the second-order induction axiom (together with the other axioms of arithmetic, which are common to first-order and second-order arithmetic) but not from the instances of the first-order induction schema (see Shapiro 1991: 110).
See also Bath, The Most Honourable Order of the; British Empire, The Most Excellent Order of the; Chrysanthemum, Order of the; Companions of Honour, Order of the; Garter, The Most Noble Order of the; Golden Fleece, Order of the; Knights of Malta; Legion of Honour, Order of the; Merit, Order of; Paulownia Sun, Order of the; Pour le Mérite; Rising Sun, Order of the; Royal Victorian Order; Saint Michael and Saint George, The Most Distinguished Order of; Templar; Thistle, The Most Ancient and Most Noble Order of the.
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See also Bath The Most Honourable Order of the British Empire The Most Excellent Order of the Chrysanthemum Order of the Companions of Honour Order of the Garter The Most Noble Order of the Golden Fleece Order of the Knights of Malta Legion of Honour Order of the Merit Order of Paulownia Sun Order of the Pour le Mérite Rising Sun Order of the Royal Victorian Order Saint Michael and Saint George The Most Distinguished Order of Templar Thistle The Most Ancient and Most Noble Order of the