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Logic: Is it a science?

Necessarily, you join our facebook group.

Seth Kurtenbach here.  The above link is a claim that can be formalized with modal logic.  Modal logic is a branch of logic that deals with necessity and possibility qualifiers on truth.  Depending on how one specifies the type of necessity and possibility, one can formalize arguments from many domains that cannot otherwise be precisely represented with regular old classical logic.  Classical logic is strictly extensional, because it was developed to explore the foundations of mathematics.  It is complicated why this is so, but trust me, it is.  An intensional logic is one that does not guarantee the substitution of identities.  For example, suppose the number of planets = 8.  Furthermore, suppose, reasonably, that 8 = 8.  In regular extensional logic, one can substitute these equivalencies, due to transitivity.  However, suppose one wants to say that necessarily, 8 = 8.  Now, one cannot validly make the substitution.  It is not necessary that the number of planets = 8.  It is possible that the number of planets = 9.  This is an example of intensionality disrupting the logic.  In the last half of the 20th century much work has gone into exploring the various types of modal logic and their rules of inference.

During the recent XKCD hubbub about all Wikipedia roads leading to philosophy, I encountered a Wikipedia page about the formal sciences.  The page indicates that these sciences are different from most other sciences insofar as the formal sciences are a priori, while the other sciences are a posteriori.  That is, the formal sciences are not empirical, but instead amass knowledge based on definitions and axioms.

Among the so-called formal sciences is logic and its various subfields.  I do work in modal logic, and so it would seem that if Wikipedia is correct, then I am a sort of scientist.  But, I have never thought of my work as being a type of science.  I usually consider logic to be a subfield of analytic philosophy, and I consider analytic philosophy to be distinct from science.  In that same vein, I am not sure any of the so-called formal sciences are actually sciences, because my impression is that all science is primarily empirical.

I would like to know what others think about these so-called formal sciences, specifically about logic.  Am I a formal scientist, or is that a misnomer?


About Seth Kurtenbach

Philosophy grad student who wandered into a computer science PhD program with a backpack full of modal logic and decision theory.

9 comments on “Logic: Is it a science?

  1. Dave Muscato
    May 29, 2011

    What I want to know is, how do we know logic always applies? How do we show that logic is always, reliably right? Where does logic come from? How can we be confident in trusting it? This would be helpful in rebutting arguments from miracles, among other things.

  2. Seth Kurtenbach
    May 29, 2011

    1. Logic applies to arguments. All arguments may be represented, to varying degrees of precision, by some logical system or another. Anything with a truth value can be represented logically. The color red does not have a truth value, so logic does not apply to it. One cannot represent redness logically. However, if one wishes to describe the sentence “the ball is red,” then one can use logic to represent the sentence. Redness as a predicate can be described logically, in the form, “x is red”, or Rx. If x is a thing that is red, then Rx is true. If not, then Rx is false. But redness itself is neither true nor false, so logic does not apply. Different domains of argument may have different systems of logic. One knows which system to use because each system is explicitly created to govern some domain or other. For example, if one is in an argument about what is necessarily or possibly true, then one appeals to modal logic of some sort to judge the validity of the argument at hand; that’s the domain modal logic was created for.

    2. We show a particular system of logic is always reliably right, relative to its domain, by showing that it is both sound and complete. It is sound if it calls only the valid arguments “valid,” and complete if it calls all valid arguments “valid.” Soundness guarantees reliability, and completeness guarantees the power to always give a right answer in its particular domain.

    3. Logic formalizes the truth behavior of sentences. The human capacity for language grounds our ability to learn the referents of words, and our ability to judge the truth values of sentences. Logic takes this ability and rigorously formalizes it, so that one can judge the truth behavior of a set of sentences based on form alone. The goal of any logic is to devise a system of formal rules that recognize all and only valid forms of argument within that logic’s domain. If the form of a set of sentences guarantees the truth of another sentence, then the inference from the form of that of sentences, stripped of its particular content, to the latter sentence, is said to be valid. We call the set of sentences ‘premisses’, and the final sentence the ‘conclusion.’ Logic aims to recognize all valid moves/inferences from premisses to conclusion as valid, and all invalid moves/inferences as invalid. This all comes from our capacity for language.

    4. We can be confident in trusting it because it is based on our capacity for language, and we trust our capacity for language. If one can learn the meanings of words and judge the truth values of sentences composed of those words, then one can trust logical systems governing arguments using those sentences.

    5. How is this related to miracles??

  3. Maggiesaurus May
    May 29, 2011

    I think the most accurate and descriptive title you could have would be: philosoraptor. I’m not giving in on this one.

  4. Tony Lakey
    May 30, 2011

    You cannot “Like” comments on here as you can on Facebook, but I still felt the need to show my support for Maggie’s comment, as I entirely agree.

  5. Joe
    May 31, 2011

    How can one use the scientific method in modal logic?

    • Seth Kurtenbach
      May 31, 2011

      That’s a good question. I think the same question applies to all of the so-called formal sciences, due to their primarily a priori nature. Much will turn on how one specifies what exactly the scientific method is. It is actually quite difficult to give a definition of the scientific method that includes all of the activities we want to recognize as scientific and only the activities we want to recognize as scientific. For example, many want to include something like “perform an experiment” as an essential part of the scientific method, but for the most part astronomers, cosmologists, and theoretical physicists (string theorists) are not performing experiments. They are usually making uncontrolled observations, in a more passive manner than one would like, or in the case of some cosmologists and string theorists, speculating in a purely a priori manner. When was the last time Stephen Hawking conducted an experiment? The discipline in which he engages, cosmology, is done mostly through the a priori manipulation of mathematics. Many cosmologists argue from thought experiments, based on mathematics, but this is not the same as an empirical experiment. Modal logicians engage in thought experiments as well, with the same degree of formal rigor. So, I suspect that if the definition of the scientific method is amended in order to include those disciplines, which surely I think they should be considered sciences (although I have heard some physicists reject the idea that string theory is a science), then there may be room for other a priori disciplines, like modal logic. However, this is just speculation. Before we can begin to answer your question, we must first arrive at an acceptable account of the scientific method.

  6. Joe
    May 31, 2011

    Excellent points, but do not cosmologists, physicists, etc. use their formulas and hypotheses to make predictions about the empirical world, which they measure against reality in a way that could be called an ‘experiment’?

    • Seth Kurtenbach
      May 31, 2011

      Good point. Modal logicians do not make empirical predictions, as far as I can tell, and my brief perusal of the wikipedia page about the scientific method indicates that prediction is an essential part of the scientific method. I think this constraint might cause some problems for theoretical physicists and string theorists, because many of the entities about which they theorize are considered to be unobservable (quantum particles, parallel multiverses, etc.).

      But, if by ’empirical world’ you don’t mean ‘observable world,’ but something more like ‘natural world,’ then the division might cut right between string theorists and physicists on the one hand, and modal logicians on the other. Although, interestingly, some modal logicians (David Kellogg Lewis) independently converged on multiverse-ish theories that align nicely with certain interpretations of quantum physics and cosmology. Insofar as the quantum physicists/cosmologists are making an empirical prediction when they say that there are infinite multiverses, so too are the modal logicians of Lewis’s school, I would think.

  7. Jerry Winn
    June 29, 2011

    It’s not generally considered essential to meet EVERY defining criterion of “science” for a field to be considered a science. Rather, we tend to informally categorize it in degrees. For example, psychology is a science, but doesn’t strictly meet the requirement for repeatable empiricism. We can not genuinely repeat a psychological experiment, which is why the field relies so heavily on statistics. In fact, it was the relatively recent advent (circa 100 years ago) of statistical methods of analyses that transitioned psychology from a humanity to a science. That said, psychology still relies on qualitative methods, which are not scientific– rather, they exploratory and interpretive, though meaningful nonetheless.

    My point, which I hope has become salient, is that it is not the discipline, but the methodology, that determines what is science. To the extent that philosophy deals with representations of mathematical theorems, either of which being an effort to express the phenomenon of our experiences and our universe through a form of language, it might be considered a science. Most would probably argue that if it lacks empiricism, then it isn’t science, I think that to the extent that math and logic are empirical (that is, they necessitate empirical support even if the support itself needs not be further demonstrated; e.g., 2+2 is 4; the Pythagorean theorem works), they are scientific.

    I generally don’t concern myself with semantics; I think they’re a lower form of rumination and that we should spend little time deciding what words should mean and simply agree on what they do mean. However, I think there are valid points to be made about the extension of logic to empiricism due to the inextricable relationship between the two.

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This entry was posted on May 28, 2011 by in Author: Seth Kurtenbach and tagged , , , , , .
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