Some Logic

(I) Introduction

One of the main points that I will stress throughout the semester is this: I don't really care what your opinions are, so long as you support them. Anyone can have any old opinion you please. But it takes a bit of rigor and discipline to be able to lay out and explain the reasons why you have the opinions you do. And being able to do this well is the primary aim of philosophy. So we will begin the course with a brief overview of some of the basic principles in logic.

In philosophy--and especially in my class--we will always be dealing with arguments. An argument is a rational process whereby someone supports a thesis (called a conclusion) with reasons (called premises). For our purposes, we can define an argument as something involving at least two propositions, one of which is the conclusion, which logically follows from the premises.

(II) Deductive Arguments

A deductive argument is one that attempts to follow a certain logical form such that if the premises are true, the conclusion must be true. If the logical form is a good one, the resulting argument will be valid. A valid argument is one in which the truth of the premises guarantees the truth of its conclusion. An invalid argument is one in which the truth of the premises does not guarantee the truth of the conclusion. In invalid arguments, the conclusion does not follow with strict necessity from the premises, even though it is claimed to.

Here is an example of a valid, deductive argument:

(a)             All people are mortal
Socrates is a person
Therefore, Socrates is mortal

The forgoing argument has the following form:

(a*)          All P are M
s is a P
Therefore, s is M

Here is an example of an invalid, deductive argument:

(b)                All people are mortal
Nacho (my cat) is mortal
Therefore, Nacho is a person

The forgoing argument has the following form:

(b*)             All P are M
s is M
Therefore, s is P

Notice the difference in form between (a*) and (b*). All we did was switch the letter in (P2) from P to M, yet given (P1), this switch was enough to turn the argument from valid to invalid. Our goal in constructing deductive arguments is to make them valid. We want it to be the case that if the premises are true, the conclusion must be true . Observe, however, that in the case of validity, it is not necessary for the premises or the conclusion to be true. Validity simply means that if it were the case that the premises were true, the conclusion could not fail to be true. So, for example, the following argument is valid, even though the premises are false:

(c)            All monkeys have blue teeth
Meg is a monkey
Therefore, Meg has blue teeth

Notice that in this example, the premises and conclusion are all false. Yet the argument is still valid because IF the premises were true, the conclusion would have to be true as well. A valid argument is important because it tells us that the form is A-OK, yet if we find out that all the premises are true, then the truth of the conclusion is guaranteed. So what happens when the premises of a valid argument are in fact true? We call a valid argument with true premises a sound argument. So argument (c) above is valid, but not sound, while argument (a) above is both valid and sound. Notice that you can NEVER have an argument that is sound, but not valid. An argument must be valid before it can be sound. Soundness tells you that (i) you’ve got an argument such that IF the premises are true, then the conclusion will be true AND that (ii) the premises are true. So…

 VALIDITY     + TRUE PREMISES  = SOUNDNESS

Below, are a couple of valid forms of arguments that we will be seeing throughout the course.

 Modus Ponens If P, then Q P Therefore, Q Modus Tollens If P, then Q Not Q Therefore, Not P Disjunctive Syllogism Either P or Q Not P Therefore, Q & Introduction P Q Therefore, P & Q

(III) Inductive Arguments

Unlike deductive arguments, inductive arguments are not truth preserving. That is, even if an inductive argument has a good logical form, it will never be the case that if the premises are true, the conclusion must be true. The most that an inductive argument can hope for is that it’s highly probable that its conclusion is true. In other words, a good inductive argument is such that if the premises are true, then the conclusion is most likely true. Another way of putting the same point: the truth of the premises does not guarantee the truth of the conclusion, but only makes the conclusion very probable.

As such, we do not speak of validity/invalidity or soundness/unsoundness when it comes to inductive arguments. Instead, inductive arguments are either strong/weak or cogent/uncogent. A strong, inductive argument is such that that it is improbable that the premises are true and the conclusion is false. Conversely, a weak inductive argument is such that the conclusion does not follow probably from the premises, even though it is claimed to. To find out if an inductive argument is strong or not, we run a similar test as that of valid, deductive arguments. That is, we consider whether, given the truth of the premises, they support the conclusion in such a way that it is highly improbable that the conclusion is false. So when faced with an inductive argument, we should ask ourselves: IF the premises are true, then does the conclusion most probably follow? If yes, then the argument if strong. If not, the argument is weak.

Here is an example of an strong, inductive argument:

(d)         This cooler contains 30 cans.
25 cans selected at random were found to be Pabst Blue Ribbon (PBR).
Probably all the cans are PBR.

Here is an example of a weak, inductive argument:

(e)         This cooler contains 30 cans.
3 cans selected at random were found to be PBR.
Probably all the cans are PBR.

Notice that unlike validity and invalidity, strength and weakness of an inductive argument admits of degrees. Thus, we can make (d) weaker by changing the number of cans selected at random from 25 to 15; similarly, we can make (e) stronger by changing the number of cans selected at random from 3 to 10.

Observe, however, that in the case of strength, it is not necessary for the premises or the conclusion to be true. (Strength is akin to validity in this respect). A strong argument simply means that if it were the case that the premises were true, the conclusion probably could not fail to be true. So, for example, the following argument is strong, even though the premises are false:

(f)         Every monkey I’ve seen so far (and I’ve checked at least 800 of them!) has blue teeth
Probably the next monkey I see will have blue teeth

Notice that in this example, the premise and conclusion are both false. Yet the argument is still strong because IF the premises were true, the conclusion would most probably be true as well. A strong argument is important because it tells us that the structure is A-OK, yet if we find out that all the premises are true, then the truth of the conclusion is most likely true.

One example of a kind of inductive argument that we will be seeing throughout the course is called an Argument by Analogy. An argument by analogy occurs whenever one makes a comparison of two or more things and concludes, because of the similarity of the things compared, and because one of the things has a certain characteristic, then the other thing(s) has this characteristic too. For an example, look at (g)

(g) Watches exhibit order, function, and design. They also were all created by a creator. The universe, much like a watch, exhibits order, function, and design. So, similarly, the universe must have been created by a creator.

The idea is that to the extent that two things are similar, similar things can be legitimately inferred from both. However, do remember that this is just an inductive argument, so the truth of the premises does not guarantee the truth of the conclusion. It does, however, admit of strength depending on the probability of the conclusion following from the premises.

So what happens when a strong, inductive argument does in fact have true premises? We call a strong argument with true premises a cogent argument. So argument (f) above is strong, but not cogent, while argument (d) above is both strong and cogent. Notice that you can NEVER have an argument that is cogent, but not strong. An argument must be strong before it can be cogent. Cogency tells you that (i) you’ve got an argument such that IF the premises are true, the conclusion will most likely be true AND that (ii) the premises are true. So…

 STRENGTH    + TRUE PREMISES   = COGENCY

(IV) A Priori vs. A Posteriori

At this point, one might wonder how one goes about showing that the premises of arguments are true or not. After all, the soundness of a valid, deductive argument (or the cogency of strong, inductive one) is dependent on whether or not the premises are in fact true. So how do we show that they are true? It is here that we should make the distinction between premises—or propositions—that are known a priori (independent of experience) and those that are known a posteriori (based on experience). A proposition is known a priori if it can be known by a subject who has no (or in some cases, only a little) experience with the world. That is, armed only with logic and language, a person can come to know certain propositions independent of experience. In contrast, propositions are known a posteriori just in case experience with the world is required to see if the proposition is true.

Here is an example of an a priori proposition:

(g) If Abe is taller than Bob, and Bob is taller than Cal, then Abe is taller than Cal.

Here is an example of an a posteriori proposition:

(h) If Abe is taller than Bob, then Abe will be better than Bob at basketball.

The distinction between premises known a priori and premises known a posteriori is a distinction that we will be using throughout the semester.

(V) Possible Worlds

Think of all the ways in which the world could have been different from the way that it actually is: There could have one more tree in the quad than there actually is; there could have been a Taco Bell on Franklin Street; there could have been purple toads; you could have been a millionaire; Arnold Schwarzenegger could have been president; George W. Bush might not have ever been born; Dinosaurs might have ruled the world, never to be extinct, prohibiting humans from flourishing.

Now imagine that all of these different ways that the world could have been is a way that some world--some possible world--is. So, if it could have been that you are in the NBA, then there is a possible world where you are in the NBA. Now, true, in such a world, several things may have to be different--you may have to be several inches taller than you actually are now, or the rest of the population may have to be shorter than it actually is, or it may have to be that you have 'flubber' feet or an uncanny ability to always make a basket no matter how hard you try to miss, etc. But if it is true that you could have been in the NBA (no matter what else might have needed to be different about the world in order for this to be true), then let us say that there is a possible world where this IS the case.

If there is a possible world for every way the world could have been, then there will be (at least!) an infinite number of worlds (For there is a possible world where just 1 bunny hopped on the quad; another where 2 bunnies hopped on the quad; another where 3 bunnies hopped on the quad, etc.). But using these worlds will help us when we are trying to figure out what is possible and what is not.

For example, on of the second day of class I talked about deductive arguments and validity. I explained that a valid argument is one such that if the premises are true then the conclusion must be true. Is the following argument valid?

Premise 1: All monkeys are mischievous
Premise 2: George W. Bush is a monkey
Conclusion: George W. Bush is mischievous

One way to determine validity is to use possible worlds: Imagine all of the possible worlds where the 2 premises are true--i.e., imagine all of the worlds in which it is true that all monkeys are mischievous and
that George W. Bush is a monkey. Among such worlds, is it ever false that George W. Bush is mischievous? No. In this way, we can use possible worlds as an imaginative tool to aid us in our intuitions about possibilia.

As we progress through the semester, you will see that we will use possible worlds often: to test for validity, to track our intuitions about material objects and identity, to create counterexamples to certain claims and arguments, etc.

For way too much information on Modal Realism, go here.

Sources

Feinberg, Joel and Shafer-Landau, Russ, Reason and Responsibility, Eleventh Edition. (Ed. Feinberg and Shafer-Landau) Wadsworth, 2002.

Hurley, Patrick J., A Concise Introduction to Logic, Seventh Edition. Wadsworth, 2000.

Pojman, Louis P., Philosophy: The Quest for Truth, Fourth Edition. Wadworth, 1999.

Sennet, Adam. Philosophy 107, Syracuse University, Fall 2001 course notes.

Page Last Updated: May 13, 2009
To Meg's Teaching Page
To Meg's Main Page