\documentclass[12pt]{article} \NeedsTeXFormat{LaTeX2e} \usepackage{principia} \usepackage{fullpage} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{setspace} \usepackage{amssymb} \usepackage{amsmath} \usepackage{pifont} \usepackage{graphicx} \begin{document} \title{Demonstration of Tests of Conversion of PM Dot Notation to Parentheses} \author{Dennis J. Darland} \maketitle ------------------------------------------------------- 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 --------------------------------------------------------------- $\pmast 2 \pmcdot 06 \pmthm \pmdottt p \pmimp q \pmdot \pmimp \pmdott q \pmimp r \pmdot \pmimp \pmdot p \pmimp r $ Version with parentheses $\pmast 2 \pmcdot 06 \pmthm ( p \pmimp q ) \pmimp (( q \pmimp r ) \pmimp ( p \pmimp r ))$ $\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 $ Version with parentheses $\pmast 3 \pmcdot 47 \pmthm ( p \pmimp r ) \land ( q \pmimp s ) \pmimp (( p ) \land ( q ) \pmimp ( r ) \land ( s ))$ $\pmast 4 \pmcdot 22 \pmthm \pmdott p \pmiff q \pmand q \pmiff r \pmdot \pmimp \pmdot p \pmiff r$ Version with parentheses $\pmast 4 \pmcdot 22 \pmthm ( p \pmiff q ) \land ( q \pmiff r ) \pmimp ( p \pmiff r)$ $\pmast 4 \pmcdot 41 \pmthm \pmdottt p \pmdot \pmor \pmdot q \pmand r \pmdott \pmiff \pmdot p \pmor q \pmand p \pmor r$ Version with parentheses $\pmast 4 \pmcdot 41 \pmthm (( p ) \pmor ( q ) \land ( r )) \pmiff ( p \pmor q ) \land ( p \pmor r)$ $\pmast 4 \pmcdot 43 \pmthm \pmdottt p \pmdot \pmiff \pmdott p \pmor q \pmand p \pmor \pmnot q$ Version with parentheses $\pmast 4 \pmcdot 43 \pmthm ( p ) \pmiff (( p \pmor q ) \land ( p \pmor \pmnot q))$ $\pmast 4 \pmcdot 44 \pmthm \pmdottt p \pmdot \pmiff \pmdott p \pmdot \pmor \pmdot p \pmand q$ Version with parentheses $\pmast 4 \pmcdot 44 \pmthm ( p ) \pmiff (( p ) \pmor ( p ) \land ( q))$ $\pmast 4 \pmcdot 87 \pmthm \pmdottt p \pmand q \pmdot \pmimp \pmdot r \pmdott \pmiff \pmdott p \pmdot \pmimp \pmdot q \pmimp r \pmdott \pmiff \pmdott q \pmdot \pmimp \pmdot p \pmimp r \pmdott \pmiff \pmdott q \pmand p \pmdot \pmimp \pmdot r$ Version with parentheses $\pmast 4 \pmcdot 87 \pmthm (( p ) \land ( q ) \pmimp ( r )) \pmiff (( p ) \pmimp ( q \pmimp r )) \pmiff (( q ) \pmimp ( p \pmimp r )) \pmiff (( q ) \land ( p ) \pmimp ( r))$ $\pmast 4 \pmcdot 88 \pmthm \pmdottt p \pmand q \pmdot \pmimp \pmdot r \pmdot \pmiff \pmdott p \pmdot \pmimp \pmdot q \pmimp r \pmdott \pmiff \pmdott q \pmdot \pmimp \pmdot p \pmimp r \pmdott \pmiff \pmdott q \pmand p \pmdot \pmimp \pmdot r$ Version with parentheses $\pmast 4 \pmcdot 88 \pmthm ( p ) \land ( q ) \pmimp ( r ) \pmiff (( p ) \pmimp ( q \pmimp r )) \pmiff (( q ) \pmimp ( p \pmimp r )) \pmiff (( q ) \land ( p ) \pmimp ( r))$ $\pmast 5 \pmcdot 33 \pmthm \pmdottt p \pmand q \pmimp r \pmdot \pmiff \pmdott p \pmandd p \pmand q \pmdot \pmimp \pmdot r$ Version with parentheses $\pmast 5 \pmcdot 33 \pmthm ( p ) \land ( q \pmimp r ) \pmiff ( p ) \land (( p ) \land ( q ) \pmimp ( r))$ From Landon D. C. Elkind's Paper in Russell: Vol. 43, no. 1, page 44 $\pmast 431 \pmcdot 441 \pmthm \pmdot p \pmor q \pmdot \pmiff \pmdot r \pmimp s$ Version with parentheses $\pmast 431 \pmcdot 441 \pmthm ( p \pmor q ) \pmiff ( r \pmimp s)$ $\pmast 431 \pmcdot 442 \pmthm \pmdott p \pmdot \pmor \pmdot q \pmiff r \pmdott \pmimp \pmdott s$ Version with parentheses $\pmast 431 \pmcdot 442 \pmthm (( p ) \pmor ( q \pmiff r )) \pmimp (( s))$ $\pmast 431 \pmcdot 443 \pmthm \pmdott p \pmor q \pmdot \pmiff \pmdot r \pmdott \pmimp \pmdott s$ Version with parentheses $\pmast 431 \pmcdot 443 \pmthm (( p \pmor q ) \pmiff ( r )) \pmimp (( s))$ $\pmast 431 \pmcdot 444 \pmthm \pmdott p \pmdott \pmor \pmdott q \pmiff r \pmdot \pmimp \pmdot s$ Version with parentheses $\pmast 431 \pmcdot 444 \pmthm ( p ) \pmor (( q \pmiff r ) \pmimp ( s))$ $\pmast 431 \pmcdot 445 \pmthm \pmdott p \pmdott \pmor \pmdott q \pmdot \pmiff \pmdot r \pmimp s$ Version with parentheses $\pmast 431 \pmcdot 445 \pmthm ( p ) \pmor (( q ) \pmiff ( r \pmimp s))$ 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 $ Version with parentheses $\pmast 431 \pmcdot 54 \pmthm (( p ) \land ( q )) \land (( r ) \land ( s )) \pmimp (( p ) \land ( s )) \land (( r ) \land ( q ))$ check longer prop name % $\pmast 5 \pmcdot 33 \pmthm \pmdottt pabc \pmand qdef \pmimp rghi \pmdot \pmiff \pmdott pabc \pmandd pabc \pmand qdef \pmdot \pmimp \pmdot rghi$ Version with parentheses % $\pmast 5 \pmcdot 33 \pmthm ( pabc ) \land ( qdef \pmimp rghi ) \pmiff ( pabc ) \land (( pabc ) \land ( qdef ) \pmimp ( rghi))$ Propositions involving quantifiers $\pmast 9 \pmcdot 2 \pmthm \pmdott \pmall{x} \pmdot \pmpf{psi}{x} \pmdot \pmimp \pmdot \pmpf{psi} {y} $ Version with parentheses $\pmast 9 \pmcdot 2 \pmthm (( \pmall{x} ) \pmpf{psi}{x} ) \pmimp ( \pmpf{psi} {y} )$ $\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} $ Version with parentheses $\pmast 9 \pmcdot 21 \pmthm (( \pmall{x} ) \pmpf{psi}{x} \pmimp \pmpf{phi}{x} ) \pmimp (( \pmall{x} ) \pmpf{psi}{x} ) \pmimp ( \pmall{x} ) \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} $ Version with parentheses $\pmast 9 \pmcdot 22 \pmthm (( \pmall{x} ) \pmpf{psi}{x} \pmimp \pmpf{phi}{x} ) \pmimp (( \pmsome{x} ) \pmpf{psi}{x} ) \pmimp ( \pmsome{x} ) \pmpf{phi}{x} $ $\pmast 9 \pmcdot 31 \pmthm \pmdottt \pmsome{x} \pmdot \pmpf{phi}{x} \pmdot \pmor \pmdot \pmsome{x} \pmdot \pmpf{phi}{x} \pmdott \pmimp \pmdot \pmsome{x} \pmdot \pmpf{phi}{x}$ Version with parentheses $\pmast 9 \pmcdot 31 \pmthm (((( \pmsome{x} ) \pmpf{phi}{x} ) \pmor ( \pmsome{x} ) \pmpf{phi}{x} )) \pmimp ( \pmsome{x} ) \pmpf{phi}{x}$ $\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} $ Version with parentheses $\pmast 9 \pmcdot 401 \pmthm ((( p ) \pmor (( q ) \pmor ( \pmsome{x} ) \pmpf{psi}{x} ))) \pmimp (( q ) \pmor (( p ) \pmor ( \pmsome{x} ) \pmpf{psi}{x} ))$ $\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} $ Version with parentheses $\pmast 10 \pmcdot 35 \pmthm (( \pmsome{x} ) p ) \land ( \pmpf{psi}{x} ) \pmiff ( p ) \land (( \pmsome{x} ) \pmpf{psi}{x} )$ $\pmast 11 \pmcdot 2 \pmthm \pmdott \pmall{x,y} \pmdot \pmpf{phi}{[x,y]} \pmdot \pmiff \pmdot \pmall{y,x} \pmdot \pmpf{phi}{[x,y]}$ Version with parentheses $\pmast 11 \pmcdot 2 \pmthm ( \pmall{x,y} \pmdot \pmpf{phi}{[x,y]} ) \pmiff ( \pmall{y,x} \pmdot \pmpf{phi}{[x,y]})$ 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]} $ Version with parentheses $\pmast 11 \pmcdot 551 \pmthm ((( \pmall{x} ) \pmsome{y} ) \pmpf{psi}{x} ) \land ( \pmpf{phi}{[x,y]} ) \pmiff ( \pmpf{psi}{x} ) \land (( \pmsome{y} ) \pmpf{phi}{[x,y]} )$ From same, page 46 $\pmast 431 \pmcdot46 \pmthm \pmdott \pmall{x} \pmdot \pmpf{psi}{x} \pmand \pmpf{phi}{x} \pmdot \pmimp \pmdot \pmall{x} \pmdot \pmpf{psi}{x}$ Version with parentheses $\pmast 431 \pmcdot46 \pmthm (( \pmall{x} ) \pmpf{psi}{x} ) \land ( \pmpf{phi}{x} ) \pmimp ( \pmall{x} ) \pmpf{psi}{x}$ 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} $ Version with parentheses $\pmast 99 \pmcdot 99 \pmthm (( \pmnot \pmsome{x} ) \pmnot \pmpf{psi}{x} ) \pmimp ( \pmall{x} ) \pmnot \pmpf{psi}{x} $ \end{document}