I have a flavour of propositional calculus with axiom schemas
1. A->(B->A)
2. (A->B)->((A->(B->C))->(A->C)
3. A->AvB   4. B->AvB   5.(A->C)->((B->C)->((AvB)->C))
6. A&B->A   7. A&B->B   8.(A->B)->((A->C)->(A->(B&C)))
9. (A->B)->((A->~B)->~A).   10.~~A -> A
and the Modus Ponens as the only inference rule, and I’m trying to check if
all the usual formulas can be infered.

The problem is with distributivity, where (A&B)vC -> (AvC)&(BvC) is easily
proved, but not the other way round. The reason appears to be that axioms 5
and 8 have the disjunction sitting on the left hand of the resulting ->, and
the conjunction on the right. Should the axiom set be expanded? Or does one
go about proving the inverse through contradiction?

Dually, it’s easy to prove (A&C)v(B&C) -> (AvB)&C, but not the other way
round. Could someone help?

Alexander.

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Hello! I am new to the group.

I have a question concerning modal logic and possible worlds.

I have been reading a paper and there is a sentence in it that I can
not fully understand. The sentence is "If we require that the real
world always be one of the possible worlds (which amounts to saying
that the possibility relation is reflexive), then it follows that you
can’t know anything false."

As far as I know, in possible worlds approach you know something if it
is true in all the possible worlds. However, in this sentence real
world is just one of the possible worlds, how can one not be able to
know anything false? Could someone explain please? Thanks in advance.

ok is this valid?
in layman’s terms:

If all the known expressions of PA and ZFC etc etc, were written in
the equivalent longhand natural language and placed in a book and a
complete dictionary containing the definition of every word of that
natural language was also included and all of this dropped among
intelligent beings on some other planet who knew nothing of earth

then:

even if they did surmise that the squiggles on the stuff called paper
had meaning, they would not be able to know for certain what that
meaning was, since in the dictionary each word is self-referentially
defined only in terms of other words in the dictionary.

They might be able to "guess" some assumptions of a possible "meaning"
for each of the squiggle-sequences (words) in such a way as to have
all of the other strings of squiggles be consistent with those
assumptions.

But on different sets of assumptions there could be other equally self-
consistent "interpretations" of the pattern of squiggles.

In the manner of Wittgenstein’s finite rule paradox.

Ocam’s razor aside, they would never be able to say with certainty,
what the meaning of any of those squiggles was, or even if they had
any meaning at all.

So even natural language itself, on its own, is not sufficient to
define the meanings which it can express.

And since (semantic) Truth is something that only meaningful strings
of squiggles (utterances) can have, then since language by itself is
not sufficient to define meaning, it is not, by itself, able to define
(the meaning of) truth either.

That seems about right to me. what do you think?

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Solutions  manual  to   Fundamentals of Applied Electromagnetics 5th
edition by Fawwaz T. Ulaby
Solutions  manual  to    Fundamentals of Fluid Mechanics, 5th By
Bruce,R. Munson, Donald
Solutions  manua  to    Fundamentals of Signals and systems using web and
matlab third edition

Solutions  manua  to     Corporate Finance 1e by Berk SM
Solutions  manual  to   Introduction to Environmental engineering and
science 2e by Gilbert M. Masters
Solutions  manual  to   Introduction to Mathematical Statistics 6/E
Robert V. Hogg
Solutions  manual  to   Introduction to Quantum Mechanics (1 & 2
Edition), By David J. Griffiths
Solutions  manua  to
Introduction_Fluid_Mechanics,_6Th_Edition_Solution_by_fox.rar
Solutions  manual  to   Linear Algebra with Applications 6 edition by
Leon
Solutions  manual  to   Mathematical Methods for Physics and
Engineering 3th By  Riley  M P Hobson
Solutions  manua  to    Mathematics for Economists Solution Manual –
Simon and Blume (ver 2)
Solutions  manual  to   Mathematics for Economists Solution Manual
(Blume, 1994)
Solutions  manual  to   Microelectronic circuits by R. Jaeger 3rd
edition
Solutions  manual  to   Numerical methods for engineers 5th  by Chapra
solution manual to  Probability and Statistical Inference( 7th
edition, Hogg & Tanis)
Solutions  manual  to   Probability & Statistics for Engineers &
Scientists, 8th by Sharon Myers , Keying Ye, Walpole
Solutions  manual  to   Probability and Statistics for Engineering and
the Sciences 7/e JAY L.DEVORE
Solutions  manual  to   Probability,Random Variables and Stochastic
Processes,4th,by Athanasios Papoulis
Solutions  manua  to    Separation Process Principles, 2nd Ed.,by
Seader, Henley
Solutions  manua  to    Statistics for Engineers and Scientists by
William Navidi
Solutions  manual  to   Transport Phenomena by Bird, Stewart &
Lightfoot, 2nd edition
Solutions  manual  to   Unit Operations of Chemical Engineering
(7th)    By Warren McCabe, Julian Smith
Solutions  manual  to   Vector Mechanics for Engineers: statics,    7th
Edition (much same as 8th),By F. P. Beer, E. R. Johnston
Solutions  manual  to   Vector Mechanics for Engineers: DYNAMICS,
8th Edition ,By F. P. Beer, E. R. Johnston
Solutions manual to     Microeconomic Theory  (By Chiaki Hara)
Solutions manual to     Principles of Economics 4th Ed by N. Gregory
Mankiw, Chapter Included: 1,2,4,5,6,7,8,10,11,13,14,15,16,17
Solutions manual to     Financial Accounting" (Libby, 2001 edition)
Solutions manual to     Fundamentals of Corporate Finance ( Ross,
Westerfield, Jordan  8th)
Solutions manual to
Solutions manual to     options, futures and other derivatives》(5th
edition by John Hull

Solutions manual to     Intermediate Miroeconomics: Homawork and
solutions (Varian
Solutions manual to      Mathematical Statistics by Shao, 2003
Solutions manual to      Econometric Analysis(Greene 6e, 2008)
Solutions manual to     Financial Accounting Harrison 6th solution manual
Solutions manual to     Financial Accounting (Libby, fifth edition)
Solutions manual to     Corporate Finance by Ross, Westerfield & Jaffe
Solutions manual to     Introductory Econometrics for Finance(Chris
Brooks 2002)
Solutions manual to     Fundamentals of Financial Management With Infotrac
Concise 4th Ed by Eugene Brigham
Solutions manual to     The Macroeconomics of Self-Fulfilling Prophecies
2th by Roger E. A. Farmer
Solutions manual to     Linear Algebra   by Jim Hefferon
Solutions manual to     Introductory Econometrics for Finance (Chris
Brooks 2002)
Solutions manual to     INTRODUCTION TO LINEAR ALGEBRA: Solutions to
Exercises (Gilbert Strang)
Solutions manual to     Operational Risk Quantification: Mathematical
Solutions for Analyzing Loss Data
Solutions manual to     Fundamentals of Financial Management With Infotrac
Concise  4th by Eugene Brigham
Solutions manual to     ENGINEERING MECHANICS: DYNAMICS by BEDFORD
5th.
Solutions manual to     computer system architecture 3rd  by M.Morris
Mano
Solutions manual to     Design Analysis in Rock Mechanics By William G.
Pariseau
Solutions manual to     Digital Design 4th  by M. Morris Mano, Michael D.
Ciletti

Solutions manual to     Engineering and Chemical Thermodynamics by Milo D.
Koretsky
Solutions manual to     Fundamentals of Clas Sical Thermodynamics 6th
edition by Van Wylen
Solutions manual to     Mechanical Vibrations, 3rd Edition, by Singiresu
S. Rao
Solutions manual to     Mechanical Vibrations, Third Edition, by Singiresu
S. Rao
Solutions manual to     Microelectronic Circuit Analysis and Design, 3ed.
by Neamen (2006) – [Solutions Manual Only] by Donald A. Neamen
Solutions manual to     Modern Control Systems 11th by Richard C Dorf and
Robert H. Bishop
Solutions manual to     Modern Organic Synthesis: An Introduction by
Michael H. Nantz, Hasan Palandoken, George S. Zweifel
Solutions manual to     Power System Analysis By John J. Grainger, William
D. Stevenson Jr.

Solutions manual to  Auditing and Assurance Services  12 th  by: Alvin
A Arens, Randal J Elder, Mark Beasley
Solutions manual to Signals and Systems: Analysis of Signals Through
Linear Systems   by: M.J. Roberts, M.J. Roberts