Stories & Soliloquies

Tag: logic

• The Philosopher’s Lexicon: Analytic and Synthetic Reasoning

  Posted at 12:30 pm by Michelle Joelle, on May 8, 2015

  Welcome back to The Philosopher’s Lexicon. My primary goal in this series is to explore common philosophical vocabulary, hopefully transforming these words from useless jargon into meaningful terms. My secondary goal is to highlight how contentious some of these terms can be – especially those which seem obvious. These definitions will not be comprehensive by any means, so please feel free to add your own understanding of each term as we go. 

  

  This week’s entry into the lexicon features yet another epistemological distinction: the difference between analytic and synthetic reasoning. These are common terms you will find in many works and texts books, but for clear and concise explanations of these terms, see Kant’s Groundwork for the Metaphysics of Morals, Nils Ch. Rauhut’s Ultimate Questions and A. C. Grayling’s An Introduction to Philosophical Logic. 

  Let’s jump right in.

  An analytic statement is one where the truth of the statement can be determined by the internal relationships between the words or symbols within the statement. Analytic reasoning looks to the internal consistency of a given set of symbols, statements, or ideas according to a particular theoretical system. Given certain rules, no outside information is needed – all of you have know is how words or symbols relate to each other. For example, take the statement: “A bachelor is an unmarried man.” You don’t have to have any knowledge of the world beyond the definition of the words involved to affirm that within the common sense of our grammar, this statement is true, and more fundamentally, because this statement is a definition, it can be used analytically regardless of whether this definition matches our typical use (though in this case, it does).

  Another example of a purely analytic statement comes from Augustine’s Soliloquies. In his attempt to ground an argument for the immortality of truth without the use of any empirical or worldly knowledge save the rules of grammar, he states (in rough paraphrase):

  If the world will last forever, then it is true that the world will last.

  This is an analytical tautology, or rather, an axiom, because it depends on no outside information to be proven true. It’s truth is self-evident. It doesn’t matter if it were known beyond the shadow of a doubt that the world were to end to tomorrow – this statement would still be true because of it’s conditional form. The relationship between the words in the proposition render it infallible. In fact, Augustine then goes on to claim:

  If the world will not last forever, then it is true that world will not last.

  From this, Augustine then deduces that whether or not the world lasts forever, truth itself will, meaning that Truth (now with a capital T) is an ontologically independent concept and not just an epistemological construct dependent on our evaluation of reality. It takes a few additional premises and extrapolations, and the validity and use-value of this argument are certainly debatable, but the main focus here is that the causal realities of the world are not relevant to the truth value of either of these statements. All that matters is the maintenance of internal consistency (and I’m deliberately leaving Tarski out of this in the name of simplicity, but feel free to take him up in the comments if you so desire).

  In contrast, synthetic reasoning requires that we have additional information from the world in order to determine the truth value of a given statement. Example: “Bill is a bachelor.” In this case, you have to know a little something about Bill himself in order to determine the truth of this statement – particularly, whether or not he is married. One thing to note is that a statement like this cannot be analytically false – while a synthetic need not be an analytical tautology, it cannot contain any logical contradictions either. If we were to say that “Bill is both married and a bachelor,” we would be saying something that could be neither analytically true, nor synthetically verified. since the terms “married” and “bachelor” (in their simplest colloquial interpretations) contradict each other.

  Similarly, questions about the long-term existence of the world require rather a grand mixture of analytic and synthetic reasoning. We would need a lot of empirical evidence – synthetically connected information – from the world, but since we would be engaging in a prediction, we would also need to analytic reasoning to help us organize this evidence and find patterns, while also keeping us from speculating too wildly.

  But this mixture needs a solid ground in one type of reasoning or the other, and it also requires a lot of awareness. Augustine’s first attempt to prove the immortality of truth in the Soliloquies makes liberal use of analytic statements and empirical observations, but because Augustine doesn’t recognize that he is mixing the two categories, he ends up in a contradiction, leading him to start over with a cleaner, more purely analytic slate. Of course, we can and do mix these types of reasoning successfully all the time, but typically only when we do so intentionally.

  As Godel proved, a system can either be axiomatically (internally) consistent, or it can be synthetically comprehensive, but it cannot be both. How we use and mix the analytic and synthetic modes of reasoning really does matter, whether we’re checking on Bill’s marital status or testing the limits of mathematical axioms – and it requires a great deal of self-awareness. Because of this, I would say that this bit of jargon is rather useful.
  Posted in Series | 10 Comments | Tagged academia, analytic truth, definitions, epistemology, kant, knowledge, lexicon, logic, philosophy, reason, synthetic truth, truth claims, vocabulary, words

• The Philosopher’s Lexicon: De Dicto/De re Distinction

  Posted at 1:00 pm by Michelle Joelle, on April 17, 2015

  Welcome back to The Philosopher’s Lexicon. My primary goal in this series is to explore common philosophical vocabulary, hopefully transforming these words from useless jargon into meaningful terms. My secondary goal is to highlight how contentious some of these terms can be – especially those which seem obvious. These definitions will not be comprehensive by any means, so please feel free to add your own understanding of each term as we go. 

  

  This week’s entry into the lexicon is the distinction between claims that are made de dicto and claims that are made de re. Literally, a “de dicto” proposition carries its meaning in the words that are said, while a “de re” proposition carries its meaning in the thing that exists behind the words. This is most easily understood in an example. For this, most explanations turn to Quine. The Stanford Encyclopedia of Philosophy renders his example thus:

  (1) Ralph believes that someone is a spy.

  This could mean either of the following.

  (2) Ralph believes that there are spies

  or

  (3) Someone is such that Ralph believes that he is a spy.

  The first meaning (2) is what results when the initial statement (1) is taken de dicto. The meaning of the proposition is found literally within the words given, in a self-contained way. The second rendering (3) is referring to some thing out in the world that is being represented by the words in the statement, meaning that what we are looking for is not just the meaning of the words de dicto, but the meaning behind the words, de re. 

  There is, of course, a lot of complexity in working out this sort of ambiguity in our language with logical notation and categorical distinctions, but what is more interesting to me is how this ambiguity plays out not just in our syntax, but in our affirmation of truths, our understanding of the world, and in our beliefs.

  This distinction between de dicto and de re beliefs has been on my mind recently because of a short comment made at a theological ethics talk I attended a few weeks ago. The topic of conversation (very loosely rendered) was whether (and of course, how) the morality of an act depended upon a person’s express belief in its morality de dicto, or upon the alignment of the particular act with an objective moral standard de re.

  This is, of course, a fairly easy dilemma to solve if we’re operating under expressly Abrahamic assumptions. If there is an objective standard of goodness against which all acts must be measured, then clearly that standard will supersede our own human understanding and linguistic representation of it. Moving into an expressly Christian framework, if a person professes to believe in the word of Christ de dicto, but acts in a way that is contrary to all Christian teachings, the acts themselves are still immoral.

  It gets a little more contentious if we shift the model around, however: if a person uses the language wrong and perhaps misunderstands the laws in their express rendering, but follows the spirit – de re – of Christ’s teachings, then she is, for many Christians, still behaving in a moral way. Of course, for many other Christians, both an express belief de dicto and a spiritual enactment of Christ’s teachings de re are important, but it’s generally clear that while de dicto belief is debatable, de re belief is not. Romans 2 gets a little tricky with the language, but drives generally at this point:

  For it is not those who hear the law who are righteous in God’s sight, but it is those who obey the law who will be declared righteous. (Indeed, when Gentiles, who do not have the law, do by nature things required by the law, they are a law for themselves, even though they do not have the law. They show that the requirements of the law are written on their hearts, their consciences also bearing witness, and their thoughts sometimes accusing them and at other times even defending them.) (Romans 2:12-15, NIV).

  

  A great exemplar of the de re form of Christianity that the ethicist who inspired this lexicon entry brought up in the informal discussion that followed her talk can be found in C.S. Lewis’s Narnia tale The Last Battle (spoiler alert, FYI). There is a scene where a man finds himself face to face with Aslan in the afterlife after having spent this life piously praising and praying to another God (Tash). In re, Tash is a cruel and oppressive deity, while Aslan is a good and forgiving figure, but in dicto, the man took his God Tash to be good, loving, and protective. In describing his exchange with Aslan, he says this:

  Then I fell at his feet and thought, Surely this is the hour of death, for the Lion (who is worthy of all honor) will know that I have served Tash all my days and not him… But the Glorious One bent down his golden head and touched my forehead with his tongue and said, Son, thou art welcome. But I said, Alas, Lord, I am no son of thine but a servant of Tash. He answered, Child, all the service thou has done to Tash, I account as service done to me… if any man swear by Tash and keep his oath for the oath’s sake, it is by me that he has truly sworn, though he know it not, and it is I who reward him. And if any man do a cruelty in my name, then, though he says the name Aslan, it is Tash whom he serves and by Tash his deed is accepted (Lewis 204,205).

  This same model remains useful if we remove it from a theological context but still maintain an objective standard de re. When it comes to mathematics, a de dicto approach would focus on notation, equations, and formulas, while a strictly de re approach would relegate mathematical language to the role of tool which merely helps us find answers. This probably seems extremely obvious, but in practice we often focus far more on the way mathematics is expressed than on the rational truths being expressed, when in truth, we really need a balance of both. In previous posts on education I’ve called this a “vocabulary-based” approach, but the de dicto/de re distinction is perhaps more precise.

  A great exemplar of this distinction can be found in an anecdote from the essay “He Fixes Radios by Thinking!” from Surely You’re Joking, Mr. Feynman!:

  While I was doing all this trigonometry, I didn’t like the symbols for sine, cosine, tangent, and so on. To me, “sin f” looked like s times i times n times f! So I invented another symbol, like a square root sign, that was a sigma with a long arm sticking out of it, and I put the f underneath. For the tangent it was a tau with the top of the tau extended, and for the cosine I made a kind of gamma, but it looked a little bit like the square root sign… I thought my symbols were just as good, if not better, than the regular symbols – it doesn’t make any difference what symbols you use – but I discovered later that it does make a difference. Once when I was explaining something to another kid in high school, without thinking I started to make these symbols, and he said, “What the hell are those?” I realized then that if I’m going to talk to anybody else, I’ll have to use the standard symbols, so I eventually gave up my own symbols (Feynman 24).

  While we can use any symbols or language we want to explain and understand principles de re, we need a common de dicto understanding for communication. Where we often run into problems, mathematically, is that we tend to treat the way that mathematics is expressed as the truth of thing – we spend our time learning nothing but mathematics de dicto and find ourselves with little to no understanding of mathematics de re. I’m sure that the same thing could be said for nearly any subject or course of study. Feynman found this to be the case even in his classes at MIT:

  I often likes to play tricks on people when I was at MIT. One time, in mechanical drawing class, some joker picked up a French curve (a piece of plastic for drawing smooth curves – a curly, funny-looking thing) and said “I wonder if the curves on this thing have some special formula?”

  I thought for a moment and said, “Sure they do. The curves are very special curves. Lemme show ya,” and I picked up my French curve and began to turn it slowly. “The French curve is made so that at the lowest point on each curve, no matter how you turn it, the tangent is horizontal.”

  All the guys in the class were holding their French curve up at different angles, holding their pencil up to it at the lowest point and laying it along, and discovering that, sure enough, the tangent is horizontal. They were all excited by this “discovery” – even though they had already gone through a certain amount of calculus and had already “learned” that the derivative (tangent) of the minimum (lowest point) of any curve is zero (horizontal).  They didn’t put two and two together. They didn’t even know what they “knew”

  I don’t know what’s the matter with people: they don’t learn by understanding; they learn by some other way – by rote, or something. Their knowledge is so fragile! (Feynman 36,37).

  What is clear to me in both the theological and the mathematical examples is that while the re is different, so long as we believe there is some objective external standard of truth to be found, this distinction is absolutely necessary, as collapsing the distinction tends to lead to an erosion of understanding. In religious frameworks we end up focusing on trivial contradictions and minute, seemingly arbitrary details at the expense of the general spirit or message a particular religion is attempting to prioritize. In mathematics, we focus so much on the process that we miss out on the end result. In taking the de dicto meaning of a proposition for the de re, we shift our focus from finding truth to merely affirming agreement.

  Of course, if there is no re that exists outside of our representation in language and symbols, then this distinction naturally falls apart. But while I’m unwilling to stake a claim on the exact nature of what is objectively true de re, I’m committed enough to its existence to find this distinction – and this particular piece of jargon – invaluable.
  Posted in Series | 16 Comments | Tagged academia, C.S. Lewis, de dicto/de re distinction, definitions, dictionary, feynman, lady philosophy, lexicon, logic, mathematics, philosophy, Romans, symbols, syntax, theology

• The Philosopher’s Lexicon: Occam’s Razor

  Posted at 12:00 pm by Michelle Joelle, on March 27, 2015

  Welcome back to The Philosopher’s Lexicon. My primary goal in this series is to explore common philosophical vocabulary, hopefully transforming these words from useless jargon into meaningful terms. My secondary goal is to highlight how contentious some of these terms can be – especially those which seem obvious. These definitions will not be comprehensive by any means, so please feel free to add your own understanding of each term as we go. 

  
  The term for this week’s entry into the philosopher’s lexicon is “Occam’s Razor,” also rendered as “Ockham’s Razor”, is the argumentative principle that says that the simplest possible answer is most often the correct one. Named for 14th century philosopher William of Ockham, this principle is taken up by mathematicians, scientists, doctors, woodworkers, teachers, and more as a way to solve dilemmas and save time.

  But ironically, as far as principles go, this one isn’t simple. There are, in my estimation, at least three ways to understand Occam’s Razor, and each of these ways come with their own set of complications.

  In the first place, Occam’s Razor hinges upon our preference for simplicity itself. Simple things are easy to grasp and understand. Complex ideas are painful to grasp and difficult to keep clear in our minds, so we – simply – like the easier ideas better. All else being equal, if you have two ways to solve a problem, and one is for more complex than the other, the more elegant version is preferable.

  In the second place, the problem with a complicated explanation isn’t just that it’s difficult to grasp; when there are more details in an explanation, there are more points where you could admit error, and more chances that parts of your explanations either conflict with each other or lead in different directions. There are more variables to control, more assumptions to validate, and more places where a moment’s thoughtlessness can lead to ruin. The simplest explanation is often more likely to be correct than a complex one because there are fewer places where it is vulnerable to change, error, and uncertainty – frankly, it is more likely to be right because there are fewer places where it can go wrong.

  In the third place, we can also view Ockham’s view of simplicity from the perspective of the Scholastic theologian (and here I hope you’ll permit me to paint with an exceedingly broad brush): simplicity itself had inherent value. That which was simple wasn’t just preferable in this world view, it was inherently superior to complexity. God, goodness, and truth were conceived to be utterly simple and thus immaterial and infallible in contrast to the unpredictable decay of material complexity. While there’s definitely some common ground here with the explanation above – that is, to say that a being made of several complex parts has in it more potential for conflict and failure than a being made up of fewer parts – the “simple” truth here is that the more simple an idea was, the closer it was to Truth itself, with a very deliberate capital T, and Goodness itself, abstracted from all of the messy complexities and failures of mortal, material life.

  Regardless of the motivation behind the search for simplicity, finding the simplest answer or simplest course of action isn’t always itself a simple task. When it comes to optimization, there are a lot of times when its actually easier and more efficient to do things the complicated, hard way. For example, running errands when all of your stops are pretty close together: you can sit down and look at maps and plot out the best way to hit each store on your list without ever doubling back, and without letting the shrimp sit in a hot car for too long, and so on, or you can just get in your car and get the errands done in the same amount of time, zig-zagging back and forth across town in a complex way. Sometimes there are constraints that make optimization obvious (say, if the post office closes first, and then there’s a long drive out to the fish market, the order of events is pretty easy), but sometimes, optimizing just isn’t optimal.

  Something else to watch out for is Occam’s Razor’s evil twin: confirmation bias. Sometimes the simplest answer or explanation is the one that fits in most easily with a person’s existing worldview. Basically, if a proposition makes sense according to what you already believe, you’ll automatically think that it is more correct than the proposition which challenges your worldview and forces you to do some complicated and difficult reconfiguration of your thought-process. In choosing the simplest answer, we may often be just choosing the answer that allows us persist in our own biases, rather than judging possibilities on their own merits.

  I think will stick with Albert Einstein’s (alleged) contribution to this principle: “Everything should be made as simple as possible, but not simpler.” Or rather, I should stick to the original quote from which this aphorism is thought to derive, taken from Einstein’s Herbert Spencer lecture delivered at Oxford, Jun. 10, 1933:

  It can scarcely be denied that the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience.

  I do apologize for my overuse of the word “simple” throughout this post. Once I started writing, I found myself unable to avoid it, and after making an attempt to reword and work out the redundancy, I decided that it would be, well, simpler, just to leave it in.

  Posted in Series | 22 Comments | Tagged academia, confirmation bias, definitions, lexicon, logic, Occam's Razor, philosophy, reason, vocabulary, William of Ockham

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