Today, we count using a decimal positional numbering system. For example, 365 represents three one-hundreds, six tens and five ones. This makes life much easier for all concerned. This is not how it used to be. For example, in Roman numerals, this would be CCCLXV. Consider trying to come up with a set of rules for adding Roman numerals: VIII + VIII = XVI. The ancient Greeks used a numbering system where the first nine letters represented the numbers 1 through 9, the next nine represented 10, 20, 30, through to 90, and any remaining letters represented 100, 200, etc. Thus, for example, α + α = β, α + β = γ and θ + θ = κζ. This, humorously enough, was actually a regression from the Babylonian numbering system, which was also positional, but where the base was not 10 but 60 (hence, 60 seconds to the minute and 60 minutes to the hour, or 60 arc seconds to the arc minute and 60 arc minutes to the degree). To add 3.25 + 723.492, you don't need to understand Babylonian, Greek or Roman counting systems. Instead, you just use the techniques from elementary school to get 726.742. It would be much more difficult to describe an algorithm for the division of two Roman numerals, so for example, what is MMDCCXXX divided by CXXX? How quickly could you get XXI? It's probably easier to convert the Roman numerals into decimal notation, perform the division, and then convert the result back into Roman numerals. The reason for this is that a positional system is simply superior to any other previous system. Is a decimal positional system the most efficient? Probably not: it has only one feature going for it, the fact that we have ten fingers. A base 12 numbering system would be more practical for real-world applications (a foot is already divided into 12 inches), but that is a discussion for another day.

Thus, while the ancient Greeks introduced the concept of logic, and began to apply it, a more reasonable system of logic and set theory was developed in the 19th century. So why do so many people continue to try to even understand or worse master the logic of ancient Greece? Let us compare and contrast the two:

For Aristotle, one major hurdle he had to overcome was accepting that this concept of true and false was actually applicable in discourse and debate. Thus, one consequence that if you define two states, true and false, and if a statement P can be either true or false, then

the truth value of P equals itself,
either P is true or it is false, and
P cannot be both true and false simultaneously.

These are called the so-called "laws of logic", but this is only because Aristotle used these to demonstrate that logic can actually be used in discourse; however, their use is limited. For example, "This tree is tall" is not a truth statement, because "tall" is subjective; however, "This tree is taller than six feet" is a truth statement. You can observe from Aristotle's arguments where he was trying to show how certain statements can have such truth values, and the reason for these arguments was that many of his peers argued against such a formal logic. Many statements about society, for example, cannot be reduced to simple truth values: "If we reduce the number of firearms, violence will go down." Is this true or false?

Next, to apply logic, Aristotle introduced four universal (applying to all objects in a collection) or existential (there exists an object in a collection) truth statements:

All P are Q.
No P are Q.
Some P are Q.
Some P are not Q.

Examples of these include "All Greeks are human," "No humans are immortal," "Some humans are Greek," and "Some humans are not Greek." Each of these statements is true; however, "All Greeks can command a ship,", "No Greek has mastered elementary mathematics," "Some Greeks live in Hawaii," and "Some Greeks are not human." At least at the time of Aristotle, all of these statements were false. We will focus on such statements when they are true.

Note that in some cases we will identify a single object, such as saying that "Socrates is a human" instead of "All Greeks are human." The first is just an abbreviated variation of saying "All things that are Socrates are human." Similarly, "Socrates is not divine" is just an abbreviated way of saying "No things that are Socrates are divine."

Now, if all P are Q and all Q are R, then all P must be R. This is what Aristotle termed a syllogism, or conclusion or inference. That is, if all P are Q and all Q are R, we may conclude (or infer) that all P are R. What Aristotle and others then did is combine all possible combinations of these, to determine which allowed one to make a valid conclusion or inference. The first would contain some relationship between P and Q, the second would contain some relationship between Q and R, and the conclusion would contain some relationship between P and R. Some examples include:

All P are Q, some Q are R, so some R are P.
All P are Q, no Q are R, so no P are R.

There are 8³ or 512 such combinations of such terms, and thus, Aristotle and his school examined all 512 of these combinations and determined which had valid conclusions. Now, to be clear, and to not strawman Aristotle, there is a hidden assumption in these four concepts, and so more correctly, the four truth statements were:

There is at least one object that has the property P, and all P are Q.
There is at least one object that has the property P, but no P are Q.
Some P are Q.
Some P are not Q.

By saying "some P are Q", this implies that the collection of objects that satisfies P must be non-empty. I point this out, because if you ignore this hidden assumption, then it is possible to create absurd statements. I will refer back to this later, but let us assume now if we say "All P are Q", there is an assumption that at least one object exists that satisfies P. For example, "All mountains of pure gold are mountains" would be invalid, as there are no mountains of pure gold; however, technically, if a mountain of pure gold existed, it would indeed be a mountain.

Looking at our two examples:

All P are Q and some Q are R, so some R are P.
All P are Q and no Q are R, so no P are R.

The conclusion of the first is false: "All penguins are birds", "Some birds can fly", and therefore "Some things that fly are penguins." Because penguins cannot fly, the statement "Some things that fly are penguins" is false." Thus, the conclusion of the first ("Some R are Q") cannot be deduced from "All P are Q" and "Some Q are R".

However, the conclusion of the second does follow: If all objects that satisfy the property P also satisfy the property Q, and nothing that satisfies the property Q satisfies the property R, then no object that satisfies the property P can also satisfy the property R. To show this, we will assume the contrary, so that there is one object that satisfices P that also satisfies R:

Assume that there is at least one object x that satisfies P that also satisfies R.
Now, if all objects that satisfies P also satisfies Q, then x must satisfy Q.
However, if x satisfies Q, and no object that satisfies Q satisfies R, then x cannot satisfy R.
This is a contradiction, as x cannot simultaneously satisfy R and not satisfy R.

Thus, the two statements and the subsequent conclusion,

All P are Q,

and no Q are R,

so therefore no P are R.

describes a valid syllogism; that is, if the first two statements are true, then the conclusion must be true.

It does not take significant wit to understand the interesting frustration of writing down all 512 such statements:

All P are Q,

and all Q are R,

so therefore all P are R.

All P are Q,

and all Q are R,

so therefore no P are R.

All P are Q,

and all Q are R,

so therefore some P are R.

and skipping 508 intermediate cases

Some Q are not P,

and some R are not Q,

so therefore some R are not P.

The last is clearly not a valid syllogism; for example, some negative integers (Q) are not positive integers (P), and some integers greater than 10 (R) are not negative integers (Q), and therefore some positive integers greater than 10 are are not positive integers? This is clearly a false conclusion, for all integers greater than 10 are also positive. Some freezing days are not above freezing days, and some warm days are not freezing days, and therefore some warm days are not above freezing days. This is of course, an absurd conclusion, for all warm days must be above freezing.

Thus, imagine going through all 512 cases, and determining which are valid and which are invalid. Not only that, you now have to somehow remember those that are valid. The way this was done in the earliest Latin schools of logic was to associate a letter with each of the statements:

[A] All P are Q.

[I] No P are Q.

[E] Some P are Q.

[O] Some P are not Q.

Thus, the valid syllogism

All P are Q,

and all Q are R,

so therefore all P are R.

is remembered by recalling the word "Barbara", and to master logic, you had to memorize all of

Barbara

Cesare

Datisi

Calemes
Celarent

Camestres

Disamis

Dimatis
Darii

Festino

Ferison

Fresison
Ferio

Baroco

Bocardo

Calemos
Barbari

Cesaro

Felapton

Fesapo
Celaront

Camestros

Darapti

Bamalip

However, even memorizing this list was insufficient, for Barbara had "All P are Q, and all Q are R, so all P are R," but Darapti had "All Q are P, and all Q are R, so some P are R." However, you had better not accidentally remember the word Camostri, as this does not represent a valid syllogism. This seems exceptionally painful and unnecessary, and it is, so we will look at how modern theories of logic and set theory account replace these with simpler concepts.

To give an analogy for the difficulty involved here, suppose you were learning algebra, and you were told to memorize that all possible expansions of a(x + y), which include ax + ay, xa + ay, xa + ya, ax + ya, ay + ax, ya + ax, ya + xa and ay + xa, but not xy + xa and not ya + yx. Instead of memorizing all eight valid patterns, you simply have to remember three rules:

multiplication distributes over addition, so a(x + y) = ax + ay,
multiplication is commutative, so xy = yx, and
addition is commutative, so x + y = y + x.

From these three rules, you can now deduce all possible expansions of a(x + y), but with these three rules (together with the rule that 1·x = x) you can also determine the expansion of (x + y)⁹. Imagine how little algebra you could actually perform if you were restricted in your knowledge to simply having memorized that all possible expansions of a(x + y) are those eight that are listed above. In a similar manner, we will show that anyone who wastes their time memorizing the syllogisms above and indeed, anyone who thinks the three "laws of logic" are the pinnacle of logical expression have a serious deficit of mastery of the subject matter. The issue was, however, that for millennia, no advances in logical thought since Aristotle occurred until the 19th century with the introduction of the axiomatic method.

A renewed interest in the axiomatic method (as begun by Euclid) began when serious investigations went into geometries that obeyed most of Euclid's postulates, but only changed the parallel postulate. For example, if you declare antipodal points on a sphere as "a point" and the line through two such points as being the great arc that passes through both points, then the parallel postulate now changes, as parallel lines always meet. This is described as spherical geometry and it has applications in the real world (navigation) just like Euclid's planar geometry. Interestingly, any proposition that does not require the parallel postulate in its proofs still holds for spherical geometries.

We are going to make the transition to more modern descriptions, so what Euclid called common notions or postulates are today called "axioms," and any truth statement that is derived from those common notions or postulates is called a "theorem." We will continue to use this terminology from this point forward. Specifically, the axioms for the integers are:

We have two operations + and · where m + n and m·n return an integer whenever m and n are integers.
Both operations commute: m + n = n + m and m·n = n·m.
Both operations associate, so ℓ + (m + n) = (ℓ + m) + n and ℓ·(m·n) = (ℓ·m)·n.
Multiplication distributes over addition, so ℓ·(m + n) = ℓ·m ℓ·n.
There are integers 0 and 1 such that 0 + n = n and 1·n = n for all n.
Every integer n written as -n such that n + –n = 0, and when we write m – n we mean m + –n.
If ℓ·m = ℓ·n and ℓ ≠ 0, then m = n.

All other properties of the integers can be derived from these seven axioms. For example, Axiom 5 says that there is one integer 0 such that 0 + n = n for all integers n, but it does not say that this 0 is unique. The uniqueness of 0 can be derived as a theorem based on this and other axioms. Similarly, these axioms do not state that 0·n = 0 for any integer n, and yet this, too, is a theorem that can be derived from these axioms. You can also prove that both 1 ≠ 0 and n + 1 ≠ n. Thus, just like Euclid's common notions, postulates and propositions, theorems about the integers can be derived from the above seven axioms.

The axiomatic method

In the 19th century, the concepts of logic were finally further refined, and this involved three separate concepts:

the idea of equality,
the concept of propositional logic, which only dealt with the operations on propositions that were true or false, and
the concept of set theory.

A. Equality

Both Euclid's common notions and the first "law" of logic introduce the idea of equality. For Aristotle, one axiom was that "P = P", while for Euclid, his common notions included:

Things which equal the same thing also equal one another.
If equals are added to equals, then the wholes are equal.
If equals are subtracted from equals, then the remainders are equal.
Things which coincide with one another equal one another.
The whole is greater than the part.

The second through fifth postulates associate equality with other operations, but if we are only considering equality, then the following four axioms are sufficient: however you wish to define equality x = y where both x and y come from a specified collection of objects , that equality has all the properties we expect from such a notion if:

each object equals itself: x = x,
x = y if and only if y= x,
if x = y and y = z, then x = z, and
if x = y, then if F is any relevant operation on these objects, then F(x) = F(y).

Thus, in a modern axiomatic approach, the idea of equality is separated from that of logic. You will note that Euclid's common notions 2 and 3 are simply interpretations of Axiom 4. Next, we will look at the concept of propositional logic. The only other point not raised previously by the ancient Greeks was the idea of symmetry: for an equality to make sense, x = y if and only if y = x. The Greeks likely did not think of equality as a binary operation, and therefore this last axiom was probably seen as obvious.

Note that equality can mean different things. We may consider two persons to be "equal" if they are of the same age, and if the requirement for players two be on the same team, then in this sense, two individuals are "equal" if they are of the same age. Another example of equality is two integers modulo a given positive integer, say 12. In this case, m = n if m - n is a multiple of 12, so -11 = 1 = 13 = 25 = 37, etc., and -7 = 5 = 17 = 24, etc. You may notice that if m = n under this definition, and k is any integer, then k + m = k + n and also km = kn under this definition of equality, too. It may seem odd, but in this case, under all operations, 5 and 17 behave the same way.

B. Propositional logic (statements about true and false)

Propositional logic simply looks at statements P, Q, R, etc., that must be either true or false. Operations on such statements include and, or, implies, and if-and-only-if. We will describe each of these:

The statement "not P" is true if P is false, and otherwise it is false. This is written as ¬P.
The statement "P and Q" is true if both P and Q are true, otherwise it is false. This is written as P ∧ Q, and you may remember this by observing that the '∧' looks like the 'A' in 'and'.
The statement "P or Q" is true if either P or Q are true, and it is false only when both are false. This is written as P ∨ Q.
The statement "P implies Q" is true if Q is true whenever P is true. This is often described as "the truth of P implies the truth of Q." For example, "it is raining implies there are clouds in the sky" is a true statement, while "there are clouds in the sky implies it is raining" is false. The first is true because if it is raining, there must be condensation of moisture, a consequence of which are clouds. However, just because clouds form does not imply that it must also be raining. This is written as P → Q.
Finally, the statement "P if-and-only-if Q" is true if P and Q are either both false or both true. For example, "lightning is striking if and only it is thundering" is true because they are both consequences of the same phenomenon; however "I see lightning if and only if I hear thunder" is false, for the lightning may not be visible due to intervening clouds or blindness, but you may still hear the thunder; or you may see the lightning, but because the event is so far away, or because you may be deaf, you cannot hear the thunder. This is written as P ↔ Q.

Now, we have both "and" and "or" operators (∧ and ∨), and the concept of if-and-only-if is also something understood by most people: the left side is true if and only if the right side is true. However, implication (P → Q) is more difficult. An implication P → Q is true unless it is ever the case that P s true, and yet Q is false. For example, the statement "There are clouds in the sky implies it is raining." This author is looking out the window right now, and can see clouds, and yet, it is not raining, so this implication is false. However, the implication "It is raining and I am outside in the open implies I am getting wet." is true, for one can hardly not get wet if one is out in the open while it is raining.

There is, however, another way of saying an implication: "If there are clouds in the sky, it is raining." is an implication, but it is a false implication. The second statement above in this format is "If it is raining and I am outside in the open, I am getting wet." is a true implication. Notice that if the left-hand side is false, I may or may not be getting wet. For example, "It is raining and I'm standing under an umbrella" or "It is raining and I am in a building" are two cases of "It is raining and I am not out outside in the open."

There is yet another means of saying an implication: "If it is raining, there are clouds in the sky" is logically equivalent to "Either it is not raining or there are clouds in the sky", so P → Q is equivalent to ¬P ∨ Q. Basically, if you are trying to argue that an implication is false, it may actually be easier to argue that both ¬P and Q can simultaneously be false. For example, one may take "Either there are no clouds in the sky or it is raining." Well, looking outside right now, there are clouds in the sky, and yet it is not raining, so both statements are false, so the statement "Either there are no clouds in the sky or it is raining" is also false. We will look at a number of other logical statements that are equivalent, but before we get to that, we will discuss tautologies: logical statements that are always true.

P ∨ ¬P

Consider the statement P ∨ ¬P. If P is true, then P ∨ ¬P must be true, for "or" requires only one to be true for the entire statement to be true: "I am shorter than six foot or there are magical unicorns flying over my head" is a true statement, for the first is true, even if the second is absurd. If, however, P is false, then ¬P must be true, so P ∨ ¬P must still be true. We'll look at a few examples:

Either there is one god, or there is not one god. The second statement is the exact negation of the first.
Either there is one god, or there are no gods. This is invalid, as the second is not the negation of the first: there may be many gods.
Either I am tall or I am not tall. This is a valid logical statement, but "being tall" is ambiguous. If you are over six foot tall, then most would describe you as being "tall", and if you are under five-foot-four, then most would describe you as being "not tall", but what if you are five-foot-seven or five-foot-ten? The statement "She is tall" is not a propositional statement.
Either I am over six feet or I am not over six feet, or equivalently, either I am over six feet or I am less than or equal to six feet tall. In both cases, the second statement is the negation of the first.

¬(P ∧ ¬P)

Next, consider the statement ¬(P ∧ ¬P). This says "it is false that both P and ¬P are true. This, too, is always true, for if P is true, then ¬P is false, and a statement like "It is raining and it is not raining" is clearly false, so the statement "It can never happen that it is both raining and not raining." is a true statement.

There are equivalent to Aristotle's second and third "laws" of logic; however, there are so many more, and so to limit your self to Aristotelian logic really limits what you can actually discuss. Humorously, even Bertrand Russell suggests that the title "Laws" are a little over-the-top for such trivial logical statements. They were a start, but only a start, and form a very small part of logical thought.

(P ∧ Q) ↔ (Q ∧ P) and (P ∨ Q) ↔ (Q ∨ P)

If you try all combinations of P and Q being either true or false, you will find that the left-hand side always has the same truth value as the right-hand side, so this says that the order in which statements are made doesn't matter, just like x + y = y + x or xy = yx.

This is called the symmetry property of both the and (∧) and or (∨) operators.

(P ∧ (Q ∧ R)) ↔ ((P ∧ Q) ∧ R) and (P ∨ (Q ∨ R)) ↔ ((P ∨ Q) ∨ R)

This says that the statements P ∧ Q ∧ R and P ∨ Q ∨ R are unambiguous, so together with the previous statement, the order in which a sequence of propositional statements are joined with either and or or. This is the same as saying: if we are adding 20 numbers together, ti doesn't matter what order we add them in. So for example, in adding 2 + 11 + 10 + 6 + 3 + 8 + 1 + 14 + 9 + 7 + 5, you can do this by calculating (2 + 8) + 10 + (6 + 14) + (3 + 7) + (1 + 9) + 11 + 5 or 10 + 10 + 20 + 10 + 10 + 16 = 76. Similarly, "Alice is in grade 1, Bob is in grade 2 and Cayley is in grade 3" is equivalent to saying "Cayley is in grade 3, Alice is in grade 1 and Bob is in grade 2."

(P → Q) ↔ (¬P ∨ Q)

We have already discussed this one: P implies Q is true if and only if the statement P is false or Q is true.

P → P

This one seems trivial, the truth of P implies the truth of P. This is indeed, true, because "if P is true, then P is true," and "If P is false, then P is false" are both true statements. If you think this one is trivial, then combine this with the previous few and we get that: (P → P) ↔ (¬P ∨ P) and (¬P ∨ P) ↔ (P ∨ ¬P). That is, P → P is equivalent to saying P ∨ ¬P, so if you think P → P is obvious, then you should also be aware that P ∨ ¬P is obvious.

In that last example, we actually used a trick that we have not yet formalized:

(P ↔ Q) ∧ (Q ↔ R) → (P ↔ R)

This says is the logical equivalent of saying: if a = b and b = c, then a = c, so a = b = c is unambiguous, as is (P → P) ↔ (¬P ∨ P) ↔ (P ∨ ¬P).

¬¬P ↔ P

Another point is that ¬¬P ↔ P is always true.

(P → ¬P)→ ¬P

This is our first non-trivial and non-obvious statement. If P implies its own falsehood, then P must be false. This is also known as reductio ad absurdum, or the reduction to the absurd. This is used to show for example that there are infinitely many prime numbers. We will not go into the details but "There are infinitely many prime numbers" is equivalent to saying "There are not only a finite number of prime numbers." Thus, "It is false that there are infinitely many prime numbers" is the same as saying "It is false that there are only a finite number of prime numbers." This second statement is essentially of the form ¬¬P, so this second statement is the same as saying "There are only a finite number of prime numbers."

Now, if there are only a finite number of prime numbers, then there must be a fixed number n of prime numbers, so "there are only a finite number of prime numbers" is equivalent to saying "there are exactly n prime numbers". The problem is, if you assume that there are exactly n prime numbers, you then then show that there must be at least n + 1 prime numbers: that is, if you assume there are n prime numbers, you can construct one more, so there must be n+ 1 prime numbers. Thus, "There are exactly n prime numbers implies there are at least n + 1 prime numbers", or in other words "There are exactly n prime numbers implies that there are not exactly n prime numbers." Thus, from this we conclude that "there are exactly n prime numbers" is false. Thus, "there are only a finite number of prime numbers is false", so "there are infinitely many prime numbers" is true.

(¬P → P) → P

Thus, we also have (¬P → P) → P, or if a statement is assumed to be false implies that the statement is true, then the statement is true.

(P → Q) ∧ (Q → R) → (P → R)

This is the central tool of Aristotle's syllogisms: if P implies Q and Q implies R, then P implies R, too. For example, "Barb is over six foot tall implies Barb is over five foot tall", and "Barb is over five foot tall implies Barb is over four foot tall" are two statements of the form P → Q and Q → R, so it immediately follows that "Barb is over six foot tall implies Barb is over four foot tall." Another example is "A person being Greek implies that person is human" and "A person being human implies that person will die", so "A person being Greek implies that person will die."

A more useful example is "If an engineer commits fraud, then this reflects poorly on all engineers" and "if something reflects poorly on all engineers, then engineers may lose the trust of society." Consequently, "If an engineer commits fraud, then engineers may lose the trust of society" is an example of a syllogism: the third statement is a conclusion of the first two.

Quick summary so far

We have seen a number of different logical statements that are always true. Here is one that is not: (P → Q) ↔ (Q → P). As an obvious example, "If it is raining, there are clouds in the sky" is not logically equivalent to "If there are clouds in the sky, it is raining."

We'll continue with other tautologies.

(P → Q) ∧ (Q → P) ↔ (P ↔ Q)

This says that P and Q being logically equivalent (the right-hand side) is the same as saying "P implies Q" and "Q implies P".

(P → Q) ↔ (¬Q ↔ ¬P)

Recall how we said that (P → Q) ↔ (Q → P) is not a tautology? However P implies Q (or P being true implies Q is true) is logically equivalent to not Q implies not P, or in other words, Q being false implies that P is false. Coming back to our simple example, "If there are no clouds in the sky, then it is not raining" is equivalent to "If it is raining, then there are clouds in the sky."

Note, however, this can sometimes be useful to argue against an implication. For example, suppose that someone claims "If a student is good, then that student gets good grades." On the surface, this seems reasonable; however, it is much easier to argue against this if you reframe it as "If a student does not get good grades, then the student is not good." In particular, I know one student who was only completing this student's degree in order to get the piece of paper. This student was very focused on one aspect of computer engineering, and knew exactly what field of engineering this student wanted to practice in. The student's grades in most courses were in the 50s or 60s, and only those courses that interested the student saw the student expend any effort what-so-ever. The student also participated with many design project groups outside of the student's courses, and the student's contribution to all of these was significant. The student also had many engineering-related hobbies that saw the student design and implement many engineering solutions to real-world problems. So while this student did not get good grades, I would argue that this student was actually a very good student. Thus, this one counterexample demonstrates that the implication"if a student is good, then that student gets good grades" is false.

For those focused on Aristotelian logic, this author asks you to show how this can be shown using Aristotle's syllogisms or three so-called laws of logic.

This particular tautology is called the contrapositive.

(P ∧ (P → Q)) → Q

If P is definitely true, and P implies Q is a true implication, then Q must be true. This is more than an implication. For example, "It is raining, and if it is raining, there are clouds in the sky; therefore, there are clouds in the sky."

((P → Q) ∧ ¬Q) → ¬P

If P implies Q is a true implication, and Q is false, then P must also be false. "If it is raining, there are clouds in the sky, and yet there are no clouds in the sky; therefore, it cannot be raining."

¬(P ∧ Q) ↔ (¬P ∨ ¬Q)

For it to be false that P and Q are true, is equivalent to either P being false, or Q being false (or both).

¬(P ∨ Q) ↔ (¬P ∧ ¬Q)

For it to be false that P or Q are true, is equivalent to both P being false and Q being false.

¬(P → Q) ↔ (P ∧ ¬Q)

For it to be false that P implies Q is equivalent to saying that P is true and Q is false.

((P → Q) ∧ (R → S)) → ((P ∨ R) → (Q ∨ S))

If P implies Q and R implies S, then P or R being true implies that Q or S must be true. "If it is raining, there are clouds in the sky, and if it is cold, I will put on a sweater" all implies that "If it is raining or cold, then there are clouds in the sky or I'm putting on a sweater."

You may remember the distributive rule from arithmetic: a(b + c) = ab + ac, so multiplication distributes over addition. In logic, both logical and and logical or distribute over the other:

(P ∨ (Q ∧ R)) ↔ ((P ∨ Q) ∧ (P ∨ R))

(P ∧ (Q ∨ R)) ↔ ((P ∧ Q) ∨ (P ∧ R))

To be fair, this is not really all that useful in conversation: "It is cold or it is wet and raining" is equivalent to "It is cold or wet, and it is cold or raining." However, the fact that logical operators behave very similarly to that of arithmetic operators is interesting.

C. The universe of discourse and predicates

Before we move on to the next topic, when we are discussing a subject and applying logic, it is useful to restrict our discussion to a well defined collection or set of items. This set is described as the universe of discourse; that is, the totality of items being discussed. For example, we may be discussing people, people in a particular country or region, animals, books, armies, etc. For any one item in the set we are discussing, we will represent that item by a variable x, y or z. x could represent one specific item being discussed, or it may be representative of an arbitrary item being discussed.

Next, we will introduce the concept of a Boolean-valued predicate. Given an item x we are discussing, a Boolean-valued predicate is a statement about x that is either true or false. These predicates will be represented by functions F, G and H. For example, if our universe of discourse is all people, and our predicate F(x) is "x is taller than 6 foot." For each person, this statement is true or false. If it is true for x, we simply write F(x), and if the predicate is false for x, we write ¬F(x). For example, if x is this author, then ¬F(x) says "It is false that this author (represented by x) is over six foot tall", and if y is Jamie Teather, then F(y) says that "It is true that Jamie Teather (represented by y) is over six foot tall.

The predicate G(x) saying that "x is tall" is not Boolean-valued, as while some persons are definitely "tall", and others are "not tall", there are many in between who cannot be definitely described as being either "tall" or "not tall." Predicates can also take the place of propositions in any logical statements. As an obvious example, if the predicate F(x) says "x is over five foot tall", and the predicate G(x) says "x is over six foot tall", then G(x) → F(x); that is, if x is over six foot tall, then x is over five foot tall. We can also apply a syllogism: F(x) says "x is a Greek", and G(x) says "x is a human", and finally H(x) says "x is mortal." If you have the two implications: first, F(x) → G(x), which says "If x is Greek, then x is a human;" and second G(x) → H(x), which says "If x is human, then x is mortal." We immediately can apply our the syllogism: ( (F(x) → G(x)) ∧ (G(x) → H(x)) ) → (F(x) → H(x)), and thus we may conclude that "if x is Greek, then x is mortal." Similarly, F(x) → G(x) is equivalent to ¬G(x) → ¬F(x), so "If x is not human, then x is not Greek;" and similarly, G(x) → H(x) is equivalent to ¬H(x) → ¬G(x), so "If x is immortal (not mortal), then x is not human." Of course, we then have the syllogism ( (¬H(x) → ¬G(x)) ∧ (¬G(x) → ¬F(x)) ) → (¬H(x) → ¬F(x)), so we conclude "If x is immortal, then x is not Greek."

D. Universal and existential quantifiers

What is missing from our previous discussion of predicates is that we cannot say "All people are over two feet tall." This also allows us to describe, for example, "For all x such that F(x) is true, it follows that G(x) is also true," or as an implication, "For all x, if F(x) is true, then G(x) is true." We could write this as "For all x, F(x) → G(x)". Of course, mathematicians preferring a more succinct notation, replace the phrase "For All" with ∀ (yes, an up-side down "A"), so we could write ∀x:(F(x) → G(x)). Formally, this is called a universal quantifier, as it specifies that the logical statement that follows is true for all x in our universe of discourse. If it is ever true that there is an x for which the logical statement is false, then we say "It is false that all people are over six feet tall," and we may write this as ¬(∀x: F(x)).

Similarly, another aspect that is missing is the ability to say "There is a person who is taller than 7 feet." It was decided to describe this using the phrase "There exists an x such that F(x) is true." That is, within our universe of discourse, there is at least one x such that the predicate F(x) is true. If F(x) says that "x is over six foot tall" and G(x) says that "x weighs less than 200 lbs", then if we are discussing humans, we could say "There exists an x such that both F(x) and G(x) are true." As above, we would like to write this more succinctly, so we replace the phrase "There Exists" with the symbol ∃ (yes, a backwards "E"). Thus, we could write ∃x:(F(x) ∧ G(x)). This is called an existential quantifier, as it specifies that the logical statement that follows is true for at least one x within our universe of discourse. As before, if no x satisfies a logical statement, we could then write this as ¬(∃x: F(x)), or "It is false that there exists an x such that F(x) is true", or equivalently, "There does not exist an x such that F(x) is true." For example, if F(x) says that "x is taller than 10 feet" and our universe of discourse is all humans, then we could easily write ¬(∃x: F(x)).

¬(∃x: F(x)) ↔ ∀x: ¬F(x)

¬(∀x: F(x)) ↔ ∃x: ¬F(x)

We will now look at the relationship between these two quantifiers. You will notice that "There does not exist an x that is taller than 10 feet" is equivalent to (the same as) saying "For all x, x is taller than 10 feet is false," or "For all x, x is not taller than 10 feet." Thus, we may observe that ¬(∃x: F(x)) ↔ ∀x: ¬F(x). Similarly, we notice that "It is false that for all x, x is taller than six feet" is equivalent to saying "There exists an x such that x is not taller than six feet." This can be written as ¬(∀x: F(x)) ↔ ∃x: ¬F(x).

Please note, we are always assuming our universe of discourse is not empty, for if we are discussing mountains made of gold, the statement "it is false that all mountains of gold are taller than 2000 m" does not mean "it is true that there exists a mount of gold that is not taller than 2000 m". Because the "universe" of all mountains made of gold is empty (meaning, none exist), the equivalencies are nonsensical. This is why it is critical to establish that the universe of discourse as at least one object within it.

Going back to Aristotle

We can now reinterpret Aristotle's terms as follows:

All P are Q versus ∀x:(P(x) → Q(x))
No P are Q versus ∀x:(P(x) → ¬Q(x))
Some P are Q versus ∃x:(P(x) ∧ Q(x))
Some P are not Q versus ∃x:(P(x) ∧ ¬Q(x))

The modern version is more clear about hidden assumptions for the first two, as our universe of discourse is explicitly assumed to be non-empty. We also immediately observe that "Some P are Q" is actually equivalent to "Some Q are P", and "Some P are not Q" is equivalent to "Some that are not Q are P". We will later look at a few other equivalent statements.

∃x:(P(x) ∧ ¬Q(x)) ↔ ¬∀x:(P(x) → Q(x))

Note that we can make some observations. Recall the tautology ¬(P → Q) ↔ (P ∧ ¬Q). We see this in the last statement:

∃x:(P(x) ∧ ¬Q(x)) so ∃x:(P(x) ∧ ¬Q(x)) ↔ ∃x:(¬(P(x) → Q(x))), however, from above we have that the right-hand side is equivalent to ¬∀x:(P(x) → Q(x)). Thus, we have ∃x:(P(x) ∧ ¬Q(x)) ↔ ¬∀x:(P(x) → Q(x)), or "Some P are not Q is equivalent to saying it is false that for all x, P(x) implies Q(x) is true."

∃x:(P(x) ∧ Q(x)) ↔ ¬∀x:(P(x) → ¬Q(x))

If we take the same tautology ¬(P → Q) ↔ (P ∧ ¬Q) but replace Q with ¬Q, we get ¬(P → ¬Q) ↔ (P ∧ ¬¬Q) ↔ (P ∧ Q). In other words, if it is false that P implies not Q, then both P and Q must be true. Thus, "Some P are Q is equivalent to saying ¬∀x:(P(x) → ¬Q(x))

In summary, Aristotle four terms are essentially equivalent to the four logical statements

∀x:(P(x) → Q(x))
∀x:(P(x) → ¬Q(x))
¬∀x:(P(x) → ¬Q(x))
¬∀x:(P(x) → Q(x))

This demonstrates remarkable insight on the part of Aristotle and his colleagues. Also, as you may guess, you can also rephrase all of these in terms of the existential quantifier:

¬∃x:(P(x) ∧ ¬Q(x))
¬∃x:(P(x) ∧ Q(x))
∃x:(P(x) ∧ Q(x))
∃x:(P(x) ∧ ¬Q(x))

There are however, other equivalent statements: recall that (P → Q) ↔ (¬Q ↔ ¬P), so the first statement is the same as:

( ∀x:(P(x) → Q(x)) ) ↔ ( ∀x:(¬Q(x) → ¬P(x)) )

That is, "All P are Q" is equivalent to "All that are not Q are not P" or "No Q are not P". Similarly, we have

( ∀x:(P(x) → ¬Q(x)) ) ↔ ( ∀x:(¬¬Q(x) → ¬P(x)) ) ↔ ( ∀x:(Q(x) → ¬P(x)) )

Thus, "No P are Q" is equivalent to "All Q are not P".

Why is this important? Because there are many different ways of saying the same idea logically, but it is sometimes difficult to tell that they are indeed equivalent:

All P are Q, no Q are not P, it is false that some P are not Q, and it is false that some that are not Q are P.
All P are not Q, all Q are not P, it is false that some P are Q, and it is false that some Q are P.
Some P are Q, some Q are P, it is false that all P are not Q, and it is false that all Q are not P.
Some P are not Q, some that are not Q are P, it is false that all P are Q, and it is false that no Q are not P.

This is a much richer environment, and with these equivalencies, it is much easier to expand on what Aristotle began. Thus, Aristotle's example:

All Greeks are men, and
All men are mortal,
Thus, all Greeks are mortal.

is in fact equivalent to:

None that are not men are Greek, and
It is false that some who are men not mortal,
Thus, it is false that some who immortal are Greek.

When you are engaging others in discourse, you must be aware of such equivalencies. However, what is worse is that all of Aristotle's list of syllogisms can be derived from three syllogisms using quantifiers, and we will call these Case 1 through 4:

{ [∀x:(P(x) → Q(x))] ∧ [∀x:(Q(x) → R(x))] } → [∀x:(P(x) → R(x))]
{ [∃x:(P(x) ∧ Q(x))] ∧ [∀x:(Q(x) → R(x))] } → [∃x:(P(x) ∧ R(x))]
{ [∃x:Q(x)] ∧ [∀x:(Q(x) → P(x))] ∧ [∀x:(Q(x) → R(x))] } → [∃x:(P(x) ∧ R(x))]
{ [∀x:(Q(x) → P(x))] ∧ [∃x:(Q(x) ∧ R(x))] } → [∃x:(P(x) ∧ R(x))]

Aristotle actually includes one more, but it is a trivial consequence of Case 1 assuming at least one x satisfying P(x) exists:

{ [∃x:(P(x)] ∧ [∀x:(P(x) → Q(x))] ∧ [∀x:(Q(x) → R(x))] } → [∃x:(P(x) → R(x))]

We will now go through all of Aristotle's syllogisms and see how each of these are derived from three trivial logical tautologies. One completely unnecessary framework placed onto syllogisms by Aristotle et al. is the concept of a "major premise" and a "minor premise." Modern logic does not place any such significance to one or the other, as the ∧ operation is symmetric.

Barbara

Use Case 1:

{ [∀x:(P(x) → Q(x))] ∧ [∀x:(Q(x) → R(x))] } → [∀x:(P(x) → R(x))]

All Greeks are men, and all men are mortal; therefore all Greeks are mortal.

Barbari

This is the trivial consequence of Barbara: if all Greeks are mortal and a Greek exists, then some Greeks are mortal, too.

Bamalip

This is a trivial consequence of Barbara: if all Greeks are mortal and a Greek exists, then some mortals are Greek.

Celarent

Replace R(x) with ¬R(x) in Case 1:

{ [∀x:(P(x) → Q(x))] ∧ [∀x:(Q(x) → ¬R(x))] } → [∀x:(P(x) → ¬R(x))]

All snakes are reptiles, and no reptiles have fur; therefore no snake has fur.

Celaront

This is the trivial consequence of Celarent: if no snake has fur and a snake exists, then some snakes have no fur.

Calemos

This is the trivial consequence of Celarent: if no snake has fur, and a furred animal exists, then some furred animals are not snakes.

Calemes

Apply the contrapositive to the right-hand implication of Celarent:

{ [∀x:(P(x) → Q(x))] ∧ [∀x:(Q(x) → ¬R(x))] } → [∀x:(R(x) → ¬P(x))]

All snakes are reptiles, and no reptiles have fur; therefore furred animals are snakes.

Camestros

Apply the contrapositive to the second and third implication in Celarent:

{ [∀x:(P(x) → Q(x))] ∧ [∀x:(R(x) → ¬Q(x))] } → [∀x:(R(x) → ¬P(x))]

All snakes are reptiles, and no furred animals are reptiles; therefore all furred animals are not snakes.

Camestros is now a trivial consequence of this: if all furred animals are not snakes, and a furred animal exists, then some furred animals are not snakes.

Camestres

Replace R(x) with ¬R(x) in Case 1:

{ [∀x:(P(x) → Q(x))] ∧ [∀x:(Q(x) → ¬R(x))] } → [∀x:(P(x) → ¬R(x))]

Apply the contrapositive to the second and third implications:

{ [∀x:(P(x) → Q(x))] ∧ [∀x:(R(x) → ¬Q(x))] } → [∀x:(R(x) → ¬Q(x))]

All horses are hooved, and no human has hooves; therefore no human is a horse.

Camestros

This is the trivial consequence of Camestres: if no human is a horse, then some humans are not horses.

Cesare

Replace R(x) with ¬R(x) in Case 1:

{ [∀x:(P(x) → Q(x))] ∧ [∀x:(Q(x) → ¬R(x))] } → [∀x:(P(x) → ¬R(x))]

Apply the contrapositive to the second implication:

{ [∀x:(P(x) → Q(x))] ∧ [∀x:(R(x) → ¬Q(x))] } → [∀x:(P(x) → ¬R(x))]

All cows walk on four legs, and no human walks on four legs; therefore no cow is a human.

Cesaro

This is a trivial consequence of Cesare: if no cow is a human, than some cow is not a human.

Darii

Use Case 2:

{ [∃x:(P(x) ∧ Q(x))] ∧ [∀x:(Q(x) → R(x))] } → [∃x:(P(x) ∧ R(x))]

Some pets are rabbits, and all rabbits have fur; therefore some pets have fur.

Dimatis

This is a trivial consequence of Darii: if some pets have fur, then sum furred animals are pets.

Ferio

Replace R(x) with ¬R(x) in Case 2:

{ [∃x:(P(x) ∧ Q(x))] ∧ [∀x:(Q(x) → ¬R(x))] } → [∃x:(P(x) ∧ ¬R(x))]

Some reading is homework, and no homework is fun; therefore some reading is not fun.

Baroco

Replace Q(x) with ¬Q(x) and R(x) with ¬R(x) in Case 2:

{ [∃x:(P(x) ∧ ¬Q(x))] ∧ [∀x:(¬Q(x) → ¬R(x))] } → [∃x:(P(x) ∧ ¬R(x))]

Some pets are not mammals, and all non mammals are not cats; therefore some pets are not cats.

Next, apply the contrapositive (R(x) → Q(x)) ↔ (¬Q(x) → ¬R(x))

{ [∃x:(P(x) ∧ ¬Q(x))] ∧ [∀x:(R(x) → Q(x))] } → [∃x:(P(x) ∧ ¬R(x))]

Some pets are not mammals, and all cats are mammals; therefore some pets are not cats.

Bocardo

Replace P(x) with ¬P(x) in Case 2:

{ [∃x:(¬P(x) ∧ Q(x))] ∧ [∀x:(Q(x) → R(x))] } → [∃x:(¬P(x) ∧ R(x))]

Some non-pets are cats, and all cats are animals; therefore some non-pets are mammals.

Next, swap the order of both ∧ operations:

{ [∃x:(Q(x) ∧ ¬P(x))] ∧ [∀x:(Q(x) → R(x))] } → [∃x:(R(x) ∧ ¬P(x))]

Some cats are not pets, and all cats are mammals; therefore some mammals are not pets.

Darapti

Use Case 3.

{ [∃x:Q(x)] ∧ [∀x:(Q(x) → P(x))] ∧ [∀x:(Q(x) → R(x))] } → [∃x:(P(x) ∧ R(x))]

Felapton

Replace P(x) with ¬P(x) in Case 3:

{ [∃x:Q(x)] ∧ [∀x:(Q(x) → ¬P(x))] ∧ [∀x:(Q(x) → R(x))] } → [∃x:(¬P(x) ∧ R(x))]

Next, apply symmetry to the right-hand and (∧) operation:

{ [∃x:Q(x)] ∧ [∀x:(Q(x) → ¬P(x))] ∧ [∀x:(Q(x) → R(x))] } → [∃x:(R(x) ∧ ¬P(x))]

No snake has fur, and all snakes are reptiles; therefore, if a snake exists, some reptiles are do not have fur.

Fesapo

Apply the contrapositive to the first implication in Felapton:

{ [∃x:Q(x)] ∧ [∀x:(P(x) → ¬Q(x))] ∧ [∀x:(Q(x) → R(x))] } → [∃x:(R(x) ∧ ¬P(x))]

No furred animal is a snake, and all snakes are reptiles; therefore, if a snake exists, some reptiles are do not have fur.

Datisi

Use Case 4 but apply symmetry to the right-hand side:

{ [∀x:(Q(x) → P(x))] ∧ [∃x:(Q(x) ∧ R(x))] } → [∃x:(R(x) ∧ P(x))]

All cows eat grass, and some cows are brown; therefore some brown animals eat grass.

Disamis

Apply symmetry to the left-hand and (∧) operation and relabel P and R in Datisi:

{ [∃x:(Q(x) ∧ P(x))] ∧ ∀x:(Q(x) → R(x))] } → [∃x:(P(x) ∧ R(x))]

Next, apply symmetry to the right-hand and (∧) operation:

{ [[∃x:(Q(x) ∧ P(x))] ∧ ∀x:(Q(x) → R(x))] } → [∃x:(R(x) ∧ P(x))]

Some cows are brown, and all cows eat grass; therefore some animals that eat grass are brown.

Festino

Replace P(x) with ¬P(x) in Case 4:

{ [∀x:(Q(x) → ¬P(x))] ∧ [∃x:(Q(x) ∧ R(x))] } → [∃x:(¬P(x) ∧ R(x))]

Apply the contrapositive to the first implication and symmetry to the two and (∧) operation:

{ [∀x:(P(x) → ¬Q(x))] ∧ [∃x:(R(x) ∧ Q(x))] } → [∃x:(R(x) ∧ ¬P(x))]

No pet is wild, and some roadrunners are wild; thus some roadrunners are not pets.

Ferison

Replace P(x) with ¬P(x) in Case 4:

{ [∀x:(Q(x) → ¬P(x))] ∧ [∃x:(Q(x) ∧ R(x))] } → [∃x:(¬P(x) ∧ R(x))]

Apply symmetry for the and (∧) operation on the right-hand side:

{ [∀x:(Q(x) → ¬P(x))] ∧ [∃x:(Q(x) ∧ R(x))] } → [∃x:(R(x) ∧ ¬P(x))]

No wild animal is a pet, and some wild animals are roadrunners; thus some roadrunners are not pets.

Freisison

Apply symmetry to ∃x:(R(x) ∧ Q(x)) of Festino:

{ [∀x:(P(x) → ¬Q(x))] ∧ [∃x:(Q(x) ∧ R(x))] } → [∃x:(R(x) ∧ ¬P(x))]

No pet is wild, and some wild animals are roadrunners; thus some roadrunners are not pets.

We have just gone through 24 of Aristotle's syllogisms, and demonstrated how they may be deduced a very small number of rules. You may note, however, that it is hardly complete. For example, often we had intermediate steps that were proved but not included in Aristotle's list.

I will now describe what I think should be taught:

Case 1. { [∃x:(P(x)] ∧ [∀x:(P(x) → Q(x))] ∧ [∀x:(Q(x) → R(x))] } → {[∀x:(P(x) → R(x))] ∧ [∃x:(P(x) ∧ R(x))]}

If there exists at least one x such that P(x) is true,

and if for all x such that P(x) is true, then Q(x) is also true,

and if for all x such that Q(x) is true, then R(x) is also true,

then

for all x such that P(x) is true, then R(x) is also true, and

there exists an x such that both P(x) and R(x) are true.

You will note that the right-hand side summarizes all three of Aristotle's consequences:

All P are R.
Some P are R.
Some R are P.

Case 2: { [∃x:(P(x) ∧ Q(x))] ∧ [∀x:(Q(x) → R(x))] } → [∃x:(P(x) ∧ R(x))]

If there exists an x such that both P(x) and Q(x) are true,

and if for all x such that Q(x) is true, then R(x) is also true,

then

there exists an x such that both P(x) and R(x) are true.

Case 3: { [∃x:Q(x)] ∧ [∀x:(Q(x) → P(x))] ∧ [∀x:(Q(x) → R(x))] } → [∃x:(P(x) ∧ R(x))]

If there exists an x such that Q(x) is true,

and if for all x such that Q(x) is true, then P(x) is also true,
and if for all x such that Q(x) is true, then R(x) is also true,

then

there exists an x such that P(x) and R(x) are true.

Case 4: { [∃x:(Q(x) ∧ P(x))] ∧ [∀x:(Q(x) → R(x))] } → [∃x:(P(x) ∧ R(x))]

If there exists an x such that both Q(x) and P(x) are true,

and if for all x such that Q(x) is true, then R(x) is also true,

then

there exists an x such that P(x) and R(x) are true.

Because we have simply stated ∃x:(F(x) ∧ G(x)), there is no need to repeat ourselves unnecessarily. Because the and operator is symmetric, it always follows that [∃x:(P(x) ∧ R(x))] ↔ [∃x:(R(x) ∧ P(x))], so while Aristotle repeated himself unnecessarily, we do not.

Substituting F(x) with ¬F(x)

Also, we can always replace any F(x) with ¬F(x), so Case 1 actually describes eight different cases:

{ [∃x:(P(x)] ∧ [∀x:(P(x) → Q(x))] ∧ [∀x:(Q(x) → R(x))] } → {[∀x:(P(x) → R(x))] ∧ [∃x:(P(x) ∧ R(x))]}

There exists a Greek, and all Greek are human, and all humans are mortal,

so all Greeks are mortal and there exists something that is both Greek and mortal.

{ [∃x:(¬P(x)] ∧ [∀x:(¬P(x) → Q(x))] ∧ [∀x:(Q(x) → R(x))] } → {[∀x:(¬P(x) → R(x))] ∧ [∃x:(¬P(x) ∧ R(x))]}

There exists a non-Greek person, and all non-Greek persons are human, and all humans are mortal,

so all non-Greek persons are mortal and there exists something that is both non-Greek and mortal.

{ [∃x:(P(x)] ∧ [∀x:(P(x) → ¬Q(x))] ∧ [∀x:(¬Q(x) → R(x))] } → {[∀x:(P(x) → R(x))] ∧ [∃x:(P(x) ∧ R(x))]}

There exists a Greek, and all Greek are not gods, and all who are not gods are mortal,

so all Greeks are mortal and there exists something that is both Greek and mortal.

{ [∃x:(¬P(x)] ∧ [∀x:(¬P(x) → ¬Q(x))] ∧ [∀x:(¬Q(x) → R(x))] } → {[∀x:(¬P(x) → R(x))] ∧ [∃x:(¬P(x) ∧ R(x))]}

There exists a non-Greek person, and all non-Greek persons are not gods, and all who are not gods are mortal,

so all non-Greek persons are mortal and there exists something that is both non-Greek and mortal.

{ [∃x:(P(x)] ∧ [∀x:(P(x) → Q(x))] ∧ [∀x:(Q(x) → ¬R(x))] } → {[∀x:(P(x) → ¬R(x))] ∧ [∃x:(P(x) ∧ ¬R(x))]}

There exists a Greek, and all Greek are human, and all humans are not immortal,

so all Greeks are not immortal and there exists something that is both Greek and not immortal.

{ [∃x:(¬P(x)] ∧ [∀x:(¬P(x) → Q(x))] ∧ [∀x:(Q(x) → ¬R(x))] } → {[∀x:(¬P(x) → ¬R(x))] ∧ [∃x:(¬P(x) ∧ ¬R(x))]}

There exists a non-Greek person, and all non-Greek persons are human, and all humans are not immortal,

so all non-Greek persons are not immortal and there exists something that is both non-Greek and not immortal.

{ [∃x:(P(x)] ∧ [∀x:(P(x) → ¬Q(x))] ∧ [∀x:(¬Q(x) → ¬R(x))] } → {[∀x:(P(x) → ¬R(x))] ∧ [∃x:(P(x) ∧ ¬R(x))]}

There exists a Greek, and all Greek are not gods, and all who are not gods are not immortal,

so all Greeks are not immortal and there exists something that is both Greek and not immortal.

{ [∃x:(¬P(x)] ∧ [∀x:(¬P(x) → ¬Q(x))] ∧ [∀x:(¬Q(x) → ¬R(x))] } → {[∀x:(¬P(x) → ¬R(x))] ∧ [∃x:(¬P(x) ∧ ¬R(x))]}

There exists a non-Greek person, and all non-Greek persons are not gods, and all who are not gods are not immortal,

so all non-Greek persons are not immortal and there exists something that is both non-Greek and not immortal.

Applying the contrapositive

Any implication can be replaced by its contrapositive, so there are eight different

{ [∃x:(P(x)] ∧ [∀x:(P(x) → Q(x))] ∧ [∀x:(Q(x) → R(x))] } → {[∀x:(P(x) → R(x))] ∧ [∃x:(P(x) ∧ R(x))]}

There exists a Greek, and all Greek are human, and all humans are mortal,

so all Greeks are mortal and there exists something that is both Greek and mortal.

{ [∃x:(P(x)] ∧ [∀x:(¬Q(x) → ¬P(x))] ∧ [∀x:(Q(x) → R(x))] } → {[∀x:(P(x) → R(x))] ∧ [∃x:(P(x) ∧ R(x))]}

There exists a Greek, and all non-humans are not Greek, and all humans are mortal,

so all Greeks are mortal and there exists something that is both Greek and mortal.

{ [∃x:(P(x)] ∧ [∀x:(P(x) → Q(x))] ∧ [∀x:(¬R(x) → ¬Q(x))] } → {[∀x:(P(x) → R(x))] ∧ [∃x:(P(x) ∧ R(x))]}

There exists a Greek, and all Greek are human, and all immortals are not human,

so all Greeks are mortal and there exists something that is both Greek and mortal.

{ [∃x:(P(x)] ∧ [∀x:(¬Q(x) → ¬P(x))] ∧ [∀x:(¬R(x) → ¬Q(x))] } → {[∀x:(P(x) → R(x))] ∧ [∃x:(P(x) ∧ R(x))]}

There exists a Greek, and all non-humans are not Greek, and all immortals are not human,

so all Greeks are mortal and there exists something that is both Greek and mortal.

{ [∃x:(P(x)] ∧ [∀x:(P(x) → Q(x))] ∧ [∀x:(Q(x) → R(x))] } → {[∀x:(¬R(x) → ¬P(x))] ∧ [∃x:(P(x) ∧ R(x))]}

There exists a Greek, and all Greek are human, and all humans are mortal,

so all immortals are non-Greeks and there exists something that is both Greek and mortal.

{ [∃x:(P(x)] ∧ [∀x:(¬Q(x) → ¬P(x))] ∧ [∀x:(Q(x) → R(x))] } → {[∀x:(¬R(x) → ¬P(x))] ∧ [∃x:(P(x) ∧ R(x))]}

There exists a Greek, and all non-humans are not Greek, and all humans are mortal,

so all Greeks are mortal and there exists something that is both Greek and mortal.

{ [∃x:(P(x)] ∧ [∀x:(P(x) → Q(x))] ∧ [∀x:(¬R(x) → ¬Q(x))] } → {[∀x:(¬R(x) → ¬P(x))] ∧ [∃x:(P(x) ∧ R(x))]}

There exists a Greek, and all Greek are human, and all immortals are not human,

so all immortals are non-Greeks and there exists something that is both Greek and mortal.

{ [∃x:(P(x)] ∧ [∀x:(¬Q(x) → ¬P(x))] ∧ [∀x:(¬R(x) → ¬Q(x))] } → {[∀x:(¬R(x) → ¬P(x))] ∧ [∃x:(P(x) ∧ R(x))]}

There exists a Greek, and all non-humans are not Greek, and all immortals are not human,

so all immortals are non-Greeks and there exists something that is both Greek and mortal.

As a quick summary, the richness of the possible syllogisms described by our four cases, together with applications of either the substitution of a predicate with its negation (replacing F(x) with ¬F(x)) or applying the contrapositive give us a much greater richness than what is provided with Aristotle. Additionally, some may try to claim there is "too much" here, but we must remember that Aristotle, in a similar fassion, provides far too much, for instead of saying "P and R overlap", we must have one syllogism when the consequence is "Some P are R", and then another syllogism has the consequence "Some R are P".

E. Set theory

The other new concept is that of a set. A set is a collection of unique objects. Given two sets, one may ask what is the union of the two sets; that is, what is the set containing all objects that are in either of the two sets. Similarly, given two sets, one may ask what is the intersection of the two sets; that is, what is the set

Case 1. { (P ≠ Ø) ∧ (P ⊆ Q) ∧ (Q ⊆ R) } → {(P ⊆ R) ∧ [(P ∩ R) ≠ Ø]}

Case 2: { [(P ∩ Q) ≠ Ø] ∧ (P ⊆ R) } → [(Q ∩ R) ≠ Ø]

Case 3: { (P ≠ Ø) ∧ (Q ⊆ P) ∧ (Q ⊆ R) } → [(P ∩ R) ≠ Ø]

Case 4: { [(Q ∩ P) ≠ Ø] ∧ (Q ⊆ R) } → [(P ∩ R) ≠ Ø]

We could go through all of Aristotle's syllogisms and see how to rewrite them in terms of set theoretic terms, but this is not worth the time. Instead, we will see how we can deduce other expressions that are true based on the above four cases. As before, if we are discussing only one object in Q, we can abbreviate the first and third cases:

Case 1'. { (p ∈ Q) ∧ (Q ⊆ R) } → {(p ∈ R) ∧ [(P ∩ R) ≠ Ø]}

Case 3': { (q ∈ P) ∧ (q ∈ R) } → [(P ∩ R) ≠ Ø]

Note also that (Q ∩ P) ≠ Ø when Q = {q} (that is, Q is a set containing just a single object), then q ∈ P is equivalent to (Q ∩ P) ≠ Ø, so the second and fourth simplify to the same idea:

Case 2': { [p ∈ Q] ∧ (p ∈ R) } → [(Q ∩ R) ≠ Ø]

Case 4': { [q ∈ P] ∧ (q ∈ R) } → [(P ∩ R) ≠ Ø]

Any set P can be replaced by its complement; that is, all objects that are not in the set P; so we can replace P with P'. The equivalence of the contrapositive is the statement (P ⊆ Q) ↔ (Q' ⊆ P'); that is, P is a subset of Q if and only if the complement of Q is a subset of the complement of P. This has much greater succinctness than either the use of logic and quantifiers described above, and Aristotle's list of syllogisms.

One elegant aspect is that all of the rules of logic have analogous rules for set theory. In the following, 𝒰 is our universe of discourse.

P ∨ ¬P

P ∪ P' = 𝒰

¬(P ∧ ¬P)

(P ∩ P') = 𝒰

(P ∧ Q) ↔ (Q ∧ P) and (P ∨ Q) ↔ (Q ∨ P)

(P ∩ Q) = (Q ∩ P) and (P ∪ Q) = (Q ∪ P)

(P ∧ (Q ∧ R)) ↔ ((P ∧ Q) ∧ R) and (P ∨ (Q ∨ R)) ↔ ((P ∨ Q) ∨ R)

(P ∩ (Q ∩ R)) = ((P ∩ Q) ∩ R) and (P ∪ (Q ∪ R)) = ((P ∪ Q) ∪ R)

(P → Q) ↔ (¬P ∨ Q)

P ⊆ Q ↔ P' ∪ Q = 𝒰

P → P

P ⊆ P

(P ↔ Q) ∧ (Q ↔ R) → (P ↔ R)

P = Q ∧ Q = R → P = R

¬¬P ↔ P

P'' = P

(P → ¬P)→ ¬P

P ⊆ P' → P = Ø

(¬P → P) → P

P' ⊆ P → P = 𝒰

(P → Q) ∧ (Q → R) → (P → R)

(P ⊆ Q) ∧ (Q ⊆ R) → (P ⊆ R)

(P → Q) ∧ (Q → P) ↔ (P ↔ Q)

[(P ⊆ Q) ∧ (Q ⊆ P)] ↔ (P = Q)

(P → Q) ↔ (¬Q ↔ ¬P)

(P ⊆ Q) ↔ (Q' ⊆ P')

(P ∧ (P → Q)) → Q

(P ≠ Ø ∧ (P ⊆ Q)) → Q ≠ Ø

((P → Q) ∧ ¬Q) → ¬P

((P ⊆ Q) ∧ Q ≠ 𝒰) → P ≠ 𝒰

¬(P ∧ Q) ↔ (¬P ∨ ¬Q)

(P ∩ Q)' = P' ∪ Q'

¬(P ∨ Q) ↔ (¬P ∧ ¬Q)

(P ∪ Q)' = P' ∩ Q'

¬(P → Q) ↔ (P ∧ ¬Q)

P ⊈ Q ↔ P ∩ Q' ≠ Ø

((P → Q) ∧ (R → S)) → ((P ∨ R) → (Q ∨ S))

((P ⊆ Q) ∧ (R ⊆ S)) → ((P ∪ R) ⊆ (Q ∪ S))

(P ∨ (Q ∧ R)) ↔ ((P ∨ Q) ∧ (P ∨ R))

(P ∪ (Q ∩ R)) = ((P ∪ Q) ∩ (P ∪ R))

(P ∧ (Q ∨ R)) ↔ ((P ∧ Q) ∨ (P ∧ R))

(P ∩ (Q ∪ R)) = ((P ∩ Q) ∪ (P ∩ R))

Not only does set theory incorporate many of the ideas of logic, it also allows us to very succinctly write our expressions: much more clearly than that of Aristotle's syllogisms.

The point?

Hopefully the above description has demonstrated three points: First, Aristotelian logic was a good first start: it wasn't ideal or any way near complete, but it was never-the-less, a good start, just like Roman numerals and Euclidean geometry were both good first starts at counting and the axiomatic methods, respectively. Second, Aristotelian logic, however, is only one small aspect of concepts such as equality, logic, and set theory, and having mastered only Aristotelian logic, a person can there can hardly be considered to have any mastery at all of concepts such as equality, logic and set theory. Indeed, Aristotelian logic really only describes one small aspect of propositional logic. Third, and finally, there really is no benefit of teaching any aspect of Aristotelian logic: it is an archaic first attempt at ideas that today we understand as equality, logic, and set theory; and to start with Aristotelian logic would be equivalent to teaching Roman numerals to count first, just because Roman numerals came before Arabic numerals.

Final comment

I first learned about Boolean logic from Profs. Ernest Frejer and Dave Calvert, both of whom had their own approaches to the subject matter, the first in the context of set theory, and the second in terms of computer science. After taking Dave's course, I became interested in reviewing Aristotelian logic. There simply was no comparison, and it seemed almost trivial and unexpressive when compared to the power of explicitly defining what it means for an operation to be considered an equality, what the logical operators are, existential and universal quantifiers, and ultimately set theory (without the Axiom of Choice). There is no analogous concept as a "major premise" or "minor premise" in modern logical theories, as there is no need for them. There is so much more to logic than Aristotle's syllogisms, and the so-called "laws of logic" are actionally nothing more than some of the most trivial tautologies. The "laws of logic" do not define logical theory, but rather are consequences of the definitions. It is frustrating when modern individuals, especially hosts of shows, focus so entirely on Aristotelian logic, suggesting there is value to even learning what they teach. They teach nothing of value, and require the expenditure of significant amounts of time. Additionally, what is most humorous is if you go to the Wikipedia page on Aristotelian logic, they go out of their way to try to demonstrate that Aristotelian logic is deep an inciteful, with images such as this attempt to show what Darii means. However, much more damning is that the only significant institution that continues to use Aristotelian logic today is the Catholic Church, sub-organizations of which refuse to acknowledge any other form of logic. One only need read the text George Hayward Joyce's "Principles of Logic." For anyone outside the Catholic Church to give any credence to Aristotelian logic (beyond what it is: a really good first attempt at understanding logic), this gives them credence they do not deserve.