Two-valued logic: Operators

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Remember the following.

true is always true,
true can never be false,
false is always false, and
false can never be true.

Here are two simple operators.

"Eqv" means "is equal to" (equivalence)
"Neqv" means "is not equal to" (nonequivalence)

These operators are relational operators as they relate quantities, as opposed to being logical operators (and, or, not, imply, etc.).

The following are true expressions.

true
true Eqv true
true Neqv false
false Eqv false
false Neqv true

The following are false expressions.

false
true Neqv true
true Eqv false
false Neqv false
false Eqv true

There are three primary logical operations.

Logical negation (i.e., Not, "Not", etc.) means not.
Logical conjunction (i.e., And, "And", etc.) means both.
Logical disjunction (i.e., Or, "Or", etc.) means one or the other or both.

Some other common logical operations include the following.

Logical equivalence or equals (i.e., Eqv, "Eqv", etc.) means both operands are the same.
Logical nonequivalence or exclusive or (i.e., Xor, "Xor", etc.) means both operands are different.

Logical negation returns the opposite of the operand.

Not True is False
Not False is True

Some other symbols for Not:

¬ (math notation)
! (C-based languages)
not (Python, Lua, Pascal, etc.)

Logical negation (i.e., "Not") is related to the complement operation of sets.

Here is a mathematical definition of complement. Complement equation

Question: What is the opposite of not in?

The opposite of not in is in

"in" is the opposite of "out".
"out" is the meaning of "not in".
So, "in" is the opposite of "not in".

See how easy it is to get mixed up, if you are not careful. Remember not to forget that!

Is the following sentence true or false?

This sentence is false.

This is a self-referential statement since it refers to itself.

Self-referential statements can cause logical problems (i.e., paradoxes).

Is "no" your answer to this question? Answer either "yes" or "no".

The English word "conjuction" comes, through French, from the Latin word "coniungere" ≈ "join".

The logical conjunction or intersection of two operands is true if (and only if) both operands are true. The logical conjunction "and" is the same as the natural language "and".

Example: To graduate, you must take and pass both MA101 (Math 101) and EN101 (English 101) and some others, ignored here. That is, you must take both courses to graduate.

Conjunction can be extended to any number of operators. There are other logical operators for two-valued logic that are not needed here.

More: Logical conjunction

The conjunction of two planets happens when the two planets are both in-line with the sun.

Example: Conjunction of Venus and the Earth.

Conjoined twins are born joined at some part of their body.

Job 38:31 Canst thou bind the sweet influences of Pleiades, or loose the bands of Orion? [kjv]

συνηκας δε δεσμον πλειαδος και φραγμον ωριωνος ηνοιξας [lxx]

… coniungere … Pliadis … Arcturi … [v]

The logical conjunction of two operands is true if (and only if) both operands are true. The English word "conjuction" comes, through French, from the Latin word "coniungere" ≈ "join".

The LXX (Septuagint) uses the word "and" for conjunction while the English uses "or" for disjunction. Most translations use the word "or" or omit any use of "and" or "or" (e.g., the NIV (New International Version)).

More: Logical conjunction

Job 38:31

All

KJV: Canst thou bind the sweet influences of Pleiades, or loose the bands of Orion?

Hebrew: התקשר מעדנות כימה או משכות כסיל תפתח׃

Greek: συνηκας δε δεσμον πλειαδος και φραγμον ωριωνος ηνοιξας

Latin: numquid coniungere valebis micantes stellas Pliadis aut gyrum Arcturi poteris dissipare

Wycliffe: Whether thou schalt mowe ioyne togidere schynynge sterris Pliades, ethir thou schalt mowe distrie the cumpas of Arturis?

Luther: Kannst du die Bande der sieben Sterne zusammenbinden, oder das Band des Orion auflösen?

Job 38:31 Canst thou bind the sweet influences of Pleiades, or loose the bands of Orion? [kjv]

συνηκας δε δεσμον πλειαδος και φραγμον ωριωνος ηνοιξας [lxx]

… coniungere … Pliadis … Arcturi … [v]

The constellation Orion and the Pleiades are more related than they appear in the sky.

Details are left as a future topic.

Logical expressions deal with how the logical values True and False are combined.

False And False is False
False And True is False
True And False is False
True And True is True

Some other symbols for And:

∧ (math notation), ∩ (set notation, intersection)
&& (C-based languages , & is the bitwise conjunction)
and (Python, Lua, Pascal, etc.)

The conjunction or intersection of A & B or A and B is true=1 if both of A and B are true=1, else the result is false=0.

Venn diagrams	Expression tree	Extended truth table
	$Expression tree for A & B$	A B \| A & B ----------- 0 0 \| 0 0 0 0 1 \| 0 0 1 1 0 \| 1 0 0 1 1 \| 1 1 1 The result is in the column under the "&" operator.

More: Logical conjunction

The word "and" is a conjunction in that both parts (on either side) are needed.

Example: You need to buy milk and cookies at the store.
Distribute as follows.

You need to buy milk at the store and
you need to buy cookies at the store.

You need to do both.

More: Logical conjunction

Logical conjunction (i.e., "And", "&", etc.) is related to the intersection operation of sets.

Here is a mathematical definition of set intersection.

Note how the set intersection operator "∩" and the logical conjunction symbol "∧" are related. They are wide at the bottom and narrow/close at the top. This did not just happen. They are related.

More: Logical conjunction

$Expression tree for A & B$	A B \| A & B ----------- 0 0 \| 0 0 0 0 1 \| 0 0 1 1 0 \| 1 0 0 1 1 \| 1 1 1 Here is the extended truth table using 0 for false and 1 for true.
Here is a tree diagram for the conjunction of logical values A and B. Values flow up the tree.	The result (under the operator "&") is true or 1 only if both A and B are 1 or true.

More: Logical conjunction

Short circuit evaluation comes from the idea that, logically, the conjunction of A and B is false if either A is false or B is false.

If one evaluates A and it is false, one need not even look at B to determine that the conjunction is false.

If one evaluates B and it is false, one need not even look at A to determine that the conjunction is false.

The order does not matter!

This follows a general principle in computer science and logic and reasoning that there are times when one can determine that one need not even bother (spend time) looking (which takes time and resources).

More: Nothing is something: actions

More: Logical conjunction

Logical disjunction is true if either or both operands are true.

False Or False is False
False Or True is True
True Or False is True
True Or True is True

Some other symbols for Or:

∨ (math notation), ∪ (set notation, union)
|| (C-based languages , | is the bitwise conjunction)
or (Python, Lua, Pascal, etc.)

Logical disjunction (i.e., "Or") is related to the intersection operation of sets.

Here is a mathematical definition of set union Equation for set union

Logical disjunction is not the same as the natural language "or", which is an exclusive "or" and not an inclusive "or".

Question: Would you like a piece of pie or a piece of cake?

A logical answer is, "Yes", That is, I would like either a piece of pie, or a piece of cake, or both. I'll take both.

The natural language answer is to pick one of the two, but not both.

As a consequence, you must always be careful when using the word "or".

The logical "or" is logical disjunction.
The natural language "or" is an exclusive "or", or nonequivalence.

The query command is something like "find all pages that contain the specified text".

Search engines on the Internet allow logical operations to be used to limit the search criteria.

Question: When might we need more than two values for logical questions?

In a modal dialog box, the user may need to decide between yes, no, and some other choices. Modal dialog box with three choices

A program that requests a user response to a question. Very often, the valid responses are yes, no, and escape (gracefully avoid the question).

Determining the sign of an integer, which can be positive, negative, or zero (neither positive nor negative).

How many choices were there for Paul Revere's famous ride?

How do you reply what asked if you would like a piece of pie or a piece of cake?

Exclusive "or": Pick one and only one.
Inclusive "or": Pick one or both. I will take both.

Most people are used to exclusive "or".

Computer programmers take the "or" as a "logical" or "inclusive" "or". The negation of a conditional (statement) is needed with an inclusive "or" in order to address the something else of the conditional (statement).