2207-103 Philosophy & Logic, Week Two

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Arguments consist of propositions. Propositions typically defined as whatever that can be true or false -- "the bearers of truth values," as the conventional wisdom goes. However, the issue of what propositions really are and whether there are any of them at all is a big controversy in philosophy of logic, and we don't want to lose ourselves there. However, we need to know a bit about propositions in order to understand fully what is going on in the logic that we are now studying. First of all, in order to understand what propositions are typically taken to be, we need to see that they are distinct from sentences.

A good way to illustrate this is to compare two different sentences from two different languages. Suppose I have the sentence, 'It is raining.' It's of course an English sentence. Suppose further that I have another sentence, 'Es regnet.' For those of you who know German, this sentence just means it is raining. So here we clearly have two sentences: One is in English; the other is in German. Nonetheless, we only have here only one proposition!

Here is another example. 'Soraj loves cats' is an English sentence, and 'Soraj rak maew' is also a sentence, but a Thai one. (I should have used the Thai alphabet, but your browsers may not be able to support Thai fonts, and even you have Thai fonts you may need to change the font settings, which are quite troublesome.) But these two sentences express the same proposition because they both mean the same thing -- that Soraj loves cats. Thus there is only one proposition here.

I think you can now see what propositions are like. They are clearly abstract, in that they are not expressible directly in any languages. A proposition cannot be in Thai, or English, or Japanese, or in any language exclusively. However, in order to express it a natural language is obviously required. We can look at it this way. Propositions are the meanings of a language. If a sentence in a language is meaningful, then it expresses some proposition. The convention in logic is such that anything that has meaning also has truth values. That is, it is either true or false. And since propositions are just meanings as we have said, they are either true or false. (Right now we are not in a position to discuss the very important questions whether other uses of language, such as commanding, requesting, asking questions, and so on, are meaningful or not. They must be meaningful, but according to the logical tradition we are discussing here, questions, commands are the like are not propositions, and thus are not meaningful. We will quietly leave this issue aside, since our focus now is on basic logic and with the relations between propositions only.)

However, in the traditional logic we are discussing, any type of proposition will not do. Examples such as 'Rabbit!' which we say when we see a rabbit running by clearly express a proposition, because we can be right or wrong about the object of our exclamation. However, this type of one word proposition will not be subject matter for us here. We need to parse it into a proposition looking like 'The thing which is running by is a rabbit.' or something like that. That is, we will limit ourselves only to categorical propositions, which are propositions which state the relation between two things. In this case, between the thing which is running by and a rabbit. Categorical propositions make it much easier and clearer to see that propositions must be either true or false, for we need to see only whether the relation between the two things stated by the proposition obtains or not. If so, then the proposition is true, but if not, then it is false.

So we have write down a formula for a categorical proposition thus:

Categorical Proposition = TERM1 + TERM2

Thus the proposition, 'The thing which is running by is a rabbit.' can be parsed as follows: TERM1 = 'the thing which is running by', TERM2 = 'a rabbit', and the plus sign in the formula represents the verb that shows the relation, and in all cases here it's verb to be. Thus we can see also that a categorical proposition is the result of joining two terms together with verb to be. Here 'terms' mean the part of language which expresses a class or a category, or an individual. Thus, 'Soraj' is a term, so are 'cats', 'dogs', 'mammals' and 'BBA students who love wearing earrings.'

Four Types of Categorical Propositions

Since categorical propositions are two terms joined together by a verb to be, traditional logic has it that they are four types of them depending on how they are applied the quantifying terms 'all' and 'some' together with the negation term 'not.' Here are the four types:

  1. All A's are B's -- (Type A)
  2. No A's are B's -- (Type E)
  3. Some A's are B's -- (Type I)
  4. Some A's are not B's -- (Type O)

According to traditional logic, the first type is called proposition of type A; the second is called type E; the third item is called type I, and the last one is type O. You can see that these are the possible permutations of the categorical proposition showing the relation between Term A and Term B together with the quantifying terms 'all' and 'some' and the negation term 'not'. In logic these four types of propositions do not mean as exactly as we might expect them to mean in natural languages.

What I mean is that, since logic aims at presenting the exact meaning of a proposition without any change for ambiguities, these four types can mean rather differently than expected by some. This is because the exact meanings of these types do not match people's actual use of them. To say that all A's are B's actually means that, for all things in the universe, if anything is an A, then it is a B. This form of proposition does not state that there is in fact an A or not. It only says what it says and no more. However, we tend to think when we encounter a proposition of this type that there should be some A's here. For example, if I say all 4th year international BBA students are lovely, then you might think that there are some 4th year BBA students. But you know there aren't any, because you are only in your sophomore year, and you are the first class of the program. However, what I said means exactly only that if anyone is a 4th year international BBA student, then he or she is lovely. I don't commit myself to asserting the existence of the class of people I mentioned.

These four types of propositions are related in many ways. However, the most important relation is the so-called contradictory relation. A pair of propositions is said to be contradictory just in case if one is true, then the other must be false, and vice versa. Thus the proposition "Soraj loves cats" is contradictory to the proposition "Soraj does not love cat." We can see that, of the four types of propositions here, type A is contradictory to type O, and type E is contradictory to type I.

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Categorical Syllogism

Classical syllogism is the type of argument which has been continuously studied for more than two thousand years now since the time of Aristotle. It is the central piece of Aristotelian logic, and is still the most visible type of argument in logic courses and textbooks today. In fact not many arguments in the real world are of this type, but since the structure of classical syllogism is so straightforward and involves talking and distinguishing universal and particular statements, it has gained a reputation almost as the representative of the type of argument to be found in logic textbooks.

Classical syllogisms, no matter what they are about, must conform to the following scheme:

  1. There are three categorical propositions, and only three.
  2. Two of those propositions are premises; the other is the conclusion.
  3. There are only three terms, each appearing only twice.

The above requirements make it the case that classical syllogisms are rather cumbersome and unnatural. However, you can see that its structure is very transparent, and one can see almost immediately the errors found in the form or in the thought structure itself.

Look at this traditional example:

     (1) All men are mortal.
     (2) Socrates is a man.
     (3) Therefore, Socrates is mortal.

This is a clear example of a classical syllogism. There are three terms -- 'man,' 'mortal beings,' and 'Socrates.' Each term appears only twice. You can then see how the terms are interlocked. What a classical syllogism does is that it shows the relation between these three terms, such that the relation between one pair and another (which necessarily a common term) necessitates the relation between the one remaining pair, if the argument is valid. At least the conclusion of a syllogism shows the relation between the two terms which have been mentioned only once. Premises (1) and (2) in the above example show that there are two relations obtaining between three terms. Premise (1) shows the relation between terms 'men' and 'mortal beings.' (This proposition could be rewritten as "All men are mortal beings.") Premise (2) shows the relation between 'Socrates' and 'man.' These two pairs share a common term together in 'man.' This sharing necessitates the relation between 'Socrates' and 'mortal beings.' If all men are mortal beings and if Socrates is a man, then it is necessary that Socrates is a mortal being. This third relation is necessitated by the first two pairs of relation, since the argument is valid. But if there is an invalid argument, the conclusion would consist of two remaining terms, but the content of the conclusion is not necessitated by the two premises.

Here is another example:

     (1) Some BBA students wear earrings.
     (2) Some persons who wear earrings are not gays.
     (3) Therefore, some BBA students are not gays.

This is a classical syllogism also. There are three propositions which could be rephrased as categorical propositions, and three terms each appearing only twice. However, this argument, as you can see, is not valid. The reason is that even though some BBA students wear earrings and some persons who wear earrings are not gays (they may be women!). This does not necessarily show that some BBA students are not gays. In fact the premises could be all true while the conclusion is false. That is the proposition "All BBA students are gays," which is a negation of (3), can be true even though some BBA students wear earrings and some persons who wear earrings are not gays.

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Testing Validity Using Venn's Diagrams

Since classical syllogisms really are about the relations obtaining among these three terms, there are some mechanical means by which we can test whether a syllogism is valid or not. One of the most popular mechanism is Venn's Diagram, and this is the method we will use in the course. When we know how to test validity or invalidity of syllogisms this way, there is actually no need for learning or memorizing the validity rules that you find in the book. Such rules, for example, illicit major or illicit minor, is highly unintuitive and you won't learn anything much about how an argument is valid or invalid that way.

Venn's diagram testing of syllogism consists of three circles overlapping one another. The circle represent the three terms and their relations among one another. We can start first by paying attention to how the relations between any two terms are represented. The following diagrams show the relations of the two terms of the four categorical propositions, so there are here only two circles for each type:

Diagrams for the Four Categorical Propositions

1. Type A Propositions (All A's are B's)

Type A Diagram

2. Type E Propositions (No A's are B's)

Type E Diagram

3. Type I Propositions (Some A's are B's)

Type I Diagram

4. Type O Propositions (Some A's are not B's)

Type O Diagram

You can see that in the diagram for Type A, the shaded area is in the part of circle A which do not overlap with circle B. What this means is that there is no A which is not a B. The shading in Venn's Diagram always mean there is nothing in there. So when you find shaded areas you can just forget that part, since nothing is allowed in there whatsoever. Furthermore, what is meant by "All A's are B's" is not necessarily that there must be some A's. In fact it is entirely possible that propositions of this type can be true even in a situation where there is not an A at all. This is a bit difficult to grasp, but if you imagine that the proposition "All unicorns have a horn" is true, then you understand that this proposition can be true even in our world where there are no unicorns. For what this proposition states is merely that, if there are such things as a unicorn at all, then they all have a horn. The proposition does not state that there is a unicorn. Thus, type A proposition can be restated as "If there is any thing which is an A, then that thing is a B." The unshaded area does not imply in any way that there must be anything in it. (Back to the diagrams.)

Similarly, the diagram for Type E has the shaded area at the part where circles A and B overlap. This means of course that there is nothing that is both A and B. Or in other words, if there is any thing which is an A, then that thing is not a B. You can see that the reverse of this proposition is also true logically. That is, if no A's are B's, then no B's are A's either. (Back to the diagrams.)

The diagram for Type I looks differently. Instead of a shading you see an X inside the parts where the two circles overlap. This has a special meaning which is specific to propositions of type I and O. These two types state, not that all things or no things are this and that, but that some things are or are not. That is to say, these two types state the existence of the things in question. So "Some A's are B's" means that there is at least one thing which is an A, and which also is a B. The reason why there has to be at least one thing is because the number cannot be less than one; otherwise you won't have a case where there is something! So in order to cover all cases where something A is a B, then we have to admit that there must be at least one thing; in this case at least one thing A which is a B. And the 'X' in the diagram just represents the one thing (at least) that is both A and B here. (Back to the diagrams.)

Similarly, the proposition of Type O states that there is at least one thing, A, which is not a B. The 'X' in circle A represents the one thing A (at least) that is not a B. (Back to the diagrams.)

Now we are in the position to use these diagrams to check the validity or invalidity of syllogisms. Since syllogisms involve three terms, there are not only two circles but three. These three circles then overlap one another in such a way that the relations between any pair of the circles, or of all three together, can be shown. Here is the diagram for the classic example of Socrates' being mortal:

Diagram for this argument

Here 'A' represents Socrates; 'B' represents men and 'C' stands for mortal beings. You can see that the relation between A and B is that all A's are B's, and the relation between B and C is that all B's are C's. (Socrates is a man, and all men are mortal.) Now consider the conclusion, that Socrates is mortal. This conclusion, as with the two premises, is a Type A. And the idea of a valid argument is that if the two premises are true, then there is no possibility that the conclusion is false. You can see that if we take the premises to be true, then according to the diagram there is just no way that the conclusion, that Socrates is mortal, can be false. This is so because if the conclusion were to be false, then there must be some way that there could be an A which is not a C. But the diagram just does not show that possibility because the only unshaded area in A lies completely within C. This shows that the argument here is a valid one.

Here is another example. Suppose the argument is as follows:

     (1) All women are mortal.
     (2) Hera is not mortal.
     (3) Therefore Hera is not a woman.

Suppose 'A' stands for women, and 'B' for mortal beings, and 'C' for Hera. Then a diagram can be drawn up below:

Diagram for this argument

This argument is also valid, as you can see. If it is true that all women are mortal, and that Hera is not mortal (in fact she is not, because she is a goddess.), then she must not be a woman. Premise (1) is a Type A, but (2) is a Type E. Then you look at the diagram to see whether there is any possibility that the conclusion, that Hera is not a woman, is false. That "Hera is not a woman" is false means that Hera is a woman (because Hera is an individual, and we just cannot divide her up to see how her parts are doing). But the diagram just does not allow for this possibility because the area where A and C overlap is all shaded out. There is no way that there can be anything which is both an A and a C.

Now look at the argument about BBA students which we have already seen:

     (1) Some BBA students wear earrings.
     (2) Some persons who wear earrings are not gays.
     (3) Therefore, some BBA students are not gays.

Here is then the diagram of the argument:

Diagram for this argument

We have seen that this argument is invalid. Let circle A stands for BBA students; B for persons who wear earrings, and C for gays. From the premises that some BBA students wear earrings and that some wearers of earrings are not gays, it cannot be validly concluded that some BBA students are not gays. It may be the case that all BBA students are gays, which if true will make the conclusion false. The diagram contains one relation of type I and the other of type O. However, since here we have three cicles instead of two the markers of 'at least one thing' has to make it possible that the thing referred to may exist either inside or outside the third circle. For example, the diagram for Premise (1) is shown by the X between circles A and B. But since we also have the circle C then we need to show that the X here can live either inside or outside C. So we put two marks for 'X', meaning that the X can move to either position. Note that you have to be careful in recognizing that here we have only one X. This comes from the idea of the diagram for Type I that uses only one marker for the existence of the 'something' in question. The line between the two X markers show that here we have only one X but it can move to either inside or outside circle C. Similarly, the diagram for Type O relation between circles B and C is represented by the letter 'Y', which can live either inside or outside circle A. What this means in real terms is that, since some persons who wear earrings are not gays, they may be BBA students or they may be not. The proposition just does not state exactly that. That is why we need to leave open the possibility. So for the 'X' in the first example.

Now how do we know from the diagram that this argument is invalid? Remember that an argument is invalid if and only if it is possible that the conclusion is false even though all the premises are true. So let's suppose, as we must, that all the premises are true. Then it is true that some BBA students wear earrings and that some wearers of earrings are not gays. But the point is that these two propositions can be true together with the proposition that all BBA students are gays, which is the negation, or the contradictory proposition, of the conclusion that some BBA students are not gays. (We have seen that Type-O propositions is contradictory to Type-A propositions.) Why are we so certain? Because in the diagram we can see that, suppose there is only one BBA student, then it is still true that that the BBA student, represented in the diagram by the 'X', does wear earrings. (The X is found where A and B overlap.) And it is also true that there is another person, represented by 'Y', who wear earrings but is not a gay. However, since we suppose that there is only one BBA student, he can live either inside or outside circle C. But if he lives inside C, then he is a gay. And since he is the only one BBA student, it is true also that all BBA students are gays. In other words, it is true that, if there is anyone who is a BBA student, he or she is a gay. But since that is a negation of the conclusion, we have shown that the negation of the conclusion can be true together with all the premises being true. So the argument is invalid. The red circle around the X inside circle C points to this possibility which makes the argument invalid.

Now consider this argument:

     (1) Any country with a sustained current account deficit is likely to
              suffer from depreciation of its currency.
     (2) Some countries in Asia are likely to suffer from depreciation of their 
     (3) Therefore, some countries with a sustained current account deficit are 
         in Asia.

Is this argument valid? Let's see its Venn's diagram:

Diagram for this Argument

You can see that this is invalid because it is possible that there are no Asian countries with a sustained current account deficit. Let A be 'countries with a sustained current account deficit;' B be 'country likely to suffer from depreciation of its currency;' and C to 'country in Asia.' Then, since the blue X in the diagram can live in two places, inside or outside of A, it is clear that there is no Asian country with a sustained current account deficit even though all the premises are true.

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