# Formal Concepts

A major part of our project will concern what may be called “formal concepts”. These are very abstract concepts of a loosely logico-mathematical character, which play a crucial role in structuring our all of our thinking and theorizing, including about the natural world. The chief examples of formal concepts that we will consider are: truth, object, infinity, collection (or set), and negation.

Our distinctive core hypothesis is that some closely related but different concepts have been conflated in ways that result in paradoxes or other pathologies. (This idea figures centrally also in the work of Kant.) Progress can be made by disentangling coherent successor concepts. This will involve carefully articulating “job descriptions” for the various successor concepts. Suppose we articulate two distinct “job descriptions”. One possible development is that we find two concepts which can do the two jobs, respectively. Another possible development is that only one of the “jobs” is found to be necessary, while the other can be abandoned. This makes it easier to find a concept that satisfies the necessary but stripped-down “job description”. Examples of both kinds of development will be given below.

### Infinity

Loosely speaking, something is infinite if it is too large to be measured. But measured by what? With the 19th century mathematical revolution due to Cantor, two different answers became available: (i) measured by a natural number, or (ii) by one of Cantor’s “transfinite” numbers. The two different answers give rise to two different notions of infinity, which were superimposed until Cantor’s discoveries allowed them to be disentangled. Unfortunately, in subsequent theoretical developments, only one of these notions is put to active use, namely the one tied to answer (i).

We argue there are good reasons to retain the other notion as well, which has a lot in common with the ancient Aristotelian notion of potential infinity. This investigation was initiated in Linnebo’s ERC project (see his CV), but further progress can be made by exploring other aspects of the resulting view, such as connections with Dummett (1991)’s influential notion of indefinite extensibility, as well as the use of intuitionistic logic to reflect the thesis that the hierarchy of sets and numbers is merely potential in its existence, never actual. We will also seek inspiration from Kant, who anticipated many of the relevant ideas.

### Collection

The concept of set–or, more generally, collection–is central to mathematics and logic. Our hypothesis is again that two distinct ideas have been superimposed: (i) a “combinatorial” conception of a collection, which holds that a collection as constituted by its arbitrarily chosen elements; and (ii) an “intensional” conception, which holds that a collection as constituted by its conceptually articulated membership criterion. We believe that mathematical and philosophical progress can be made by disentangling these concepts. Instead of operating with a single “job description” for a concept, which no single concept of collection can satisfy, we can articulate two distinct “job descriptions”, each of which is satisfied by one of the mentioned concepts of collection.

In order explore and hopefully confirm this hypothesis, we need to:

- develop a good theory of intensional collections (a theory of properties)
- explore the use of intuitionistic logic for this purpose
- explore the phenomenon of nominalization and the relation between how properties figure as semantic values of predicates and as referents of singular terms

Furthermore, we distinguish two conceptions of generality which mirrors that between two conceptions of collection:

- a “combinatorial” conception, on which “All Fs are Gs” reduces to a conjunction of facts about each and every F to the effect that it is G. This validates classical logic.
- an “intensional” conception, on which truths of the form “All Fs are Gs” hold because it is in the essence or nature of Fs to be G. This likely validate no more than intuitionistic logic.

This distinction between two forms of generality is potentially important in connection with another central concern of our project, namely the analysis and significance of “generics”.

### Truth

Our hypothesis is once again that two distinct notions of truth have been conflated and that progress can be made by separating them:

- a deflationary notion, on which truth is merely a device for “disquotation”. (Thus, “snow is white” is true just in case snow is white.)
- a genuinely semantic or representational notion, on which a statement is true just in case it represents reality as it is. (The Tarskian approach to truth provides a good first approximation.)

Our hypothesis is that the deflationary notion of truth is in fact dispensable. This allows us to articulate a “job description” for a concept of truth from which deflationary requirements have been hived off. We believe the semantic concept of truth satisfies this stripped-down “job description surprisingly well.

The concept of truth figures in the analysis of many other concepts (so-called “remote interlocking concepts”) This restricts our ability to fix or replace the concept of truth. We will explore whether the proposed emphasis on one of the constituents of the ordinary notion of truth is legitimate.

Further, we explore whether the notion of indefinite extensibility (from 5b) can be brought to bear on the analysis of truth and the semantic paradoxes. This may yield a powerful alternative to the “contextualism” about truth developed by T. Burge, C. Parsons, and others. We explore accounts of propositions understood as the fundamental bearers of truth. Our hypothesis is that, for any definite domain, we can find propositions that take us beyond this domain. However, when the domain is understood as indefinite, our hypothesis is that this move can be blocked.

### Object

People have been conflating many different ideas under the heading of “object”, which has caused much confusion in metaphysics and the philosophy of science. The different ideas include:

- concreteness (that is, being located in space and time, and being causally efficacious)
- being a physical body, with spatiotemporal cohesiveness and natural boundaries
- a purely logical notion of object, subject only to the logical principles concerning identity
- validating some version of Leibniz’s principle of identity of indiscernibles
- capability of being “tracked” across different moments of time and possible worlds

We propose to explore the conception of ontology (the study of what there is) that results when the different ideas are disentangled and the procrustean bed of a single, demanding notion of object is abandoned. We believe this will shed light on:

- old philosophical puzzles concerning the relation between objects from different categories (such as my mind and my body, or my cup and the particles that compose it).
- objects that aren’t part of the fundamental “furniture of reality”, such as my cup, abstract objects (such as numbers and sets), and social objects (contracts, marriages).

### Negation

In the history of logic two concepts of negation have been distinguished: (i) a negation of a term or property; and (ii) a negation of a sentence or proposition. While (ii) is thought to have existential import, (ii) is not. Kant’s critical revision of metaphysical concepts shows how many purported insights into the nature of objects is really based on invalid inferences in which these two kinds of negations are conflated. Since much of our reasoning is based on indirect proofs with the use of negation, Kant’s critical engagement with such proofs is a promising starting point for the revival of the debate whether we need a renewed attention to the different functions of negation.