Copyright © 2006-2010 - Curt Hill Mathematical Logic An Introduction.

Copyright © 2006-2010 - Curt Hill

Mathematical Logic

An Introduction


History

• Classical logic is based on natural language and may be closer to philosophy

• Symbolic logic is based on mathematics

• What is the difference?


Difference• Classical logic is greatly influenced

between the connection between the statements and underlying facts– Lawyers and debaters are main users– Two lawyers can reason from the same

facts to two different conclusions– They do this by emphasizing different sets

of facts and reasoning

• Symbolic logic is mostly concerned with values that are removed from their underlying propositions– The proofs generated here are

incontrovertible


Statements• Def: A statement is a sentence that is

either true or false. It cannot be both.– Statement is in natural language– It should state a fact even if that fact is not

true– AKA proposition

• Examples:– Bill Clinton was president in 1999.– VCSU graduated 10,000 students in 2000.– There are exactly 10 million dust particles in

this room at this time.


Non-examples:

• What time is it?– Does not state a fact

• All generalities including this one are false.– Is this true or false?

• 12 + x = 5– Neither true nor false until a value is

given for x


Proposition• A statement is also known as a

proposition– We usually assign a lower case letter

to such statements of fact, starting with p so that they may be variables as well

• Why are we doing this?– There are numerous examples of

faulty logic– Everyone who ever died of cancer ate

mushrooms, thus mushrooms cause cancer.

The Goal

•To be able to mathematically treat such a set of statements and determine if they are valid or not

• How will we do this?• Three steps:

– Translate into symbolic form– Simplify the symbolic form– Optionally translate back into English



Truth values

• A statement or proposition may have one of two values: true or false

• We may not know whether a particular statement is true or false we just have to know that it must be one of the two

• Two important mathematical areas, fuzzy logic and probability can deal with non discrete values but they are not covered here


Human Thinking

• There was a belief once that all human reasoning could be expressed in logic

• Unfortunately this is not so– People believe things that are true

and false– They also operate on incomplete

information– They put varying levels of confidence

on the "facts" at their disposal


Operators

• There are a number of connecters or operators that can be applied to logical values

• Each of these takes one or two Boolean values and produces a Boolean result

• These are mostly based on English words so should be somewhat intuitive


Common Operators• You are probably familiar with the

common ones:• Disjunction (or) • Conjunction (and) • Negation (not) ¬

– Also sometimes ~ or !


Other operators

• Equivalence ≡– Also

• Discrepancy or Exclusive Or • Implication • Consequence • NAND |• NOR


Precedence

• Boolean operators have an order of operation just like arithmetic operators– Highest is anything in parenthesis – Not (¬)– And () or () are usually the same– Implication ()

– Equivalence (≡) is lowest


Equivalence and Equal• There is often some confusion

between equivalence (≡) and equal (=)

• Equivalence only takes Booleans

– 55 is not allowed– Equality may take any type of

operands• Equivalence is associative but

equality is not– (5=5) = true is OK but– 5 = (5 = true) is not


Completeness

• Is there a subset of operators that could be used to express all others?

• Yes, several sets– And, Or and Not are complete– NAND is complete in itself– As is NOR


Operator Definition

• How do we define the operation of such operators?– Informally– In terms of other operators– Truth tables– Venn diagrams– Axiomatic proofs

• We will use the latter three


Informal Definitions• We have an informal notion of what

they do from English• This is handy for understanding

– It is fortunate that George Boole was a native English speaker

– Our intuitive notion is often correct

• However, this will not help us much in proofs and calculation

• The informal definition is often ambiguous, eg. Inclusive or exclusive OR


Construction

• We may declare that an operator is related to a previously well known operator

• Consider the exclusive or– One or the other but not both

• We may define this in terms of the negation of the equivalence


Truth tables• Since a binary operand can only

have four possible inputs we may enumerate them

• Truth tables are not necessarily the best thing for proofs either since our proofs very often involve a lot more than two variables and each additional variable doubles the size

• They are very handy for visualization and understanding


Venn diagrams• Aka Euler Circles• An equivalent technique• Set theory and Boolean algebra

are isomorphic– This means that anything in one can

be translated into the other– This includes the proof on a step by

step basis– Each proposition (statement of fact)

in logic becomes a set membership in set theory


Axiomatic proofs• An axiom is an unproven assumption or

definition• An axiom should be self evident or a

fundamental definition– Each axiom must be consistent with all

other axioms in the system– From our axioms we may prove theorems

• Our proofs must use valid reasoning– This type of system is at the heart of all

mathematics• We will use all three (truth tables, Venn

diagrams, and axiomatic proofs) but only the latter is actually needed– Proofs are the most rigorous

Now What?

• Truth tables will be covered in one presentation

• Venn diagrams will be covered in one presentation

• Axiomatic definition will include several presentations

• Of course there will be the needed exercises along the way


Copyright © 2006-2010 - Curt Hill Mathematical Logic An Introduction.

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