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A Primer on Logic

Part 3:

Aristotelian Logic

Jason Zarri

1. Introduction

In the last Part we learned how to determine the validity of argument schemas, and in order to do so we had to become acquainted with a small fragment of formal logic. The branch of logic we dealt with is called propositional logic [1], which deals with statements considered as wholes and the ways in which they are related to each other. Propositional logic is good as far as it goes, but it doesn’t go far enough. There are many valid arguments whose validity it cannot account for. To find out which arguments these are and why they are valid, we must examine not only statements but the elements of which statements are composed and the ways in which they are related to each other.

2. A New Kind of Validity

Consider the following argument:

All birds are animals.

All robins are birds.

All robins are animals.

This argument is valid, but can we show that it is using the resources of Part 2? First, we need to symbolize the premises and conclusion. Let’s symbolize “All birds are animals” as ‘A’, “All robins are birds” as ‘B’, and “All robins are animals” as ‘R’. To test this argument for validity we can use the same method we used in Part 2, because it works just as well for arguments as it does for argument schemas. We conjoin the premises A and B and form the appropriate conditional:

“(A & B) ⊃ R”

Using the short way, we get this partial truth table:

A B R A & B (A & B) ⊃ R T T F T F

The reasoning that yields this result is as follows. If “(A & B) ⊃ R” is false then “A & B” must be true while R is false. And if “A & B” is true then so are A and B. This is as far as we can go. We have assessed every truth-functional compound and determined the truth value of every atomic statement. Furthermore, the truth values assigned to the atomic statements do not entail that any truth values should be assigned to the truth functional compounds other than those they already have. We can consistently assign truth values to each of the above statements, so the conditional is not a tautology, and the argument is truth-functionally invalid. But it is clear that the argument is valid, so if it is not truth-functionally valid it must be valid in some other way.

Must we conclude, then, that the above argument is not valid in virtue of its form? The answer is “no”, but to vindicate that answer we will have to enrich our notion of form. For example, this argument has the same form as the first:

All politicians are deceptive people.

All congressmen are politicians.

All congressmen are deceptive people.

Although they deal with different subjects, these arguments are similar in two main respects. First, each statement in the arguments says something about all the members of a certain category. Second, in the arguments there is a pattern concerning what is said of what. In the first premise of each argument all members of a category are said to have a certain property; in the second premise all members of a second category are said to be members of the first category; and in the conclusion all the members of the second category are said to have the property attributed to all the members of the first category.

From the above it is apparent that sometimes the validity of an argument depends not only on the way its statements are combined, but also on the internal structure of the statements themselves. In this case it depends on the meaning of “all” and the relations between what is said of each category. In Part 2 the atomic statement letters we used concealed this structure because they stood in for whole statements, making them something akin to logical black boxes. What we must do now is peer inside them to see how they function.

The first logical theory that was intended to give an account of validity based on the logical structure of statements—indeed, the first sophisticated logical theory of any kind, at least in the West—was the one developed by the philosopher Aristotle, who lived in Greece from 384 to 322 BCE. In the next section I will give a (very) rough sketch of Aristotelian logic.

3. A Brief Introduction to Aristotelian Logic

In order to explain Aristotle’s logical theory, we must introduce a new kind of letter, the general term letter, or term letter for short. These letters are like schematic letters in that they need not stand in for any particular general term. A general term is a word or phrase that applies to entities or groups. The general terms (minus the plural ending) which occur in the preceding arguments—‘bird’, ‘animal’, ‘robin’, ‘politician’, ‘deceptive person’, and ‘congressman’—are general because they can apply to more than one entity or group. There are many birds, animals, robins, politicians, deceptive people, and congressmen—far too many of the last three—and the corresponding general terms apply to each member of these categories. In this respect general terms are unlike singular terms, which can only apply to a single entity or group without becoming ambiguous. Thus many people are named ‘John’, but the name ‘John’ does not mark the people it applies to as being members of a kind, so the name is ambiguous. ‘John’ is the name of a person, not a kind of person. Similarly, ‘The Clayton Book Club’ might be the name of a particular group of people, and there may be other book clubs in other cities called ‘Clayton’ which have the same name. Nevertheless, ‘The Clayton Book Club’ is still a singular term because it is the name of a club and not a kind of club. By contrast, the term ‘human’ applies to each and every human, but it is not ambiguous because it marks the entities that it applies to as being members of a kind. There are many humans, but as far as I know none of them is named ‘Human’.

In Aristotelian logic, the basic form of argument is the syllogism, of which the preceding two arguments are examples. Every syllogism contains three statements; that is, two premises and a conclusion. Every syllogism also has three distinct general terms, each of which occurs in it twice. Because there are only three distinct terms in any syllogism we can get by with only three term letters. Using the capital letters ‘F’, ‘G’, and ‘H’ as term letters, both of the above arguments can be depicted as being of the same form:

All F are G.

All H are F.

All H are G.

Each statement in a syllogism must have one of four forms, called ‘A’, ‘E’, ‘I’, and ‘O’:

A: All F are G. Universal Affirmative

E: No F are G. Universal Negative

I: Some F are G. Particular Affirmative

O: Some F are not G. Particular Negative





According to tradition, there are a number of important logical relationships between these forms, which can be represented by this famous diagram:

The Square of Opposition





A: All F are G. E: No F are G.

___________________________________________

| \ /|

| \ / |

| \ / |

| \ / |

| \ / |

| \ / |

| / \ |

| / \ |

| / \ |

| / \ |

| / \ |

| / \ |

|/_________________________________________\ |

I: Some F are G. O: Some F are not G.

Statements of the forms A and E cannot both be true, though they can both be false. These are called contraries. Thus if “All birds are animals” is true—as it most certainly is—“No birds are animals” is false. Conversely, if we pretend that it is possible for birds not to be animals, if “No birds are animals” were true “All birds are animals” would be false. However, both statements would be false if, contrary to fact, some birds were animals and some were not. Statements of the forms I and O cannot both be false, though they can both be true. These are called subcontraries. As things are, “Some birds are animals” is true and “Some birds are not animals” is false. We can suppose that the reverse is true by suppressing what we know about ornithology, but according to the Square we cannot suppose that neither statement is true without suppressing what we know about logic itself. A and O statements are contradictories: If a statement of form A is true the corresponding statement of form O is false, and conversely. So if “All birds are animals” is true, “Some birds are not animals” is false, and if “Some birds are not animals” is true, “All birds are animals” is false.[2] E and I statements are also contradictories. Tradition also has it that A statements entail their corresponding I statements and that E statements entail their corresponding O statements.[3] Thus I statements are said to be subalterns of A statements, and O statements are said to be subalterns of E statements. “Some birds are animals” is a subaltern of “All birds are animals”, and “Some birds are not animals” is a subaltern of “No birds are animals”. A statements can also be called superalterns of I statements and E statements can be called superalterns of O statements.

In any given statement that may occur in a syllogism, a term letter that is in the same position that ‘F’ is in in the above schematic syllogism and in the four forms of statement is the subject term, and a term that is in the same position that ‘G’ is in is the predicate term. It is necessary to invoke the position of a term letter, and not just the letter itself, because a term letter that is in one position in one statement within a syllogism can be in another position in another statement within that syllogism. Intuitively, a term is in subject position when it picks out the members of the category which is said to be either included in another category or excluded from it, and a term is in predicate position when it picks out the members of that other category; i.e., the one that the first category is said to be included in or excluded from. Thus, in our schematic syllogism’s first premise, ‘F’ is in subject position and ‘G’ is in predicate position. In the second premise ‘H’ is in subject position and ‘F’ is in predicate position. In the conclusion, ‘H’ is again in subject position and ‘G’ is again in predicate position.

In any given syllogism there is a major term, a minor term, and a middle term. The major term is the term that is in predicate position in the conclusion. The minor term is the term that is in subject position in the conclusion. The middle term occurs in both premises but not the conclusion. In our schematic syllogism, ‘G’ is the major term, ‘H’ is the minor term, and ‘F’ is the middle term. Also, in a syllogism the premise containing the major term is called the major premise, and the premise containing the minor term is called the minor premise.

4. Classifying Syllogisms

Syllogisms are classified by the forms of statement they contain, how many statements of each form there are, and the order in which they occur. Thus the schematic syllogism we have been examining so far is of the form AAA. That means that the major premise, the minor premise, and the conclusion—in that order—are all universal affirmatives, or ‘As’.

There are several other valid forms. One example is:

No F are G.

All H are F.

No H are G.

This is of the form EAE, because its major premise is an E statement, its minor premise is an A statement, and its conclusion is an E statement. Another example is:

All F are G.

Some H are F.

Some H are G.

This is of the form AII.

So far we have classified syllogisms according to their mood; i.e., the form of their major and minor premises and their conclusion. AAA, EAE and AII are all moods of syllogisms; although there are several more, as we shall see. A syllogism of a given mood can also be classified according to its figure. The figure of a syllogism is determined by what position the major term is in in the major premise and by what position the minor term is in in the minor premise. There are four figures, and to illustrate them I will show what a syllogism in the mood EIO looks like in all of them:

First Figure:

No F are G.

Some H are F.

Some H are not G.

Second Figure:

No G are F.

Some H are F.

Some H are not G.

Third Figure:

No F are G.

Some F are H.

Some H are not G.

Fourth Figure:

No G are F.

Some F are H.

Some H are not G.

In the first figure the major premise has the major term in predicate position and the minor premise has the minor term in subject position. In the second figure the major and minor terms are both in subject position in their respective premises. Similarly, in the third figure the major and minor terms are both in predicate position in their respective premises. Lastly, in the fourth figure the major premise has the major term in subject position and the minor premise has the minor term in predicate position. If you’ve noticed that the term letters ‘H’ and ‘G’ are in the same position in the syllogism’s conclusion in each figure, there is a good reason for that: According to the definitions of major and minor terms, the major term is the one in predicate position in the conclusion and the minor term is the one in subject position in the conclusion. Of course we could have put the term letters in different positions in the conclusion, but that would only make ‘H’ become the major term and ‘G’ become the minor term. For example, one could get this result, which is a modified form of EIO in the first figure:

No F are H.

Some G are F.

Some G are not H.

Note that we had to substitute an ‘H’ for a ‘G’ in the first premise and a ‘G’ for an ‘H’ in the second premise in order to preserve validity.

Also, it is important to remember that the placement of the major and minor terms in each of the four figures is not peculiar to the mood EIO; a syllogism of any mood that is in the first figure would have its major and minor terms in the same positions, and the same is true of the other figures.

5. Assessing Syllogisms

I will now present a method for determining the validity of syllogisms. This method involves what I call term diagrams, so called because they make use of term letters. As was said above, a general term which is represented by a term letter applies to each member of a certain category. Term diagrams use term letters and grouping symbols to represent categories. Consider the following three expressions:

{F—F}

(G—G)

[H—H]

(By convention, two ‘F’s are always enclosed in curly braces, two ‘G’s are always enclosed in parentheses, and two ‘H’s are always enclosed in square brackets.) I will call these complete categoricals, or CCs. The first CC listed above represents all the things in the category F, i.e., all the things that the general term corresponding to the term letter ‘F’ applies to. The second CC represents the entire category of Gs, and the third represents the entire category of Hs. The two dashes in between the term letters mark “gaps” where other term letters can be “plugged in”. In addition to complete categoricals, there are also partial categoricals, or PCs, for which we use a single term letter, either ‘F’, ‘G’, or ‘H’. These represent an unspecified portion of their corresponding categories. From this point on, I will sometimes refer to both CCs and PCs as categoricals.

Term diagrams are a kind of model. A model, in general, is something used in logic to show which statements entail which.[4] A model is said to satisfy a statement when it represents a situation in which that statement would be true. One can show an argument to be valid if one can demonstrate that any model which satisfies its premises also satisfies its conclusion. Because syllogisms must consist of statements having one of the forms A, E, I, and O, we must have a model for each of these four forms. The models of these forms are:

A: (G{F—F}G)

E: {F—F}– (G—G)

I: (GF–G)

O: F— (G—G)

These models represent the relations of inclusion and exclusion that hold among categories by means of the spatial relations of the categoricals. In the model of form A, the F CC is inside the G CC, which means that all F are G. In the model of form E, the F CC is entirely outside the G CC, which means that no F are G. In the model of form I, an F PC is inside the G CC, which means that some F are G. Finally, in the model of form O, an F PC is outside the G CC, which means that some F are not G.

To assess syllogisms with term diagrams one must construct a model which satisfies both premises and see whether the resulting model also satisfies the conclusion. Modeling the premises proceeds in two steps. First, in the major step one models the major premise. Second, in the minor step one takes the model constructed in the major step and modifies it so that it also models the minor premise, filling in gaps as needed. One can then tell by inspection whether the completed model satisfies the syllogism’s conclusion. If it does the syllogism is valid, and if not, not.

I will go over several examples to show how the process of evaluating syllogisms works. Let’s start with the syllogistic form AAA:

All F are G.

All H are F.

All H are G.

In the major step we construct a model of the form A, because that is the form of the major premise:

(G{F—F}G)

Next, in the minor step we plug the H CC into the gaps in the F CC to accommodate the information that all H are F:

(G{F[HH]F}G)

We can now tell that all H are G, because the H CC is inside the F CC, and hence also inside the G CC. We may thus conclude that the syllogistic form AAA is valid.

Next, we will evaluate the form EIO:

No F are G.

Some H are F.

Some H are not G.

In the major step we place the F CC entirely outside the G CC, because neither category is included fully or partially in the other:

{F—F}– (G—G)

In the minor step, we augment the model so that it represents that some H are F. We do this by placing an H PC inside the F CC:

{FH–F}– (G—G)

It follows immediately that some H are not G, for an H PC is inside the F CC, and hence outside the G CC.

We now come to the syllogistic form EAE:

No F are G.

All H are F.

No H are G.

In the major step, since the categories F and G are disjoint, we get:

{F—F}– (G—G)

And since all H are F, in the minor step we get:

{F[HH]F} (GG)

We can see that no H are G, for the H CC is inside the F CC, and so outside the G CC.

The next example is somewhat more problematic. Consider the syllogistic form IAI:

Some F are G.

All F are H.

Some H are G.

The major step is easy enough. We write:

(GF–G)

It is not so clear what we are to do in the minor step, because there is more than one way of modifying the model to accommodate the minor premise, and each satisfies some statements that the others do not. We can get around this problem by adapting a form of argument called reasoning by cases. This form is:

p ∨ q

p ⊃ r

q ⊃ r

r

If you know that either of two statements p or q is true—even if you don’t know which—and if you also know that they both imply some statement r, then you know that r is true irrespectively of whether it is p or q or both that are true. This generalizes to disjunctions of arbitrarily many disjuncts; as long as each disjunct implies some proposition r, r must be true if the disjunction is.

So, to solve our problem we will consider every way of modifying the model. Such ways are called cases. A statement counts as a valid conclusion of a schematic syllogism only if it is satisfied by all cases. Given our current model, that is “(GF–G)”, case one is:

(G[H{FF}H]G)

Case two is:

[H(G{FF}G)H]

And case three is:

[H{F(GF}H]G)

(As you can see, in case three there is multiple overlap.) So, what can we conclude? We cannot conclude that all H are G, for that only holds in case one; in cases two and three at least one H PC is outside the G CC. Nor can we conclude that some H are not G, for that only holds in case three. Note that we also cannot conclude that some F are not G for (a) that only holds in case three, and (b) by convention ‘F’ is the middle term of a syllogism, and cannot appear in a conclusion! The only conclusion we can validly draw is that some H are G, because it is satisfied by all three cases. That some H are G is precisely what the conclusion of the form IAI asserts, and so that form is valid.

“But how,” you might be wondering, “is ‘some H are G’ satisfied by case two? To answer that question I must now say something about conversion. A given form of statement is said to convert if switching its subject and predicate terms necessarily yields a statement with the same truth value. Take the E form, which is “No F are G”. One instance of this form would be the statement “No reptiles are plants”. If we suppose this is true (which it is!), could its converse, “No plants are reptiles”, nevertheless be false? It could only be false if its contradictory, namely “Some plants are reptiles” were true, and given that “Some plants are reptiles” clearly isn’t, “No plants are reptiles” must be true. Thus, if an E statement is true, its converse must be true too. Similar reasoning should convince us that if an E statement is false its converse is also false.

The fact that E statements convert should also be evident from their corresponding term diagrams. For “No F are G” is rendered as,

{F—F}– (G—G)

while “No G are F” is rendered as,

(G—G) — {F—F}

which are merely notational variants of each other. What is not so evident from their corresponding term diagrams is that I statements also convert. For “Some F are G” is rendered as,

(GF–G)

while “Some G are F” is rendered as,

{FG–F}

Nevertheless, consider a statement of the I form, say “Some pizzas are round things”. Could it be true that some pizzas are round things, and yet false that some round things are pizzas? A little reflection should convince us that the answer is “No.” So if some pizzas are round things, some round things are pizzas. Also, a little more reflection should convince us that if it is false that some pizzas are round things, it is also false that some round things are pizzas. So “Some F are G” is logically equivalent to “Some G are F”, and I statements convert. That is why “Some H are G” is satisfied by case two above: The G CC is inside the H CC, meaning that all G are H. Recall that, according to tradition, if all G are H it follows that some G are H, and since the I form converts it follows that some H are G.

In contrast to the above, A statements and O statements do not convert. For example, the A statement “All squares are rectangles” is true, but its converse, “All rectangles are squares”, is certainly false. And while “Some cats are not mammals” is quite false, “Some mammals are not cats” is quite true.

The fact that I statements convert makes it more difficult to interpret some other term diagrams. Consider EIO in the third figure:

No F are G.

Some F are H.

Some H are not G.

In the major step we get:

{F—F}– (G—G)

But what are we to do in the minor step? We could enclose one of the F PCs inside an H CC, but that would force use to consider three cases, one where an H PC is inside the G CC and two where both H PCs are outside the G CC. It is easier to put an H PC inside the F CC, like so:

{FH–F}– (G—G)

Because I statements convert, “{FH–F}” is logically equivalent to “[HF–H]”, and so the above term diagram satisfies both “Some H are F” and “Some F are H”. Thus the conclusion “Some H are not G” still follows, and EIO is still valid in the third figure.

It will be important to remember that I statements convert when dealing with other term diagrams. I will give one more example. This is the mood AAI in the third figure:

All F are G.

All F are H.

Some H are G.

We model the major premise with this diagram:

(G{F—F}G)

As for the minor premise, we must consider three cases. The first case is:

[H(G{FF}G)H]

And the second is:

(G[H{FF}H]G)

Finally, the third is:

(G[H{FF}G)H]

It is necessary to consider these cases because the major and minor premises do not settle whether all G are H or all H are G, or whether some H are G and some H are not. What is common to all three cases is that some H are G: In the first case, at least one G PC is inside the H CC; so some G are H and by conversion some H are G. In the second and third cases at least one H PC is inside the G CC, and hence it follows immediately that some H are G.

I hope that the basic ideas behind this method for evaluating syllogisms are now clear. If you use it correctly, you should find that the following table lists all of the valid syllogistic forms, and no others.[5] The presence of an ‘x’ in a cell means that a syllogism of the given mood and figure is valid:

Figure: 1st 2nd 3rd 4th Mood: AAA x AAI x x x AEE x x AEO x x AII x x AOO x EAE x x EAO x x x x EIO x x x x IAI x x OAO x

6. Conclusion

Having learned what syllogisms are and how to assess them for validity, we now know as much of Aristotelian logic as we need to for our purposes. Before we move on to the next Part, there will be an Interlude in which I explain some of the limitations of Aristotelian logic. The quest to surmount these limitations is what led to the discovery of predicate logic, which is the subject of Part 4.

Bibliography

Parsons, Terence, “The Traditional Square of Opposition”, The Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/entries/square/>

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