Mathematical Logic, Peano, proposition calculus.doc

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Mathematical Logic, Peano, proposition calculus www.mpantes.gr

Mathematical Logic, Peano, propositional calculus

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Introduction .

In its birth Mathematical Logic was the theory of

classes (Boole). The first person to have the opinion that

the theory of statements (propositions) is more important

, was the Hugh McColl (18 !"1#$#).who in a gro%p of

papers on &the calc%l%s of e'%ivalent statements p%t

forwar the belief that the only b%siness of logic is with the

theory of statements an that the chief statement

connection is some sort of implication. The tho%ght that the theory of

statements an not that of classes is the root of Mathematical Logic , an the

tho%ght that implication of one sort or another is the chief relation to be given

attention in logic, soon became the normal beliefs of those at the hea of this

fiel .

Peano (18*8"1# +) ma e %se of the logic of statements for ma ing

clear the arg%ments of every ay mathematics , an so viewing logic as an

instr%ment for getting clear an tight reasoning in s%ch mathematics. -e was

the first to give the new logic the name of Mathematical Logic beca%se

he consi ere it as a tool of mathematics/ for him Mathematical Logic is the

logic of mathematics. (Nidditch )

That0s beca%se the eano pointe o%t that the implication was the

main relationship in mathematics, all or almost all propositions in a

mathematical system eg geometry, are implications . Th%s concl% e that it

was possible with the %se of logic to establish the mathematical proposals ( all

propositions in mathematics, not only in arithmetic) , in the form of a lang%age

of symbols, an emonstrations of all theorems can be an interchange of

s%ch symbols that relate to propositions an their relationships. This

lang%age is of logical terminology, is a generali2ation of mathematical

terminology, which changes in every fiel , b%t remain common the feat%res of

the pro %ctive system, %n er each iscipline/

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I have in icate by signs all the i eas which occ%r in the f%n amentals of arithmetic.

The signs pertain either to logic or to arithmetic . I believe, however, that with only these

signs of logic the propositions of any science can be e3presse , so long as the signs which

represent the entities of the science are a e . Peano

eano e icate his life to improve the rigor of mathematics,improving their logic fo%n ations, isplacing int%ition. The pro %ctive str%ct%re

of mathematics is a logical str%ct%re, an mathematics is an interpretation.

4o Mathematical Logic has two si es. 5ne in its roots in the past is the

science of the e %ction form of reasoning, the mathematics of logic .1 It was

in this that Boole an others intereste in the algebra of logic were wor ing ,

aime the mathemati2ation of logic . The other si e of Mathematical Logic is

the logic of mathematics , what is hi en behin the pro %ctive systems ofmathematics an not only, something eeper in eano6s irection, with res%lts

%ntil metamathematics an the fo%n ations of maths. 4o the other e3perts in

Mathematical Logic en e %p being intereste in logic for itself or were

intereste as 7rege was, in t%rning a bit of mathematics into logic. They i

not see the val%e of logic as eano, as a tool for every ay math, b%t as a

philosophy.

The propositional calculus .

The propositional calc%l%s is the %ltimate generali2ation of all

mathematical systems. It employs the post%lational metho . small set of

ta%tologies serves as the post%lates of the evelopment an then several

formal r%les are given in accor ance with which all other ta%tologies can be

obtaine from the selecte few.

9hat have in common the gro%p theory, :%cli ean geometry an

vector spaces ; They are all pro %ctive formal systems, where their formal

image is to e3tract theorems from a3ioms or previo%s theorems. 7or this

pro %ction we %se reasoning ( inferences ) in the form if so an so then s%ch

an s%ch . The propositional calc%l%s is an abstract generalization of this

process of all pro %ctive systems, an this generali2ation ( the leit"motif of1 s the plane geometry of :%cli is mathematics of the straight line an circle, the

st% y of < is the mathematics of real n%mbers, even the probabilities are the mathematics of

ran omness, etc

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mathematics ) it is in the region of logic, the last ref%ge of rational thin ing.

The logical inference is now the ob=ect of o%r st% y. 4o it is the last a3iomatic

system which escribes all the others an so it m%st be regar e as

f%n amental . It is the formalism of all formalisms propose in mathematical

theories, from :%cli %ntil to ay.

To escribe e itorially the lang%age of propositional calc%l%s ( as a

pro %ctive system ) as evelope in rincipia mathematica of


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The ship ( at the time ) approaches Chios or Mytilene , kok

an later in symbolic logic represente by the letters p, ', r ...

The concepts in these propositions are merc%ry, metal , ay, light, ship.

The symbolic lang%age of propositional calc%l%s evelope by eano

an on, b%t the fo%n ations were set by the calc%l%s of Boole, o who ha

alrea y intro %ce the logic of the propositions, an more ol stoic ( stoic >th

cent%ry B?) .

ropositions are the basic cell of speech. 9hat is the concepts on the

si e of thin ing , are the propositions on the si e of the e3plicit e3pression

( apano%tsos ) . Their logical interpretation is the fin ing of tr%e or false +.

Com inations of propositions ( operations ).

In Mathematical Logic, we have the following combination of

propositions /

1.p @ ' ( rea p an ') is calle conjunction

It raine on the mo%ntain and roppe hail on the plain .

I too the train and went to thens .

-ere ( in common lang%age ), the an connects two proposals that

are relevant to each other. In propositional calc%l%s e3presse @ which

connects any two proposals witho%t referring to their possible relationship.

con!uncti"e sentence is typically true #hen and only #hen both proposals p

and $ is true eg

4%mmer is a season of year an the fifth agreement compose

-omer. ( false) I come from the city an on the top (is ) cinnamon. (tr%e )

A is a octor an the n%mber > is o ( false)

+. pv' (p or ') is calle disjunction

2 CDEFGH JKF NDOPQ F PRFKE SDQ K EP FNHU V K W X VY GUF Z [C\ ] ristotle

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( v symbol from the Latin vel)

The ship ( at the time ) approaches Chios or Mytilene . (pv')

I will ret%rn to my homelan alive or ea . (pv')

gain 0or 0 in or inary lang%age has two interpretations. Means the

meaning of the Latin a%t (p or ' b%t not both ), or the meaning of the Latin

again vel (p or ' or both). In symbolic logic we %se the notion of vel, altho%gh

in many cases the concept of a%t is the only one possible. gain the two

engage proposals can be %nrelate . The classical logic calls the first

sentence rela3e is=%nction (at least one will happen , the first sentence )

an the secon e3cl%sive (ma3im%m one will happen , the secon sentence) .

The tr%th of a is=%nction / the p"$ is true if and only if at least one of the t#o

statements is true. The two sentences above are tr%e.

p ^ ' ( if p then ') is calle implication

If ay then there is nat%ral light.

the proposition p % $ represents a proposition that is false #hen and only

#hen p is true and $ false .

If the angles of a triangle s%m to two right then agontai two parallel point"to"

line ( false)

If 4ocrates is immortal then the s'%are has e'%al si es (tr%e ) .

That is/ a false proposition implies any proposition .

> p _ ' (p if an only if ') is calle e!ui"alence

the tr%th is that the e$ui"alence is true if and only if p and $ are both true

or false.

:g 4ocrates is immortal _ the s%m of the angles of a triangle is *$$th (tr%e ) .

*. p` ( not p) is calle the negation of p or contra iction

represents the enial or negation of p. Ie the p` represents a sentence tr%ewhen p is false an p` is false when p is tr%e. 9e m%st not conf%se a ref%sal

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by an apophatic proposal. The negation of the above proposal is/ it is not

tr%e that if 4ocrates is immortal then the s'%are has e'%al si es . 5bvio%sly

the negation of the negation of a sentence is the proposition itself .

The most complete of the logical constr%ctions is the argument . It is a

series of interrelate proposals forme to ma e clear ( to prove ) the tr%th of

a r%le. The process or metho by which the min evises an arg%ment, calle

reasoning . ( apano%tsos ) all propositions lea ing to the final are the

premises , an the final sentence conclusion . 5%r reasoning in the form of

implication if this an this , then so an so b%t sho%l not be conf%se with

the implication ^ which is an operation, b%t the arg%ment is a relationship

( T2o%varas ) .

... The reasoning is logical and phrasal complex where just put some proposals ( the premises ) ensues with logical necessity , another proposal different from the former,

it is true for the sole reason that those premises are true ... Aristotle) .

These in ristotle

B%t in a typical a3iomatic system s%ch as the propositional calc%l%s,

where from a3ioms we pro %ce theorems , what interests %s is the typical

val%r of reasoning , ie if the implication involve is tr%e regar less of the tr%th

or falsity of the premises an the concl%sion . Then the reasoning is callevali otherwise invali . Tr%e or false are only the propositions . This is the

path of formalism in formal logic .

n example of reasoning in this symbolic lang%age is/

p @ (p ') ^ ' (we rea

p an ( if p then ') implies '. for e3ample

If p is a s%nny ay

an ' the rain oesn6t fall then we rea the reasoning /is a s%nny ay an on a s%nny ay the rain oesn6t fall so the rain oes not

fall . Is a vali reasoning ;

Truth ta les .

The cleaner an more %niform metho to s%mmari2e the typical

reasonable interpretations of con=%nction, is=%nction, implication ,

e'%ivalence, enial an arg%ment is the %se of truth ta les . In these tables

signifies that the correspon ing proposition is tr%e an d false. n

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interpretation correspon s tr%th val%es to the propositions an e3pan s on

comple3 form%las by %sing a tr%th table which is s%mmari2e below/

Base on this table, we wo%l loo if the previo%s reasoning is vali or

invali . 9ill form the final form of the proposition step by step loo ing at the

or d.

otice that the last col%mn has tr%th val%e ( tr%e) . This means that

the reasoning is vali ( the arg%ment typically tr%e) regar less of the tr%th of

the premises p an '. ropositions of this in are calle tautologies and ha"e a special

role in the study of logic . 7in ing the ta%tological character of the e3pressions

is ma e on the basis of the tr%th table . The val%e of tr%th table lies precisely

in this/ that it is a control algorithm of tautologies ( T2o%varas ) In propositional

calc%l%s there is no istinction between ta%tology an vali reasoning. The

ta%tologies are calle an logical tr%ths or tr%ths of logic beca%se they can

be recogni2e as tr%e thro%gh the principles of propositional calc%l%s anwitho%t reference to any a itional information. 4o the ta%tology is a

necessarily correct proposition beca%se of the logical form%la , an not %e to

faithf%l performance of the ob=ective worl ( the typical tr%th) an can not be

%se for a false reasoning. ta%tology in common lang%age is 7ree om is

being free .

the mathematical tautologies do not describe real world situations but rather

show the logical form of the world ( Ramsey )

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The ta%tologies of propositional calc%l%s reflect the correct laws of

correct reasoning , not only in mathematics b%t also any other science ( .

Trigger epartment of Mathematics , niversity of ?rete). :3amples of

ta%tologies we have in the following table where we meet the nown laws of

ristotelian logic. The control of ta%tology

is oing with tr%th tables .

Logicall# e!ui"alent sentences /

If yo% b%il tr%th tables of both propositions p ^ ' an p ^ ' , we see that

they agree between the res%lts of tr%th an %ntr%th . The two propositions m

an n are calle logically e$ui"alent proposals an the proposal m _ n is a

ta%tology , or a law of logic.

-ere are some pairs of logically e'%ivalent proposals (_) ( from :ves)

that give the meaning of e'%als ( ) in the pro %ctive str%ct%re of the

propositional calc%l%s . they are easily emonstrate with tr%th tables . The

importance of e'%ivalent proposals lies in the fact that we can alternate them

in an e3pression witho%t affecting the tr%th val%e of the initial e3pression.

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$ . %xioms .

The a3ioms as in all pro %ctive systems are s%ggestions that apply to all

mo els of the system ( eg in all gro%ps) . In propositional calc%l%s mo els are

vario%s assessments of the tr%th val%e of the propositions. ( T2o%varas ) so

the a3ioms will be tr%e propositions for all assessments , so they are

ta%tologies . ?hoose some of which are pro %ce other important .In

rincipia mathematica there are fo%r a3ioms (-owar :ves) that are

ta%tologies /

1. a3iom of tautolog# (pv') _ p

+ . a3iom of addition ' ^ (pv') the is=%nction is tr%e when an agent is tr%e

. a3iom of permutation (pv') ^ ('vp)

>. a3iom of summation (' ^ r) ^ (pv') ^ (pvr) as in algebra if a j b then

a k c j b k c

&. 'ules .

To erive ta%tologies from given ta%tologies. These are

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).Theorems

Theorem 1 / (' ^ r) ^ (p ^ ') ^ (p ^ r) ( transitivity of implication )

9e have (' ^ r) ^ (pv') ^ (pvr) ( a3iom >).

(' ^ r) ^ (p`v ') ^ (p`v r) ( r%le +)

(' ^ r) ^ (p ^ ') ^ (p ^ r) ( logically e'%ivalent proposals (' ^

') an (pv'), have the same tr%th table ) .

Theorem $ / p ^ (p v p)

9e have ' ^ (p v ') ( a=iom + )

p ^ (p v p) ( r%le 1 replacement of p instea of ')

Theorem & / p ^ p (the implication is refle3ive )

(' ^ r) ^ (p ^ ') ^ (p ^ r) ( theor. 1 )

(pvp) ^ p ^ p ^ (pvp) ^ (p ^ p) ( replace the pvp instea of ' an p

instea of r)

(p v p) ^ p ( Theorem 1 )

p ^ (p v p) ^ (p ^ p) (


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These theorems give %s the general nat%re of propositional calc%l%s .

5ne feat%re of the calc%l%s is that is an e3ample of a Boolean algebra . If yo%

interpret the classes of Boole6s algebra to mean propositions , an 1 an $ as

ta%tology an its opposite , respectively ( contra iction ), the a3ioms of

algebra of Boole ( from a previo%s article) are theorems of propositional

calc%l%s (-owar :ves sel.+*! ). :ach theorem then in Boole6s algebra lea s

to a correspon ing theorem of propositional calc%l%s.

B%t the propositional calc%l%s cannot be erive from the calc%l%s of

classes since the classes o not contain the logical feat%res of the

combinations of propositions. In Boole, logic e3ists an is ill%strate on

mathematics, in eano logic is to interpret mathematics is a generali2ation .

4o the properties of propositions m%st be ass%me first an cannot be

erive from any other system, the propositional calc%l%s is a f%n amental

e %ctive system.

+ources .

Logic ( apano%tsos , o ona )

hilosophy of mathematics ( JN K FG N JPZ , Technical ?hamber of

reece)

:lements of mathematical logic ( th.T2o%varas , qiti )

Logic ( . Trigger epartment of Mathematics , niversity of ?rete, Internet)

The history of mathematics (

Mathematical Logic, Peano, proposition calculus.doc

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