Johnston diagrams, which look similar to Euler or Venn diagrams, illustrate formal propositional logic in a visual manner. Logically they are equivalent to truth tables; some may find them easier to understand at a glance. By overlaying one Johnston diagram on another, deductions can be made from sets of propositions. Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler [oilər] (April 15, 1707 - September 18, 1783) was a Swiss mathematician and physicist. ... Venn diagrams, Euler diagrams (pronounced oiler) and Johnston diagrams are similar-looking illustrations of set, mathematical or logical relationships. ... The propositional calculus is a formal deduction system whose atomic formulas are propositional variables. ... Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ...

Suppose that it is desired to compose logical statements describing the present state of current events in the world (or perhaps about imaginary situations in an imaginary world). Let the universal set contain (as elements) all the possible states which the world might find itself in. Only one of a variety (perhaps infinite) of elements represents the actual state of the world. All other elements represent alternative states of the world — "possible worlds". Thus, the universal set represents the space of all logical possibilities. Logic (from ancient Greek λόγος (logos), originally meaning the word, or what is spoken, but coming to mean thought or reason) is the study of arguments. ... In mathematics, and particularly in applications to set theory and the foundations of mathematics, a universe or universal class (or if a set, universal set) is, roughly speaking, a class that is large enough to contain (in some sense) all of the sets that one may wish to use. ...

Then, the objective of a logical statement should be to say something about the state of the actual world. The way this will be done — using Johnston diagrams — is to blacken out regions of the universal set which contain elements which represent alternative states of the world which could not possibly be the state of the actual world.

So black regions on a Johnston diagram are "regions of impossibility", whereas white regions are "regions of possibility": one (and only one) of the elements in the regions of possibility describes the "world" as it actually is. The objective is to narrow down the region of possibility as much as possible, up to a single point which describes reality. Black is a color with several subtle differences in meaning. ... White is a color (more accurately it contains all the colors of the spectrum and is sometimes described as an achromatic color—black is the absence of color) that has high brightness but zero hue. ...

Let the universal set be represented by a rectangle. Start out by drawing a closed curve (e.g. a circle) inside the universal set. The circle separates the universal set into a pair of regions. Let the circle be called A. Points inside or on the circle are members of A; points outside the circle are not members of A, but are members of , the complement of A. In geometry, a rectangle is a defined as a quadrilateral polygon in which all four angles are right angles. ... In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, called the centre. ... The word point can refer to: a location in physical space a unit of angular measurement; see navigation point is a typographic unit of measure in typography equal inch or sometimes approximated as inch; on computer displays it should be equal to point in typography if the correct display resolution... In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ...

Now let the region of the complement of A be blackened out (see Figure 1).

Figure 1. Johnston diagram representing the statement "A is true".

Then the region of possibility has become equivalent to set A, so Figure 1 is a Johnston diagram representing the propositional statement A. Johnston diagram representing the statement A is true. This image was made by AugPi. ...

But if, instead, the region inside A is blackened and the region outside it whitened, then the region of possibility will be equivalent to the complement of A (see Figure 2) and the diagram will represent the propositional statement : "not A".

Figure 2. Johnston diagram representing the statement "A is not true".

Draw another circle — intersecting the first circle — and call it B. Points inside this second circle are members of B, and points outside it are members of . Johnston diagram representing the logical statement A is not true. ...

If the region inside B is whitened and the region outside it is blackened (see Figure 3), the resulting diagram is equivalent to the statement B,

Figure 3. Johnston diagram representing the statement: "B is true".

but if the region inside B is blackened and the region outside it is whitened (see Figure 4), the resulting diagram is equivalent to the statement ("not B"). Johnston diagram representing the logical statement B is true. This image was made by AugPi. ...

Figure 4. Johnston diagram representing the statement: "B is not true".

A pair of statements can be combined by means of the logical AND operator. To combine a pair of Johnston diagrams using the AND operator, superpose them so that elements (points) that end up on top of each other (in the superposition) are identically equivalent and represent the same possible state of the world. Johnston diagram representing the statement B is not true. This image was made by AugPi. ... AND Logic Gate Logical conjunction (usual symbol and) is a logical operator that results in true if both of the operands are true. ...

Then blacken out the combined diagram as follows: if a point belongs to the impossibility space of at least one of the two component statements, then it belongs to the impossibility space of both statements. So, combining Figures 1 and 3 by means of the AND operator produces Figure 5, equivalent to the propositional statement ("A and B"), and Figure 5's possibility space is the set ("A intersection B"). In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ...

Figure 5. Johnston diagram representing the statement: "Both A and B are true."

A pair of statements can also be combined by means of the logical OR operator. To do so, superpose their Johnston diagrams, and blacken out the combined diagrams as follows: if a point belongs to the impossibility spaces of both component diagrams, then it belongs to the impossibility space of the combined diagram. Otherwise, if it belongs to at least one component possibility space, then it belongs to the combined possibility space. Johnston diagram representing the logical statement A and B are both true. This image was made by AugPi. ... OR Logic Gate Logical disjunction (usual symbol or) is a logical operator that results in true if either of the operands is true. ...

So, combining Figures 1 and 3 by means of the OR operator produces Figure 6, equivalent to the propositional statement ("A or B"), and Figure 6's possibility space is the set ("A union B"). In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...

Figure 6. Johnston diagram representing the statement: "A or B is true." (Either A or B (or both) are true.)

It is also possible to apply the logical NOT operator to a Johnston diagram to obtain its negation. To do so, swap the possibility and impossibility spaces of the given diagram. This means to whiten black regions while simultaneously blackening white regions. The resulting diagram will represent a statement which negates the statement represented by the original diagram. Johnston diagram representing the statement A or B is true. ... Negation, in its most basic sense, changes the truth value of a statement to its opposite. ... Negation, in its most basic sense, changes the truth value of a statement to its opposite. ...

As an example, applying the NOT operator to Figure 1 yields Figure 2: statement A becomes statement . Another example is to apply the NOT operator to Figure 6, obtaining Figure 7 whose impossibility space is the set and whose impossibility space is the set , and which represents the logical statement which is equivalent — due to De Morgan's law — to the statement ("not A and not B"). In logic, De Morgans laws (or De Morgans theorem) are the two rules of propositional logic, boolean algebra and set theory not (P and Q) = (not P) or (not Q) not (P or Q) = (not P) and (not Q) which allow us to move a negation over a...

Figure 7. Johnston diagram representing the statement "Neither A nor B is true".

Notice that Figure 7 can also be obtained by combining Figures 2 and 4 by means of the AND operator. Johnston diagram representing the logical statement Not A and Not B. This image was made by AugPi. ...

Statements A and B can also be combined to form the statement ("A implies B"). To represent this with a Johnston diagram, let its possibility space be equivalent to the set . Thus, the statement can be represented by combining Figures 2 and 3 by means of the OR operator. The result is shown in Figure 8, viz.

Figure 8. Johnston diagram representing the statement "A implies B" or "if A then B" or "A is true only if B is true."

By looking at Figure 8 one can clearly see that IF the actual state of the world is described by a member of set A, THEN this member also belongs to set B (the "actual world" can only lie within the possibility space shown in white). Johnston diagram representing the logical statement A implies B. This image was made by AugPi. ...

Similarly, statements A and B can be combined to form the statement ("B implies A"). The Johnston diagram for this statement must have a possibility space equivalent to the set . Thus, the statement can be represented by combining Figures 4 and 1 by means of the OR operator. The result is shown in Figure 9, viz.

Figure 9. Johnston diagram representing the statement "B implies A" or "if B then A" or "A is true if B is true."

Alternatively, the set in Figure 9 can be expressed as : the complement of the subtraction of A from B. Johnston diagram representing the logical statement B implies A. This image was made by AugPi. ...

Finally, the pair of statements and can be combined into the single statement ("A if and only if B"). The corresponding Johnston diagram can be formed by combining Figures 8 and 9 by means of the AND operator, resulting in Figure 10, viz.

Figure 10. Johnston diagram representing the statement "A is true if and only if B is true" or "A is equivalent to B".

The possibility space of this Johnston diagram is the set Johnston diagram representing the logical statement A is true if and only if B is true. ...

or, equivalently, the set

i.e. the complement of the symmetric difference between A and B. In mathematics, the symmetric difference of two sets is the set of elements which are in one of either set, but not in both. ...

Then there are two relatively trivial cases: the tautology and the contradiction. The tautology is the statement whose Johnston diagram has no black region of impossibility: it is all white, and its region of possibility is equivalent to the universal set. Every axiom of logic must necessarily be a tautology. A tautology does not say anything about the state of the actual world, because tautologies are true in all the possible worlds — the actual and all its alternatives. It says nothing about the contingent state of affairs in the actual world. Tautologies are either self-evident (axioms) or can be deduced (as theorems) from other tautologies. Thus, all tautologies can be deduced a priori, but the contingent state of the actual world can only be obtained a posteriori through observation. In logic, a tautology is a statement which is true by its own definition, and is therefore fundamentally uninformative. ... Broadly speaking, a contradiction is when two or more statements, ideas, or actions are seen as incompatible. ... In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ... A theorem is a statement which can be proven true within some logical framework. ... A priori is a Latin phrase meaning from the former or less literally before experience. In much of the modern Western tradition, the term a priori is considered to mean propositional knowledge that can be had without, or prior to, experience. ... Empirical or a posteriori knowledge is propositional knowledge obtained by experience. ...

An example of a tautology can be obtained by combining Figures 1 and 2 by means of the OR operator (see Figure 11).

Figure 11. Johnston diagram representing the statement "Either A is true or A is not true."

This corresponds to the axiom of (classical) propositional calculus ("A or not A"), which is called tertium non datur ("a third [possibility] is not given"). Johnston diagram representing a tautology. ...

On the other hand, the contradiction is the statement whose Johnston diagram is all black: its impossibility region is equivalent to the universal set, and its possibility region is the empty set. A contradiction says too much. In fact, a contradiction is the most one can ever say: a contradiction ANDed to any other statement produces a contradiction, but it can never be true, because the world does exist, and it has a state, which is its actual state. At least one element in the universal set must describe the actual world, so the region of possibility cannot be null.

A contradiction can be obtained by combining Figures 1 and 2 by means of the AND operator (see Figure 12).

Figure 12. Johnston diagram representing the contradictory statement "A is true but A is not true."

This corresponds to the contradictory statement ("A and not A"), which is the negation of the tautology . The negation of every tautology is a contradiction. This suggests a method of proof called reductio ad absurdum: to prove a theorem, assume its negation, then show that it leads somehow to contradiction. Once the contradiction has been reached, the proof is finished: enough said. Johnston diagram representing a contradiction. ... Reductio ad absurdum (Latin for reduction to the absurd, traceable back to the Greek ἡ εις το αδυνατον απαγωγη, reduction to the impossible, often used by Aristotle) is a type of logical argument where we assume a claim for the sake of argument, arrive at an absurd result, and then...

In summary, a Johnston diagram is a way of representing logical statements (of propositional calculus) by means of sets. Thus, logical operators can be transformed into set operations, using the following table: In logical calculus, logical operators or logical connectors serve to connect statements into more complicated compound statements. ...

It is also possible to, in like manner, transform inferences into logical statements involving sets, viz. This article is primarily concerned with truth as it is used in the evaluation of propositions, sentences, and similar items. ... Logic (from ancient Greek λόγος (logos), meaning reason) is the study of arguments. ...

Johnston visualization can also be applied to inference rules. An inference rule always has two premises and one conclusion, and can be represented generically as In logic, especially in mathematical logic, a rule of inference is a scheme for constructing valid inferences. ...

where P₁ and P₂ are the premises and C is the conclusion. This inference rule transforms into the statement

where P₁, P₂ and C have become sets. For any such sets, the following statements are always true:

∴

To each logical statement corresponds a "possibility set", namely the set which is equivalent to the region of possibility in the Johnston diagram of the statement. One may say that the amount of information contained by a statement is — roughly speaking — inversely proportional to the size of the statement's possibility set. (Then the information contained by a contradiction would be infinite; however, such information would never be obtained, as a contradiction is unprovable) Information theory is a branch of the mathematical theory of probability and mathematical statistics, that quantifies the concept of information. ...

If , then A is smaller or equal in size to B, so that A contains greater or equal information than B. Then, since , then , where function m measures the amount of "information" contained by a set.

From this last inequation it immediately follows that the strongest possible inference rule is the "conjunction introduction": Conjunction introduction is the inference that, if p is true, and q is true, then the conjunction p and q is true. ...

and that the weakest possible inference rule is the "disjunction introduction":

All other inference rules, including modus ponens, have a "strength" somewhere between these two bounds — conjunction and disjunction. Modus ponens (Latin: mode that affirms) is a valid, simple argument form (often abbreviated to MP): If P, then Q. P. Therefore, Q. or in logical operator notation: where represents the logical assertion. ...

External link

• LogicTutorial.com (http://logictutorial.com) provides interactive illustrations of Johnston diagrams, and also raises some relevant philosophical issues.