# Logicians do it with Models

A self-guided jaunt in philosophical logic

## Archive for October, 2010

### Clarity In Talking About The Status Of Attributes

October 13, 2010

When the ontological status of a property or attribute is in question, the central issue is the ontological status of an object’s possessing or exemplifying a property.  When the ideological status of a property or attribute is in question, the central issue are the kinds of predicates that express the property.  The following example brings this out starkly:  A property can be mental, but not physical (ideologically), but can be ontologically physical, but not mental.

If a mental predicate $\textup{A}(x) \in \Psi$ is not lawlike coextensive with a physical predicate $\textup{B}(x) \in \Phi$, where $\Psi$ and $\Phi$ are appropriate vocabularies, the attribute expressed by $\textup{A}(x)$ is mental, but not physical.  Nevertheless, by the principle of physical exhaustion (PE), the attribute is physical, but not mental (ontologically), as it is possessed solely by physical objects.

Entity-wise, every attribute is mathematical-physical, but only those attributes are mathematical-physical that are expressible by mathematical-physical predicates.  So, if physical reductionism is false (which is likely the case), there are attributes which are mathematical-physical entities, but are not mathematical physical attributes.

The confusion arises when terms like “mental” or “material” are used to call out properties or attributes without clearly setting out whether one is talking about the ontological or ideological status of the attribute.  This is an important point by Hellman, and we will see how this distinction and physicalist materialism in general are useful in clarifying questions of reductionism and materialism in the philosophy of mind when we turn to an analysis of Chalmers’ anti-materialist claims.

### The Ontological and Ideological Status of Attributes

October 12, 2010

With the principle of Physical Exhaustion (PE: $(\forall x)(\exists \alpha)(x \in \textup{R}(\alpha))$ where $\textup{R}(\alpha)$ is a rank in the hierarchy of the physical), which allows us to say that everything is exhausted by the physical, every attribute is mathematical-physical (i.e., existing somewhere (possibly high-up) on the set-theoretic hierarchy; see this and this). Thus the ontological status of attributes is mathematical-physical.  Now, for a given vocabulary $\psi$, and for any attribute that expressed by a predicate that makes essential use of members of $\psi$, call that attribute a $\psi$-attribute. So, for example, attributes expressed by physical predicates are physical attributes and those expressed by psychological predicates are psychological. This is their ideological status.

The ontological status of a thing has to do with which extensions of predicates (of ontological kind) under which the thing falls. The most encompassing of such predicates is “is mathematical physical’.  But there are other, narrower ways to distinguish the ontological status of things.  For example, some metaphysical predicates, “is abstract”, “is concrete”, or some scientific ones, “is an elementary particle”, “is an event”, “is a person”, “is a social process”, “is a physical magnitude”, etc. Some other ontological kind predicates have empty extensions: “is a soul”, “is a phenomenally raw feel not identifiable with any entity in the hierarchy of the physical”.  But if something were to satisfy these predicates, the predicates would indicate the ontological kind of these things.  So the important semantic relationship here is satisfaction of ontological kind predicates.

This is different than in the case of the ideological status of attributes, where the important semantic relationship is expression of an attribute by a predicate –i.e., the relationship between the argument and value of a universalizing function. Every entity has an ontological status, but only universals have ideological status determined by the types of predicates for which the universal under consideration is the value of the universalizing function.

So the ideological status of an attribute, is given by the predicates that express it.  These predicates themselves can be classified, for instance, according to scientific discipline, psychological-predicates, physical-predicates, Economic/Sociological-predicates, etc.  These classifications are historical and are subject to change due to a variety of factors –not all of them scientific.  Hellman explains that attributes themselves may have more than one ideological status: for the predicate ‘is in pain at t‘ is coextensive in a law-like fashion with a complex physical predicate then ‘being in pain’ is both a psychological and a physical attribute since it is expressed by both psychological and physical predicates.

In the next update I’ll go into Hellman’s discussion of the confusion that results when the ideological and ideological status of attributes is not clearly stated in debates concerning materialism and the mental.

### Introduction To Hellman’s Treatement of Universals

October 12, 2010

Universals, for our purposes, are sets, predicates, properties, relations and attributes –the same universal can instantiate or subsume numerically distinct things.  Among universals we distinguish between those that are extensional, like sets and predicates and those that are intensional, like properties, relations and attributes.

Roughly, the distinction between these two types of universals is the following.  Those that are extensional obey the principle that equivalence implies identity, while those that are intensional are in violation of this principle.  In cases of non-deformity, the property of being a creature with a kidney is extensionally the same as the property of being a creature with a heart insofar as the same things have each, but the properties are different.  Sets and predicates, on the other hand, do satisfy the principle of extensional equivalence: the set of creatures with a kidney and the set of creatures with a heart have the same members, and are thus identical.

Now, universals in general are taken to fall within the range of functions from the set $\textup{P}$ of predicates of a language $\textup{L}$ to the universal they give expression to $f: \textup{P} \mapsto \textup{U}$, where $\textup{U}$ is the set of universals of $\textup{L}$.  Hellman explains that among such functions there are some that assign universals with much discrimination: two predicates are mapped to the same universal only if the predicates are identical. And others do so with less discrimination: coextensionality is enough. Hellman imagines a partial ordering of universals based on discrimination criteria with predicates being among the most discriminating and extensions among the least.

Since Hellman is concerned with properties and relations used in science, so the first thing we need to do is set up identity criteria for these attributes. For example, we want to say that temperature is the very same magnitude as mean molecular kinetic energy. So two predicates $\textup{F}$ and $\textup{G}$ are mapped to the same universal $u$ just in case it is a scientific necessity that these two predicates apply to the same things.  In other words, $\textup{F}$ and $\textup{G}$ express the same attribute only in case in every model in $\alpha$, $\textup{F}$ and $\textup{G}$ have the same interpretation.  Remember that $\alpha$ is the set of structures modeling scientific possibility.

So, how do we express attributes given this way of thinking about them?  Hellman suggests using the technique from modal logic, such that the function from members of $\alpha$ to the extensions assigned to that predicate by each model in $\alpha$ is identified with the attribute expressed by that predicate.  In this way, two predicates express the same attribute in case they do so in every structure representing a scientific possibility; thereby getting the desired necessity.

If we need to represent universals that discriminate differently than scientific attributes, we can use sets or collections of structures different from $\alpha$ insofar as the universalizing function is no more discriminating than those functions that assign predicates to the same universal in case the predicates are logically equivalent.

For these universals, the identity conditions are the following:  functions (i.e., attributes) are identical just in case they assign the same arguments (i.e., members of $\alpha$) to the same values (i.e., extensions of predicates).   Or simply say that two predicates $\textup{F}$ and $\textup{G}$ express the same attribute if, and only if, $\forall (x_{1}, \dots, x_{n})(\textup{F}x_{1}, \dots, x_{n} \leftrightarrow \textup{G}x_{1}, \dots, x_{n})$ is a law of science.

This last bit will be what links the discussion of attributes to that of reducibility –since we need to make use of explicit definability.

In the next set of notes I’ll go into the difference that Hellman notes between the ideological and ontological status of universals.
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