sums will set you free

how to teach your child numbers arithmetic mathematics

writing down sets and set logic equations

site map

• truth tables for connectives
• connective/operator symbols
• working out compound truth values - logic equations
• logic and probability
• examples
  tautology
  contradiction
  contingency
• logic and programming computers
• end notes

truth tables for connectives

NOT

AND

AND truth tables (&   •   )
state	state	and state		state	state	and state
True [T]	True [T]	True [T]		1	1	1
True [T]	False [F]	False [F]		1	0	0
False [F]	True [T]	False [F]		0	1	0
False [F]	False [F]	False [F]		0	0	0

OR (inclusive OR)

OR (inclusive OR) truth tables (v   )
state	state	or state		state	state	or state
True [T]	True [T]	True [T]		1	1	1
True [T]	False [F]	True [T]		1	0	1
False [F]	True [T]	True [T]		0	1	1
False [F]	False [F]	False [F]		0	0	0

XOR (EXCLUSIVE OR)

implication

truth table [implies]
a	b	a b
True [T]	True [T]	True [T]
True [T]	False [F]	False [F]
False [F]	True [T]	True [T]
False [F]	False [F]	True [T]

equivalence

truth table [equivalence]
a	b	a b
True [T]	True [T]	True [T]
True [T]	False [F]	False [F]
False [F]	True [T]	False [F]
False [F]	False [F]	True [T]

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connective/operator symbols

NOT (also sometimes called complement)  ~   ¬
AND (also called conjunction)   &   •  
inclusive OR (also called inclusive disjunction)   v  

You can then build up compound formulae such as (~ A) B  but this format can be reduced to ~ A B. This is because, just like arithmetic connectives, an order of precedence is defined:

~ / NOT is most cohesive (applied first)
then resolve what is inside brackets (brackets are also used to emphasise structure)
then connectives like AND and OR,
which are, in turn, more cohesive than [implies] and [equivalence].

working out compound truth values - logic equations

You can indicate variables by A, B, p, q, greek letters or whatever you choose. I am going to use capital letters in what follows.

How to solve the following equation, [(A ~ B) v (C • ~ D)] [~(A D) v (~ C v E)], step by step.
Note that truth values for the connectives are obtained by reference to the tables above.

If you find this example too daunting, then go to the examples section first, then return here.

Assume A, B and C are true and D and E are false.
These values are substituted in the first line.

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[(A

~

B)

v

(C

•

~

D)]

[~

(A

D)

v

(~

C

v

E)]

[(  T

~

T)

v

( T

•

~

F )]

[~

(T

F

v

T

F

First substitute in the truth values of the variables

[(A

B)

v

(C

•

D)]

[~

(A

D)

v

(

C

v

E)]

Resolve the NOTs / ~s immediately bound/attached to variables

[(T

F)

v

( T

•

T) ]

[~

(T

F )

v

(

F

v

F)]

Three NOTs resolved by reversing, one NOT remains outside brackets

[

F

v

T

]

[~

(

F

)

v

(

F

)]

Resolve the connectives in the inner brackets

[

F

v

T

]

[

T

v

F

]

Now resolving the remaining NOT

T

T

Resolving the last connective

T

The equation resolves to TRUE

The last connective to be resolved, in this case the equivalence connective , is sometimes called the main or prime connective. Compare this with the prime connective in the next example. In that case, the prime connective is the last NOT / ~ to be resolved.

logic and probability

As you will see from the truth tables above, if there is one variable, as in the NOT table, there are two possible states: true or false (T or F, 1 or 0). If there are two variables such as A and B, as in the rest of the tables, there are four states: True True, True False, False True, False False.

So, with one variable, there are 2¹(that is 2) outcomes,
with two variables, there are 2² (that is 4) outcomes,
with three variables, there are 2³ (that is 8) outcomes.

So four variables or 2⁴ has 16 outcomes. And so on.

This is the same arithmetic as when you toss a coin. The coin will come down either heads or tails, that is a 50% chance of either heads or tails. toss the coin again and the odds remain the same - 50% heads, 50% tails. This is one chance out of two - ½.

If you toss two coins, you have one chance in four, 25%, that both will come down heads. That is ½ x ½, which is ¼. Referring to the two variable tables you can see the same pattern.

Toss a third coin and there is one chance in eight that all three coins will come down as heads. There is another one chance in eight that all three will come down as tails.

For that matter, heads can represent True or 1, and tails can represent False or 0.
So there is one chance in eight that all three coins will be heads or tails, or True or False, or 1 or 0, or On or Off. Recall the switches (link).

One in eight chances of three heads, and one in eight chances of three tails, now that’s two eighths - ah, where are the other six-eighths? Well, you can see them all laid out in the table above. They are the six cases where there is a mixture of head and tails (True and False, 1 and 0). The whole eight cases make up 100% of the possibilities.

Hey, now you are at the beginning of learning about probability and laying the groundwork for binary-based arithmetic!

A cubic, six-sided die can come down in one of six different ways each time it is thrown. Thus, Any one number has a one-sixth chance of showing up on any one throw. A die is a six-sided coin.

Enough of this for the moment, but here is a way to join the concepts in logic and probability together; and make your learning more interesting.

examples

And now we go through the wormhole into cloud-cuckoo land, so fasten your safety belt!

With copious attempts to match symbolic logic to natural language, for instance written or spoken English, and to structures of the game of mathematics/arithmetic, ‘they’ have so far settled upon eighteen structures they regard on fundamental.

I am going to start by outlining three of these.

There is so-called tautology, whence whatever values you ay substitute in for the variables. The result will always come out True. The idea of a tautology is that whatever data you may feed in to it, you extract no more information than you put in. The problem involves the false belief in the ideas of same and equals. (If you want to go further, go to the error called ‘equality’.) The idea of a tautology is, in fact, empirically unsound, but to show you this, I would have to go far further than this page intends. In everyday language, a tautology may be “wet water is wet”.

The following equation and truth table illustrates a tautology. Again, it is empirically unsound. (For those who want to dig further: the error of the excluded middle.)

The next structure is called a contradiction. Again it is empirically unsound. (For those who want to dig further see the error of the excluded middle.) With a logical contradiction, whatever values you put into the variables, the sentence will come out False.

The equation A • ~A has a result that is false, whether A is assumed to be True or False.

A third structure is called a contingency. Here, the results depend upon the values substituted for the variables.

A

(B

v

A)

A

(B

v

A)

A

(B

v

A)

A

(B

v

A)

A is True, B is True

A is True, B is False

A is False, B is True

A is False, B is False

T

(T

v

T)

T

(F

v

T)

F

(T

v

F)

F

(F

v

F)

First substitute in the truth values of the variables

T

T

T

T

F

T

F

F

Resolve the OR / v immediately bound/attached to a variable

T

F

F

T

Resolve the connective, the prime connective

Another way to write the result of this equation

T

T

T

T

T

T

F

F

T

T

F

F

T

T

F

F

T

F

F

F

This method puts the truth values for both variables and connectives on one line

A contingency is nearer to the real world. In terms of an English sentence: “If it is raining, the road will be wet”. If it is raining, this will come out True. If it is not raining, then it will come out false. But don’t try this when driving through a road tunnel! And if it is raining in Singapore, the roads may not be wet in the Mojave desert.

logic and programming computers

If you should tumble into this maelstrom, which I regard as nonsense, I remind you again of my opening notes. However, should you persist in a wish to immerse yourself further in this wonderland, I refer you again to Logic and Philosophy: A Modern Introduction.

As said, my objective is to give you enough for understanding computer logic.

Those who play with this knitting try to convert idiomatic connectives into supposed logical forms; that is, words such as AND, IF, IMPLIES.

In computing, we are concerned with much more mundane, practical matters.

Does a potential employee have school-leaving qualifications,
AND is under 87 years old,
AND is capable of lifting 56 lbs/25 kg?

If all these qualities are so, put YES in column 6.
If anyone of these conditions is not met, then put NO.

 

In computer logic terms this will be

column 6 equals A AND B AND C.

But the computer will, in fact, put 1 or 0 (one or zero) in an assigned location representing column 6.

Unless you are going to be a developer, or work for a company like Intel or AMD, all this will be made much easier for you, so that you do not have to know what is going under the computer bonnet. For example, you may write

IF Johnny Simpson UpperAge AND Qual AND Lift THEN col.6 = T, or YES, or is TRUE.

Computer languages vary considerably in their keywords and structures.

Remember the truth tables.

People even give you degrees or prizes for this stuff if you can make it look complicated enough. Any screed of nonsense will be published in ‘academic’ journals, if you carefully wrap it up in the tinsel of this sort of mumbo jumbo!

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end notes

1. Die is the singular word (the word for one object) for which the plural is dice. It is very common that the plural (describing more than one object), dice, is misapplied to one die.
   
   

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