I've been meaning to write on this subject for awhile now, but recent conversations seem to expedite the necessity of getting clear on what I mean when I use the word 'interpretation'. So, without further fan fare, hand waving, or other particularly useless and time wasting linguistic devices, let's go!
We all interpret. Simple right? We all have a way of looking at the world that frames our experiences along certain lines, which then enables us to express things about how our experiences and observations are related to one and other, and we generally express ourselves using some "natural language" or another. Our natural languages tend to obscure the formal language which underlies our statements. When we express ourselves in a natural language, we are actually also using an underlying layer of reasoning and logic which is necessarily about our interpretative strategies. Put differently, if we are making any sort of statements beyond mere logical formalisms, then reason and logic apply only within our interpretations, but we are unable to use reason or logic in the absence of an interpretation with respect to our use of a natural language about the world.
So, what do we mean by an interpretation? Well, we sort of mean the way we use the word 'interpretation' colloquially: like, for example, when we say things such as "That's only your interpretation" or "Let me offer my interpretation of this painting," and, in some sense, we also are using an interpretative structure when we translate languages from one specific language to another specific language. This latter, though, is more of a "meta" interpretive structure: an interpretation as a function that assigns elements in one interpretation to elements in another interpretation--we are less concerned about this sort of interpretation than we are about specific interpretations, like those we most often refer to when we use the word 'interpretation' in common speech.
Referring to my notes from Logic I with Dr. Mark Gardiner taken many, many moons ago, we will set out on a more than less thorough presentation of the formal and rigorous definition of an interpretation. Don't worry too much though, I will do my best to translate along the way from the formal definition into a more natural language understanding of such a thing; so, simply plow through the formalisms as best as you are able, and it is my hope that once we come out the other side of the natural language interpretation of the formalisms, the reader will be able to go back to the formal definition with a deeper sense of understanding. So, fasten the seat belts on your brains, then, we're in for a bit of a bumpy ride.
In logic, an interpretation is defined in the following way:
1. An interpretation requires a set of linguistic symbols.
In logic these are referred to as 'wffs' or, as I pronounce it, and I think others do as well, "woofs." This is simply an abbreviation for "well formed formulas." Now, without trying to teach or instruct the reader from scratch, what we mean by a "woof" is simply a string of linguistic symbols that also satisfies the rules of their construction. In symbolic logic, for example 'A & B' is a "woof" but 'AB &' is not a "woof." Another example, 'A -> B' is a "woof" but '->AB' is not a "woof." What we mean, then, by a "well formed formula," is a string of linguistic symbols that conform to specific rules in their construction in such a way that they are able to represent a clear and precise statement that we are then able to use within our system of logic. And what we mean by "a set of linguistic symbols" is all and only those strings of linguistic symbols that conform to the rules governing their construction.
Now, to be entirely as clear as I am able, let's look at the same idea as applied to English.
The sentence, "Sally had a nice day at the beach," would count as a "woof" in English. The sentence "Beach a the at Sally nice had day" is not a "woof" in English. Another example, "Seven is the number that occurs between eight and nine," is a "woof" in English (even though we might observe it is a formula that, when interpreted in terms of English qua mathematics, we see is not true--more on that in a bit). The sentence "Eight the is seven nine occurs and number between that," is not a "woof." So what we mean by a "well formed formula" is that it follows the rules of a grammar such that the sentence is sensible within our structure of interpretation; that is, the string of linguistic symbols occur with a specific sort of order that allows them to be interpreted.
Getting the picture? Good. Let's continue.
2. An interpretation requires a structure.
What the heck do we mean by that?
Here is the formal definition of a structure, U:
U must consist of precisely two things:
1. a nonempty set of individuals or elements, which together constitute a Universe of Discourse also known as a Domain.
2. one or more relations defined on this Universe of Discourse.
Any clearer? No, not really, right? And in order to present this formally and in such a way that is understandable, the reader really needs to have sat through a formal logic class. And it is not my intention to subject anyone to that against his or her will. So, I will refer again to my notes taken a few days later, where we discuss these formal matters in a somewhat less formal way. These notes, by the way, were taken on April Fools day in 1999 (probably the right year, anyway--I can't be particularly bothered to figure it out exactly): make of that what you will. This section of my notes bears the title "Metaphysical Musings."
And I quote:
A structure consists of only two types of things:
1. objects, and
2. sets of objects.
End quote.
So, in reference to the formal definitions in (1) and (2) above, the "nonempty set of elements" is simply the things that are counted as things within our Universe of Discourse. That is, it is the scope of things that our Universe of Discourse is able to "talk" about. In a natural language such as English, this is all the objects that our words refer to: dogs, cats, stars, protons, people, places, black holes, clouds, the weather, colours, and on, and on, and on, etc..
Now with regards to (2), what we mean by "one or more relations defined on this Universe of Discourse" are the relations that can be expressed in such a way that all and only those things which bear a specific property occur in some specific set of objects; in other words, the definitions of the relations that establish that the set of objects actually have some specific property.
To return to the formal definitions in terms of Predicate Logic, which is a more complex and developed version of Symbolic Logic, we define these sets of objects in two ways:
(I) By Intension, and
(II) By Extension.
"By intention" is simply a list of a set of necessary and sufficient
conditions which any object in our Universe of Discourse must satisfy in
order to have the property in question.
"By extension" is simply the set of every object that has the property in question.
Now, to return to some simplified versions of these things with respect to the natural language of English, we can consider the following as ways to understand what we mean by intensional and extensional definitions of objects that are in turn collected in some set of things.
Let's use the property of "blueness," as in, things (objects) that have the property of being blue.
Now, a definition by Intension (and this is by no means to be considered as a complete definition of the Intension regarding the property of blueness, but merely to give an idea of what such a definition would consist of) might include things like:
(A) The objects in question must reflect light at wavelengths that fall between 450–495 nm and at a frequency of 606–668 THz that have a photon energy of 2.50–2.75 eV.
(B) The objects in question must conform to what we mean by "being blue."
(C) The objects in question are in fact coloured blue.
We see that by using this list we offer a set of conditions which any object must satisfy in order to be a member of the set of all things that have the property of blueness. The objects that do not satisfy the definition by intension are not to be taken as members of the set of all things which have blueness.
A definition by extension, then, is simply the list of all the objects in our domain that also have the property of blueness. This set includes, for example (and again, not to give a complete list, but only to give the reader an idea of what is meant by "definition by extension"): bluejays, blue balloons, the billiard balls typically numbered with either a '2' or a '10', cars that are blue, the sky (sometimes), lakes (sometimes, but more and more less frequently, perhaps), garments that are blue, and so on and so on, etc..
So, it is my hope that the reader has a bit more sense of what we mean by the formal definition of a structure. We mean, then, the set of all objects that our Universe of Discourse is able to discuss and all the subsets of this set that are defined in terms of the objects having specific properties. We can note, then, that the subsets are a way of dividing up the objects in the Universe of Discourse in many different ways, which will necessarily entail that some objects are members of several subsets, but not all objects that are members of a given subset will necessarily occur in all the same subsets as one and other. For example, and returning to our subset of things with blueness: our blue billiard balls occur in the same subset of blueness that bluejays occur in, but bluejays do not occur in the set of all round things in the same way that billiard balls do not occur in the set of all birds. Some of the elements in the set of balloons that are blue will occur in the set of all round things along with the set of all billiard balls that are blue; but, the set of all balloons that are blue will not occur in the set of all things we use to play billiards, where, obviously, the set of all billiard balls, regardless of their colourings, will occur as a member of that set.
Getting the hang of this? Good.
Finally, but certainly not least:
3. An interpretation requires a set of rules for associating symbols with things.
This is not nearly as mystifying as it might seem, although when we look at it formally below, it will seem quite mystifying indeed.
All this really means is that there is some function that, for a given structure U, takes the objects and sets of objects and maps them to the strings of symbols which make up our language.
Formally, this is called an "interpretation function," and here is its formal definition:
For a given structure U, an interpretation function assigns:
1. to each constant of Predicate Logic an individual belonging to the Universe of Discourse of U, and
2. to each k-ary predicate symbol of Predicate Logic, a set of k-tuples consisting of individuals belonging to the Universe of Discourse of U.
Yes, because that is so clear now, right? Again, this is the formal definition and without knowing what the heck we mean by things like "k-ary predicates," "constants" with respect to PL, and "k-tuples," well, it's pretty much senseless. Again, without actually having sat through a formal Logic class, it is difficult to really get a grasp on these things formally.
So, let's turn back to the section we called "Metaphysical Musings" and see what the notes have to say on this matter.
I quote:
An interpretation bridges "language" and "the world" by:
1. Associating single objects with constants, and
2. Associating sets of objects with predicates.
End quote.
So, what we are getting at by the formal definition of (1) above regarding the constants of PL and the individual members of the Domain of Discourse of U, is simply a symbolic representation in language that picks out only the given object we are considering.
In the natural language of English we might, from the general set of all bluejays in our Domain of Discourse, give a specific name to a specific member of that set, let's say 'Bob,' so that when we discuss "Bob the bluejay" we know that we are referring to the same and only that particular bluejay we've called Bob.
Back to the informal talk about the formal language, an interpretation function assigns a name, say 'c1' to one and only one object from the Domain of Discourse so that there is no confusion as to what we are referring to when we come to evaluate the statements that are made in our formal language by way of operators acting in conjunction with predicates and the constants associated with these predicates. And this is, essentially, related to our association of sets of objects with specific predicates, which, more formally, is about the "k-ary predicate symbols" and the set of "k-tuples."
To go back to English, then, what we mean is that when we say something like:
"Bob the blue bluejay is a pretty bird and he is sitting on the thick branch of that tall tree."
We have a series of words that correspond to not only objects or sets of objects (Bob, bluejays, birds, branches, trees) in our Domain of Discourse, but also words that correspond to predicates, which are properties, (blueness, prettiness, sitting, thickness, tallness) in our Domain of Discourse. We also have "logical operators" which link up various individual claims or statements regarding the objects and their properties. This gets quite complex when we go from a formal language to a natural language. Indeed, some logicians spend some of their time trying to translate philosophical statements from the natural language in which they occur into a formal Predicate Logic version in order to accurately and mechanically evaluate their truth claims. Most efforts are difficult, slow going, and open for debate regarding the translations--especially with respect to which logical operators might best suit a natural language sentence.
In regards to the sentence above, there is basically only the repetitive occurrence of the logical operator & (or at least we will agree to look at it this way for ease of presentation and understanding), which we in natural language, funnily enough, call "and." Taking apart the above sentence and representing all the occurrences of & in it, is, given the simplicity of the statement, pretty much straight forward, so here it is:
1) Bob is a bird & he is blue.
2) Bob is a bird & he is blue & he is a bluejay.
2) Bob is a bird & he is blue & he is a bluejay.
3) Bob is a bird & he is blue & he is a bluejay & he is pretty.
4) Bob is a bird & he is blue & he is a bluejay & he is pretty & he is sitting.
5) Bob is a bird & he is blue & he is a bluejay & he is pretty & he is sitting & he is sitting on a branch.
6) Bob is a bird & he is blue & he is a bluejay & he is pretty & he is sitting & he is sitting on a branch & that branch is thick.
7) Bob is a bird & he is blue & he is a bluejay & he is pretty & he is
sitting & he is sitting on a branch & that branch is thick & that thick branch is part of a tree.
8) Bob is a bird & he is blue & he is a bluejay & he is pretty & he is
sitting & he is sitting on a branch & that branch is thick & that thick branch is part of a tree & that tree is tall.
See how much is contained in a relatively simple statement? Bet you had no idea that when you utter a simple phrase there is a whole bunch of background formalism present in terms of what we mean by logic or logical structure.
We are now, I feel, in a position that we can, in some sense, understand what we mean by an interpretation and how an interpretation functions in terms of things like: making claims, assessing truth, discussing objects, and etc..
An Interpretation that would allow us to make sense of the statement:
"Bob the blue bluejay is a pretty bird and he is sitting on the thick branch of that tall tree,"
needs the following objects to be present in its Domain of Discourse:
{the set of birds, the set of trees}.
The Interpretation must also have the following predicates:
{blueness, prettiness, thickness, tallness}
The Interpretation must have a function which assigns strings of symbols to these objects and the set of objects that have any given specific property. Let's call this function in this example E.
Now, E takes things from our set of objects and maps them to a set of words.
E, in our example, maps from the set of all birds that are also bluejays, to the set of words that make up our English language such that:
'Bob' picks out a specific bird, that is also a specific bluejay, from the set of all birds that are also bluejays.
E would also pick out the predicates, or properties, associated with Bob, namely:
That Bob is a bird, that Bob is a bluejay, that Bob is pretty, and that Bob is sitting.
Notice the interplay between the name "Bob," how it is we identify Bob from the set of all birds, and how the predicates as they relate to Bob help us in deciding which bird is meant by 'Bob'. Taken altogether, we hope we have an Interpretation that is rich enough to point to that specific bird that we are calling, in this instance, "Bob." In other words, our "interpretation function" is such that it allows us to pick out what we have named "Bob" as distinct from all other things named "Bob," in such a way that we understand some things about what Bob actually is and how Bob actually is, namely, Bob is pretty, Bob is sitting, and etc..
OK, let's turn to what we mean by Truth and we shall see how (1) truth only exists inside an interpretation and (2) what is taken to be a "fact" only exists inside an Interpretation relative to the truth-value that the Interpretation establishes in regards to statements about objects that are recognized in the Domain of Discourse of the Interpretation.
Back to our relatively simple statement:
"Bob the blue bluejay is a pretty bird and he is sitting on the thick branch of that tall tree."
We have seen how this statement breaks down into a series of conjuncts, the totality of which is reasonably captured by:
Bob is a bird & he is a blue & he is a bluejay & he is pretty & he is
sitting & he is sitting on a branch & that branch is thick &
that thick branch is part of a tree & that tree is tall.
A "conjunct" is simply any of the statements which occur on either side of any given occurrence of the logical operator &. So, the conjuncts of our statement are:
1) Bob is a bird.
2) Bob is blue.
3) Bob is a bluejay.
4) Bob is pretty.
4) Bob is pretty.
5) Bob is sitting.
6) Bob is sitting on a branch.
7) That branch is thick.
8) That branch is part of a tree.
9) That tree is tall.
In order for our statement to be true in our Interpretation, each of the conjuncts also has to be true. If any one of the conjuncts is false, then the whole statement is false by the rules of logic. The interpretation, then, requires that we are able to consider each term with respect to the conditions that make it either true or false. This is to say, that for each of the conjuncts we need to be able to understand what makes any one of them true or what makes any one of them false. Let's simply look at one of the conjuncts, namely, "Bob is a bird."
In our Interpretation there are two possibilities: either "Bob is a bird" is a true statement or "Bob is a bird" is a false statement. So, if our Interpretation correctly assigns the word 'Bob' to the object that is a bird, and this bird is exactly the one we mean by the string of symbols 'Bob', then on our interpretation the statement "Bob is a bird" is true. However, if our Interpretation instead assigns the word 'Bob' to our uncle, then "Bob is a bird" is false in our Interpretation, and so, with regards to our whole statement:
"Bob the blue bluejay is a pretty bird and he is sitting on the thick branch of that tall tree,"
our Interpretation would evaluate it as a false statement, since Bob is a human being and not a bird. Now, every other conjunct about Bob our uncle could be true: he could be blue, he could be pretty, he could be sitting on a branch of a tall tree, but since he is not a bird and specifically not a bluejay, our claim about Bob as a blue bird that is also a bluejay that is sitting, pretty, and etc., is in fact false.
Without an Interpretation which assigns words to things and establishes the relationships about the things in terms of their properties, well, there simply isn't any sense of "truth" to the matter. We'd have only words and only things but no bridge between them; that is, without an interpretation, we have no way to make claims about anything because we can't talk about the things and we can't associate things with their properties in any sensible, as in utterable, manner.
We can easily see how "facts" fall out of this. If there is a correspondence between our language and the objects in our language's Domain of Discourse and the statements we make about these objects are also true, then that is what we call a "fact." If Bob really is a bird, and if by the word 'Bob' and 'bird' our Interpretation establishes a relationship between those things such that is is true that "Bob is a bird" corresponds to Bob actually being a bird, then we say that it is a fact that Bob is a bird.
Again, without an Interpretation, then, there are no facts of the matter, there are only objects that have properties which are not in any way sensibly able to be discussed; put differently, the facts of the matter are uninterpreted without an association between our language and its objects of Discourse. What we mean by "fact," then, is something that only comes to be when considering true statements within an Interpretation.
To go way back to the beginning, consider the statement:
"Seven is the number that occurs between eight and nine,"
We know this to be false simply because we have an Interpretation that establishes that the object referred to by the word 'seven' occurs in a mathematical structure such that the object seven has its place between the objects referred to as "six" and "eight". Thus, it is our Interpretation that establishes that the fact of the matter does not correspond to our statement about seven in terms of the property of it occurring between the numbers we call "eight" and "nine". Without the Interpretation we have no facts of the matter because we have no way to understand the properties corresponding to the object seven, and we have no way to assess the statement about that object in its relation to other objects.
Thus, we see clearly how truth and facts are entirely dependent upon an Interpretation.
So, turning to reason and the use of logic, we can also see clearly that we are only able to reason about things insofar as these things correspond to some Interpretation which allows us to understand things about the objects in the Universe of Discourse in such a way that we are, in fact, able to reason about these things in terms of the referents and the properties that these objects have as picked out by our Interpretation. Logic is simply the rules by which we understand how the relationship between statements and the statements' objects and properties are evaluated in terms of being either true or false. Without some set of sentences and rules by which we can logically manipulate and assess those sentences, well, there's no logic to be had at all.
An important thing to note is that all interpretations are open to the use of reason and logic, but we are also able to use reason and logic on an uninterpreted set of sentences whereby the "woofs" do not correspond to any objects of a domain of discourse other than the "woofs" themselves. Such practices are called "pure logic" in the sense that the analysis and results that come out of such a study are about Logic itself. This, however, is a purely formal exercise and is not associated with any natural language; that is, when we reason about logic and use logic to analyze itself we must necessarily not be referencing anything that has to do with objects beyond the objects of logic itself.
So, when I make the claim that "reason and logic apply only within our interpretations, but we are unable to use reason or logic in the absence of an interpretation," that is not quite the case, it is, in a sense, a colloquial way of speaking, of being a bit loose with the language. What I mean is that any time we are reasoning about things in the world or using logic with regards to our statements about things in the world, then we must necessarily do so with respect to an interpretation. Without an interpretation reason and logic can only ever be about themselves.
To put this more concretely, then, when I make a claim like:
"I do tend towards having a distaste for certain 'breeds' of atheists who
declare logic and reason as their allies and claim these support their
lack of belief. They are mistaken. Logic and reason apply only within
interpretations and their interpretations assume their lack of belief..."
What I mean is that whenever a person is using logic and reason to assess their beliefs, he or she can only use it insofar as to reason from the assumptions within their Interpretation and use logic to assess the truth-value claims of how these assumptions are interconnected via logical operators (such as 'and', 'or', 'not', 'if...then...', and 'if and only if') with respect to the assumptions of a given interpretation. Logic and reasoning alone cannot establish anything about any belief or lack thereof as without an interpretation there simply is no such thing as belief or lack thereof.
In order to preempt atheists who might wish to say, due to their misunderstanding of the above presentation, that since lack of belief entails that there is no belief, we can thus use pure reason and pure logic alone to support that lack of belief since you (being me) asserted that belief does not exist in terms of reasoning about pure logic alone. I will simply point out that reasoning about logic with logic has nothing to do with any of the other beliefs or facts that exist only in their Interpretations by which they have derived or otherwise assumed the conclusion regarding their lack of belief.
If you, the reader, actually made it to the end of this article, congratulations: you now probably have a better grasp on logic, reasoning, and interpretation than most of your peers, at least, those of your peers who have never taken a course on Logic and have not been trained in ways of reasoning in a formal sense (i.e, mathematically, for example). I hope you found the ride not too bumpy and that your brain made it through without too many bruises. Now, amidst a short burst of fanfare with maybe an accompanying drum roll and perhaps some hand waving, I bid you adieu.
