Ce magnifique processus naturel se forme dans des circonstances bien spécifiques, sur les branches de certains arbres pourris et seulement dans quelques endroits sur Terre.
As in Peano arithmetic the natural numbers are generated by 0 and the successor operation [ltr][size=18]ss[/ltr]. The elimination rule states that these are the only possible ways to generate a natural number.[/size] We write [ltr][size=18]f(c)=R(c,d,xy.e)f(c)=R(c,d,xy.e)[/ltr] for the function which is defined by primitive recursion on the natural...
Martin-Löf type theory has four basic forms of judgments and is a considerably more complicated system than first-order logic. One reason is that more information is carried around in the derivations due to the identification of propositions and types. Another reason is that the syntax is more involved. For instance, the well-formed formulas (types) have to be generated simultaneously with the provably true formulas...
Intuitionistic type theory (also constructive type theory or Martin-Löf type theory) is a formal logical system and philosophical foundation for constructive mathematics. It is a full-scale system which aims to play a similar role for constructive mathematics as
We conclude with some remarks on a question that was briefly mentioned above in connection with the Special Composition Question but that pertains more generally to the underlying notion of parthood that mereology seeks to systematize. All the theories examined so far, from M toGEM and...
The algebraic strength of GEM, and of its weaker finitary and infinitary variants, is worth emphasizing, but it also reflects substantive mereological postulates whose philosophical underpinnings leave room for considerable controversy well beyond the gunk/junk dispute. Indeed, all composition principles turn out to be controversial, just as the decomposition principles examined in Section...
Let us now consider the second way of extending M mentioned at the beginning of Section 3. Just as we may want to regiment the behavior of P by means of decomposition principles that take us from a whole to its parts, we may look at composition principles that go in the opposite direction—from the parts to the whole. More generally, we may consider the idea that the domain of the theory ought...
There is a way of expressing the supplementation intuition that is even stronger than (P.5). It corresponds to the following thesis, which differs from (P.5) in the consequent:
We may also ask the opposite question: Are there any stronger ways of expressing the supplementation intuition besides (P.4)? In classical mereology, the standard answer is in the affirmative, the main candidate being the following:
It is convenient at this point to introduce some degree of formalization. This avoids ambiguities stemming from ordinary language and facilitates comparisons and developments. For definiteness, we assume here a standard first-order language with identity, supplied with a distinguished binary predicate constant, ‘P’, to be interpreted as the parthood relation.[size=12][
With these provisos, and barring for the moment the complications arising from the consideration of intensional factors (such as time and modalities), we may proceed to review some core mereological notions and principles. Ideally, we may distinguish here between (a) those principles that are simply meant to fix the intended meaning of the relational predicate ‘part’, and (b) a variety of additional, more substantive...
Mereology (from the Greek μερος, ‘part’) is the theory of parthood relations: of the relations of part to whole and the relations of part to part within a whole.[1] Its roots can be traced back to the early days of philosophy, beginning with the Presocratics and continuing throughout...
Metaphysical theories often rely on resources which are taken as “primitive”: roughly, theoretical resources which are not to be further explained or analysed. Different theories of the same subject matter will often take different resources to be primitive, and while it is a difficult question to decide whether one set of primitives is better or worse than another, evaluation of the relative simplicity, naturalness,...
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