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\title{Demonstration of Tests of Conversion of PM Dot Notation to Parentheses}
\author{Dennis J. Darland}
\maketitle

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SECTION 0. VERIFICATION TESTS (of dot to paren dot icn)

For each proposition is given:

1: the PM notation with dots.

2: the notation with parentheses 

3: the Polish (with Lukasiewicz symbols) notation

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$\pmast 2 \pmcdot 06 \pmthm \pmdottt p \pmimp q \pmdot \pmimp \pmdott q \pmimp r \pmdot \pmimp \pmdot p \pmimp r $

$\pmast 3 \pmcdot 47 \pmthm \pmdottt p \pmimp r \pmand q \pmimp s \pmdot \pmimp \pmdott p \pmand q \pmdot \pmimp \pmdot r \pmand s $

$\pmast 4 \pmcdot 22 \pmthm \pmdott p \pmiff q \pmand q \pmiff r \pmdot \pmimp \pmdot p \pmiff r$

$\pmast 4 \pmcdot 41 \pmthm \pmdottt p \pmdot \pmor \pmdot q \pmand r \pmdott \pmiff \pmdot p \pmor q \pmand p \pmor r$

$\pmast 4 \pmcdot 43 \pmthm \pmdottt p \pmdot \pmiff \pmdott p \pmor q \pmand p \pmor \pmnot q$

$\pmast 4 \pmcdot 44 \pmthm \pmdottt p \pmdot \pmiff \pmdott p \pmdot \pmor \pmdot p \pmand q$

$\pmast 4 \pmcdot 87 \pmthm \pmdottt p \pmand q \pmdot \pmimp \pmdot r \pmdott \pmiff \pmdott p \pmdot \pmimp \pmdot q \pmimp r \pmdott \pmiff \pmdott q \pmdot \pmimp \pmdot p \pmimp r \pmdott \pmiff \pmdott q \pmand p \pmdot \pmimp \pmdot r$

$\pmast 4 \pmcdot 88 \pmthm \pmdottt p \pmand q \pmdot \pmimp \pmdot r \pmdot \pmiff \pmdott p \pmdot \pmimp \pmdot q \pmimp r \pmdott \pmiff \pmdott q \pmdot \pmimp \pmdot p \pmimp r \pmdott \pmiff \pmdott q \pmand p \pmdot \pmimp \pmdot r$

$\pmast 5 \pmcdot 33 \pmthm \pmdottt p \pmand q \pmimp r \pmdot \pmiff \pmdott p \pmandd p \pmand q \pmdot \pmimp \pmdot r$

From Landon D. C. Elkind's Paper in Russell: Vol. 43, no. 1, page 44

$\pmast 431 \pmcdot 441 \pmthm \pmdot p \pmor q \pmdot \pmiff \pmdot r \pmimp s$

$\pmast 431 \pmcdot 442 \pmthm \pmdott p \pmdot \pmor \pmdot q \pmiff r \pmdott \pmimp \pmdott s$

$\pmast 431 \pmcdot 443 \pmthm \pmdott p \pmor q \pmdot \pmiff \pmdot r \pmdott \pmimp \pmdott s$

$\pmast 431 \pmcdot 444 \pmthm \pmdott p \pmdott \pmor \pmdott q \pmiff r \pmdot \pmimp \pmdot s$

$\pmast 431 \pmcdot 445 \pmthm \pmdott p \pmdott \pmor \pmdott q \pmdot \pmiff \pmdot r \pmimp s$

From same, page 54

$\pmast 431 \pmcdot 54 \pmthm \pmdott p \pmand q \pmandd r \pmand s \pmdott \pmimp \pmdott p \pmand s \pmandd r \pmand q $ 

check longer prop name

% $\pmast 5 \pmcdot 33 \pmthm \pmdottt pabc \pmand qdef \pmimp rghi \pmdot \pmiff \pmdott pabc \pmandd pabc \pmand qdef \pmdot \pmimp \pmdot rghi$

Propositions involving quantifiers

$\pmast 9 \pmcdot 2 \pmthm \pmdott \pmall{x} \pmdot \pmpf{\psi}{x} \pmdot \pmimp \pmdot \pmpf{\psi} {y} $

$\pmast 9 \pmcdot 21 \pmthm \pmdottt \pmall{x} \pmdot \pmpf{\psi}{x} \pmimp \pmpf{\phi}{x} \pmdot \pmimp \pmdott \pmall{x} \pmdot \pmpf{\psi}{x} \pmdot \pmimp \pmdot \pmall{x} \pmdot \pmpf{\phi}{x} $

$\pmast 9 \pmcdot 22 \pmthm \pmdottt \pmall{x} \pmdot \pmpf{\psi}{x} \pmimp \pmpf{\phi}{x} \pmdot \pmimp \pmdott \pmsome{x} \pmdot \pmpf{\psi}{x} \pmdot \pmimp \pmdot \pmsome{x} \pmdot \pmpf{\phi}{x} $

$\pmast 9 \pmcdot 31 \pmthm \pmdottt \pmsome{x} \pmdot \pmpf{\phi}{x} \pmdot \pmor \pmdot \pmsome{x} \pmdot \pmpf{\phi}{x} \pmdott \pmimp \pmdot \pmsome{x} \pmdot \pmpf{\phi}{x}$

$\pmast 9 \pmcdot 401 \pmthm \pmdotttt p \pmdott \pmor \pmdott q \pmdot \pmor \pmdot \pmsome{x} \pmdot \pmpf{\psi}{x} \pmdottt \pmimp \pmdottt q \pmdott \pmor \pmdott p \pmdot \pmor \pmdot \pmsome{x} \pmdot \pmpf{\psi}{x} $

$\pmast 10 \pmcdot 35 \pmthm \pmdottt \pmsome{x} \pmdot p \pmand \pmpf{\psi}{x} \pmdot \pmiff \pmdott p \pmandd \pmsome{x} \pmdot  \pmpf{\psi}{x} $ 

$\pmast 11 \pmcdot 2 \pmthm \pmdott \pmall{x,y} \pmdot \pmpf{\phi}{[x,y]} \pmdot \pmiff \pmdot \pmall{y,x} \pmdot \pmpf{\phi}{[x,y]}$

One Step in proof of 11.55
I wanted example of 2 adjacent quantifiers - hard to find.

$\pmast 11 \pmcdot 551 \pmthm \pmdottt \pmall{x} \pmdottt \pmsome{y} \pmdot \pmpf{\psi}{x} \pmand \pmpf{\phi}{[x,y]} \pmdot \pmiff \pmdott \pmpf{\psi}{x} \pmandd \pmsome{y} \pmdot \pmpf{\phi}{[x,y]} $

From same, page 46


$\pmast 431 \pmcdot46 \pmthm \pmdott \pmall{x} \pmdot \pmpf{\psi}{x} \pmand \pmpf{\phi}{x} \pmdot \pmimp \pmdot \pmall{x} \pmdot \pmpf{\psi}{x}$

Other Tests

$\pmast 99 \pmcdot 99 \pmthm \pmdottt \pmnot \pmsome{x} \pmdott \pmnot \pmpf{\psi}{x} \pmdot \pmimp \pmdot \pmall{x} \pmdot \pmnot \pmpf{\psi}{x} $


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