Names and First Order Logic

This post is a summary or brief overview of something I hope to delve into more detail about. It is about how we are to characterize our thoughts at a most basic level, logically speaking. What I mean by ‘logically speaking’ concerns how our thoughts track what is true, or what we take to be true. I am going to show one potential problem for First Order Logic concerning how it is able to explain our most basic logical thoughts about names.  After showing the problem as I see it, I will show how providing a fast and quick account of names is able to circumvent the problem. In the end, First Order Logic can translate everyday statements about names.

Part of First Order Logic has to do with quantification. I will be considering our thoughts about things, but it should be obvious that my considerations generalize to statements about things as well. To begin, let’s say I have the thought that some of the grass in my yard is green. This is expressed in First Order Logic with a variable (e.g. x) that is bound (i.e. fixed by or attached to) the quantifier “There is at least one _” where “_” is replaced by a variable of choice (Note that this is a quantifier because it specifies a quantity—at least one).

Translating the thought that some of the grass in my yard is green into First Order Logic, we say there is at least one x such that x is grass and x is in my yard and x is green. The way that this expression is satisfied (i.e. found to be true) is that all of the objects in the vicinity of what I am discussion (i.e. The domain of discourse) are named, and then each and every one of those names is substituted for x. As the quantification states, if at least one of those names substituted for x yields a true sentence with the quantification dropped, then the expression, and hence the original thought, is true.

To illustrate, let’s say that some object in my yard is named ‘g’; and so, ‘g’ will substitute ‘x’ in the translated expression, minus the starting quantification, like so: “g is grass and g is in my yard and g is green.” This is an expression that is clearly either true or false. If g is indeed a blade of green grass, then this expression is true and since we have found at least one true expression when a name substitutes x, the original thought about green grass being in my yard is true. So far so good.

Notice that the translation of my thought into a First Order quantified statement already assumes some apparati that is used to arrive at its truth. So for example, objects in my yard will have various names (or be capable of being named) so that they can come to be incorporated in quantified statements. That we have the notion that some blade of grass amongst the vast amount of them in my yard can be named as a particular to be incorporated into a simpler sentence about whether or not that particular is the way we think it is is easy enough to follow.

Yet we also may consider names themselves as a tool we use in communication. We may think that we can have all sorts of quantified thoughts about names themselves. So, for example, we easily have the thought that the name ‘Bill’ has four letters and that it is a shortened version of the name ‘William’.

Now to the potential problem: At first glance, it does not look like First Order Logic can handle such statements about names, aside from simply translating it as a simple proposition P, where its underlying logical structure will be left unclarified. This is because substituting the name ‘Bill’ for x in a statement about x will end up being a statement about Bill and not about his name. For example, if we put in the name ‘Bill’ to substitute x in “x has four letters”, we will have “Bill has four letters.” The problem is that we are trying to say something about Bill’s name, and not Bill the man. We will claim that Bill the man has flour letters, which is nonsense.

And so, the problem is that First Order Logic (from now on FOL) was able to translate one thought, about the grass in the yard being green, but not another thought, about a name having some letters. These thoughts are of course different on the surface, but their underlying logical structure—picking something out and ascribing a property to it—should be identical. So First Order Logic is idiosyncratic in that it illuminates the same type of thinking for one domain (outdoor medium-sized objects), but not others (names). Yet, if they indeed share the same type of thinking with the same underlying logical structure that FOL illuminates, then FOL’s inability to translate this structure in some cases casts doubt on the notion that FOL is the correct translation of the underlying logic of our thinking.

It may readily be thought that one answer is to name Bill’s name something, and then talk about that name. But how do we do that? If we say that ‘Bill’ = ‘b’, then this is saying that Bill’s name, ‘Bill’, is the same as ‘b’, which is clearly false. If we say that Bill = b then we are saying that two objects are identical, but nothing is said about names. So simply naming a name would be a function that is not straightforwardly of FOL. I have trouble coming up with a translation of one name picking out the same object as that of another that uses merely the variables, names, and regular connectives (e.g. and, or, etc.) of FOL. Another issue on the face of it is that Bill’s name as something that functions in an FOL statement is not something to pick out with a name, because names function to name elements in the domain of discourse (more or less the subject matter of what is being discussed), and so the domain of discourse would not itself have any names.

Despite this diagnosis, naming Bill’s name is on the right track. The straightforward solution, I believe, is to name the components of Bill’s name and then talk about such components as we do anything else. After all, such components are able to be our domain of discourse. This is how we will circumvent our problem. Once we are clear about what a name is, then we will realize that it is amenable to FOL after all, and that even the artificial naming process that FOL engages in with respect to the quantifiers is actually more down to Earth than it may first appear.

To begin down this path, a name is a label or a tag that is connected with something. Before saying how it is connected, let’s get somewhat clear about what labels or tags are. Labels or tags compose objects in the world. When we write our names, the names are composed of certain marks made of certain objects, such as marker on a whiteboard or ink on a piece of paper. These marks are shaped a certain way according to the letters. The way the mark is connected with something has to do with our ability, psychologically speaking, to affix one object with another. The primary psychological connection, as far as I can discern, is one of attention. The name brings attention to the object that has it. Entire sentences that contain the name bring attention to the situation as it includes the one named. So mentioning “Friedrich is late to class” in the context of a classroom brings attention to the fact of Friedrich’s tardiness.

Further than this, names are understood to be affixed to an object regardless of whether or not the name is being manifested. Friedrich is named ‘Friedrich’ even when no one mentions his name—i.e. When the name is not marked on a board next to him or is not being spoken near him. This is because, once a name is learned (which is no more than affixing a mark in enough proximity to the object that it fixes our attention on it), then it is known that there is a way of picking him out with the sights or sounds that compose his name. This is no different from knowing what color shirt Friedrich is wearing when you are not directly looking at him—the knowledge of the memory of what shirt he wears is enough without having to look again. So too for his name—when it is not spoken or written, it is nonetheless known which sounds or marks are the tool to bring attention to him. A name functions the same as a pointing finger, but where a finger differentiates where it points in space, a name differentiates where it points by the sound or shape of the mark.

So knowledge based on a memory (in the same sense of recalling what shirt someone wears when not looking right at it) is key with knowing names (i.e. knowing which objects have which names). Names function primarily to draw attention to the object that has the name. If I am right, then names are composed of two ingredients, memories and objects. (Note that my distinction is not between the physical and the mental. This is because it may turn out that mindless physical matter is a nonsensical notion, or that memories are really a special type of object and so aren’t incompatible with them, or something else (I would defend the second point at least, that memories are a special type of spatio-temporal object).)

With this background in place we are now ready to solve the problem for how FOL can talk about names. This is because the components of names can themselves be given names, and then included in FOL statements. For example, “Frederick’s name starts with an ‘F’” may be translated as:

There is an x such that x is a psychological mark and x is psychologically affixed to Friedrich and x starts with an ‘F’. We will take each of these components in turn, but first note that this is a rather higher-level translation; that is, perhaps a more explicit translation would make explicit what, exactly, the psychological mark or psychological affixation is (e.g. a dispositional property of members of a linguistic community). Also, as we have covered, the mark does not have to be present in the environment outside of the psychology of individuals. Yet there is no denying that there is such a psychological mark. This focus on psychology as being a locus for names may be controversial for some philosophers, so to prove this simply think of Friedrich’s name. But there is not really any Friedrich. Yet we’ve been using this name, which may be considered one in other contexts, as a heuristic. So all of our learning has made use of merely a considered name. Yet let’s say Fridrich actually exists and is in front of you. Suppose also that all marks of his name disappeared. Yet your ability to consider his name would by no means disappear. Yet what would you be considering?—there are no marks in the environment!  Bold-but-most-reasonable claim: a name is usually no more than a psychological property or particular (i.e. once it has been learned). Friedrich has his name just for the psychology of individuals who use a psychological particular in order to easily direct their attention to Friedrich. After all, names are much more often used in thought than in an environmental mark (I think of e.g. my Father much more often than I speak with him one-on-one). It is not typical of me to veer this far away from the external world when speaking of things, but the thing here is a component of language—which is the speaking itself and the thoughts that undergird it. Finally, that x starts with an ‘F’ means that the psychological component is a part of class—that which is developed or learned from those marks of the environment that start with an ‘F’—i.e. that are shaped like an ‘F’. Does this look good as a possible way to talk about names in FOL? Are there alternative accounts?

In the end, under my analysis of names, thoughts about names can be straightforwardly brought into FOL statements in a way that mirrors the underlying logical structure of similar thoughts that involve ordinary objects of the environment. The way I accomplished this in general is by analyzing the components of names and then talking about those components.

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s