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    4. Logic

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    • 06/04/2004 http://www.galilean-library.org/site/uploads/

    By /index.php?/user/4-hugo-holbling/">Paul Newall (2004)

    Logic comes up often in philosophical discourse and even more frequently in informal discussions, typically involving a claim that "logic says x". Logic, however, is the study of reasoning and so (as we’ll see) doesn’t really say anything in and of itself. In another sense, a logic (as opposed to logic in general) is a set of rules governing such reasoning. (This is why we have words like biology, for instance, and others with similar endings.) Both understandings have an historical importance to philosophy that continues today.

    Making an Argument

    Suppose we take an example of an argument:

    Of course Hugo can fly - after all, he has wings.

    Although this may read as an offhand remark, we can cut it into pieces to expose the underlying argument. Set out in stages, it would be:

    1. Hugo has wings.

    2. Therefore, Hugo can fly.

    There is something missing, though: another line making explicit how we get from 1 to 2. Thus:

    1. Hugo has wings.

    2. Critters with wings can fly.

    3. Therefore, Hugo can fly.

    This is the form of the argument. Notice that the conclusion in 3 is derived, as it were, from the information in 1 and 2. We call these the premises, being the statements from which we obtain the conclusion. We can therefore rewrite our argument in a standard form (one which we’ll use later in our series):

    P1: Hugo has wings;

    P2: Critters with wings can fly;

    C: Therefore, Hugo can fly.

    The important thing to notice about this argument is the obvious one: it is ridiculous. We know that not every critter with wings can fly and we would presumably be doubtful that Hugo has a working pair at all, unless he is on his way to a fancy dress party or some other occasion for flapping. It fails to convince us, but nevertheless there is something about its structure that seems harder to dismiss.

    To take account of this, we say that an argument is valid if the conclusion follows from the premises. An airborne Hugo does follow from P1 and P2 (regardless of whether we accept them or not), so our example is valid. To draw attention to the other aspect, however, we say that an argument is sound if it is valid and its premises are true. If not, we say it is unsound. In a subsequent entry in our series we’ll consider truth, but for now we can just say that P1 is false (Hugo does not have (functioning) wings) and so is P2 (not all winged critters can fly), so our argument is valid but unsound.

    Bad arguments

    Later in our series we’ll look in depth at those arguments known as fallacies. These are reasoning gone wrong somewhere (although not always, as we’ll see), where the premises and conclusion(s) may seem plausible but where one or more mistakes have crept in (deliberately or otherwise). These errors are common enough that we can study them systematically.

    Generalising

    If we look again at the argument above, we can note that it would stay almost the same if we spoke of someone else. Likewise, it would work for any pair like "wings = flight". We can generalise it completely by removing most of the content as follows:

    P1: A has property B;

    P2: Anything with property B has property C;

    C: Therefore, A has property C.

    Here "A" stands for "Hugo", "B" is "wings" and "C" is "flight". Going a step further, we could have:

    P1: B;

    P2: B implies C;

    C: Therefore C.

    This time "B" is shorthand for the subject of the argument (Hugo) having the property B. This increasing level of generalisation is perhaps something that scares people off logic, but instead we just need to teach ourselves to read in a different way (as discussed previously).

    Some other terminology to learn for use in philosophy occurs in our argument, which is alluded to above. Take P1:

    • Hugo has wings.

    Here "Hugo" is the subject of this statement; that is, it tells us something about Hugo. The other information provides this detail; that is, the remark "has wings" (or "is a winged creature"). This property of Hugo (being a winged creature) is called the predicate. We term the combination of the two a proposition. We can take P2 as another example:

    • Critters with wings can fly.

    This is a proposition with "critters with wings" as its subject and "flight" as its predicate. Similar arguments can be split up in the same way.

    Quantifiers

    In modern times, the study of logic changed significantly with the invention by Gottlob Frege (whom we’ll cover later) of the logical quantifiers. There are two: the existential quantifier, ∃, meaning "there exists"; and the universal quantifier, ∀, meaning "for all". These are also employed extensively by mathematicians and, as before, are merely new terminology to simplify and, more importantly, clarify propositions.

    By way of an example, take a proposition like the following:

    • All men are mortal.

    If we let M stand for the property of being mortal, we can write this as

    • ∀(x) Mx

    Translated into so-called "everyday speech", this says that for all x (the "∀x" part) we can say x is mortal (the "Mx" part). Here x is used in the same way as in algebra, to stand in for something (in this case, men). If we wanted to instead say that there exists at least one man who is mortal, we would write

    • (∃x) Mx

    The combination of these two quantifiers can clear up uncertainty in how we are supposed to read a proposition. Suppose we now take the following example:

    • Every wedded woman is married to a man.

    We understand this as saying that being married for a woman means having a man as her husband (or a piece of paper noting as much, some suggest), but are we thinking of a particular man for a particular woman, as we would expect, or one man for all the women? After all, the latter sense would work for our proposition: one specific man (luckily or unluckily, depending perhaps on his perspective) could be married to all the women we call wedded. In this event, the proposition would still work. Which is it?

    If we write the two possibilities using the quantifiers, the difference becomes plain. Let x be a woman, y a man, and M now the property of being married. In the first instance, we would have

    • (∀x)(∃y) Mxy

    This says that for each individual woman there exists a man such that x is married to y. On the other hand:

    • (∃y)(∀x) Mxy

    The order is different here, so now it says that there exists a man for all women such that x is married to y. By altering the ordering in our use of quantifiers we have changed the meaning, so we can employ either to explain which possibility we intend to speak about.

    Depending on how deeply we wanted to delve, we could pick out more shades of meaning and hence use additional letters and quantifiers to make the proposition unambiguous. Nevertheless, the principle remains the same: we use these logical symbols to clarify, not to confuse and, like any other language, they take time to understand.

    Deduction and Induction

    Another point to make about our argument is that it is deductive; that is, from the information we are given (the premises) we deduce the conclusion - after the fashion of Sherlock Holmes, almost. The alternative to a deductive argument is an inductive one, leading to the famous problem of induction (which we’ll cover soon). An inductive argument is often called ampliative because it goes beyond (or amplifies) the information contained in the premises. Consider the following example:

    P1: Hugo has wings;

    P2: Every critter with wings we have seen can fly;

    C: Therefore, Hugo can fly.

    This replaces our old P2 with a new version, according to which every instance thus far of a creature with wings was also a creature that can fly. Notice, however, that the conclusion this time does not follow from the premises unless we also assume something else:

    P3: We have seen all the critters there are.

    This additional premise rescues the situation because P2 and P3 together are the equivalent of our old P2. Nevertheless, the argument in the form we had does not say that we have examined all the creatures there are and thus does not justify the conclusion as it stands. We have to assume something extra (P3) and make an inductive step from the particular facts we have about critters to a general statement (the conclusion). We’ll return to the problems for reasoning that induction appears to lead to later in our series.

    Unpacking Arguments

    It’s rare to find an argument as explicit as the one we’ve been looking at. Typically we have to unpack the informal form we find it in, drawing out the premises and conclusion(s). In this section we’ll take an example and see how the process unfolds.

    Suppose someone said the following:

    It’s my business what I do in my own home, not the government’s.

    We can try to take this claim apart to see how we might argue for or against it. To begin with, the implication seems to be that the government has no right to create legislation which impacts on our private lives. Thus:

    P1: The public and private spheres of life are separate;

    P2: What goes on in our homes is private;

    C: Therefore, the government has no right to create legislation affecting what goes on in our homes in private.

    This gets at what was said but not in enough detail. The argument as it stands is invalid, because the conclusion does not follow from the premises; after all, it goes from a statement about the separation of public and private (which we could assume to be accurate for now) to an assertion regarding the lack of a right. We could add another premise to account for this:

    P1: The public and private spheres of life are separate;

    P2: The government may only legislate on public areas of life;

    P3: What goes on in our homes is private;

    C: Therefore, the government has no right to create legislation affecting what goes on in our homes in private.

    The argument now reads more plausibly but there remain some difficulties. We still go from a statement of what the government can or cannot do to the assertion that it lacks a right, so it seems our initial formulation was flawed here. We could alter the argument without it losing any force by changing the conclusion to "the government may not...", which leaves the sense unaltered but avoids this problematic issues of rights (something we consider later in our series). The resulting argument appears to be valid: P1 ensures that there is no overlap between public and private, which might have allowed the government some leeway; P2 tells us what the government may do; and P3 states that the home lies within the private sphere. It follows, then, that the government may not act upon the private sphere.

    Now that we have a valid argument, we need to ask if it is sound. As we learned above, that means asking if the premises are true or not. Here we run aground very quickly. To start with P1, that there is no overlap between public and private is not clear at all and it is easy to think of situations that would fall into both (for example, if we were to assault someone in the privacy of our own home it would still be assault, for which public legislation exists; or if we smoke at home, should we be permitted to harm others with the smoke?). For P2, where and when the government may legislate is a function of the type of government we have. In particular, the prohibition from law-making in a specific area would have come from a government in the first place, so it seems reasonable to suppose that this could be changed or lifted. Lastly, we might agree that some or most of what occurs in our homes in private without conceding that every possible circumstance would fall under that rubric.

    Our argument is thus valid but unsound. We would need to expand on it considerably to provide sub-arguments for our premises, which is beyond the scope of this introduction. Even so, we can see that this more formal approach has the effect of breaking a claim into smaller pieces, rendering it easier to investigate and evaluate. As before, the purpose of so doing is to clarify what has been said, not to confuse or complicate.

    Laws of Logic

    We rarely have to travel far in philosophical territory to hear talk of the "laws of logic", often accompanied by a suggestion that rejecting them is tantamount to insanity. They are not really so scary, however, nor inescapable. Traditionally there were three:

    • The Law of Identity

    • The Law of the Excluded Middle

    • The Law of Non-contradiction

    The first says "A if and only if A"; or "Hugo has wings" if and only if "Hugo has wings". The second gives us "either A or not-A" (i.e. the negation of A), which means that either A holds or it does not, these being the only options. A proposition like "Hugo has wings" would therefore have to be either true or false, rather than an alternative (such as "undecided"). The last states "not A and not-A", which is just to say we cannot have A and not-A at the same time. In our example, that would mean "Hugo has wings" and "Hugo does not have wings" not both holding simultaneously.

    These so-called laws have been challenged in relatively recent times. Intuitionists rejected the Law of the Excluded Middle (the most famous being Luitzen Brouwer, a Dutch mathematician who rewrote a host of mathematical proofs to remove reliance on it). A simpler example is the three-valued logic of von Neumann which allows for another possibility in addition to "true" or "false" when discussing a proposition: undecided (or undecidable). This is helpful when considering a proposition like "there is no A in the universe", for some A. We could easily show this to be false if we find a single instance of A somewhere, but to prove it true would require checking the entire universe at the same time (after all, we could look in one place and move on, only for an A to appear). It would seem to make sense to call a proposition like this undecidable.

    The law of non-contradiction is discarded by dialethic logic (from di and alethic, meaning "two truths"), or dialetheism. The idea that we should challenge the convention that a proposition is either true or false (but not both) has a long history in philosophy. A more recent motivation came from sentences like "this statement is false". We’ll come back to this later in our series but we can examine it for a moment. If it is true that "this sentence is false", then it must be false; and, likewise, if it is false that "this sentence is false" then it must instead be true. This is the so-called liar paradox. Although we’ll return to some alternative analyses of it, one way to avoid the paradox is to accept that some propositions can be both true and false, violating the law of non-contradiction.

    Lest it be thought that this example may indeed be a difficulty for philosophers but dialetheism does not really impact on the rest of us, consider a point on a doorframe. Is the point inside or outside the room adjacent? Since the door is the boundary between inside and out, we could meaningfully insist that the point is both. A proposition like "the point is outside the room" could then be both true and false, and this would apply similarly to a person stood in the doorway. Situations like this one involving boundaries are studied by dialetheists today.

    There are many other possibilities and logics, with some people arguing that all the laws above should be rejected or else others added. As a result of these and other factors, there is no longer much insistence within philosophy on the laws of logic.

    Logic and Philosophy

    The value of logic in philosophy (or philosophical logic, to be more accurate) is thus quite plain: we can use it to take arguments apart and study their structure in more detail than a cursory glance would otherwise allow. We can also take a more formal approach to reasoning, which will be of great benefit to us as we proceed.


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