The inherent risks in using a name-forming function at object language level (2015)

On the Paradox of the Adder (2011)

Unfortunately there are typos in the published version, this is the corrected
text.

A few comments on Roy T. Cook’s “Curry, Yablo and duality” paper

This paper is again about so called "Yablo's paradox". - October, 2010

On Russell’s Paradox with Nails and Strings - 1985

The Liar Paradox in BASIC language - 1985

Finite State Machines as Models of Reality and Explanation - 2005

Abstract

The model provides an arithmetic translation of propositional logic, as the model of one language in another language. With the help of this translation the analysis also discusses an electronic model of propositional logic. This simply means the use of an electronic spreadsheet software. In the course of this, I discuss the basic features of Mealy’s theory of Finite Automata (Finite State Machines), and then approach the truth problem from this perspective. The reason why I have opted for this way of analyzing these problems is because the delineation of a philosophical problem with the help of a spreadsheet is an apt example of what is considered to be an easily intelligible explanation in the 21st century. The cybernetic model as such does not form part of the printed version of this text. The explanation and description of the workings of the model do not equal the model itself. The latter exists via living, practical contact with the user, and therefore the contingent time of everyday life.

The central theme is Feedback and Paradox. Every truth-function corresponds to an isomorphic digital circuit. Consequently, the logical structure of every proposition can be presented within the range of propositional logic as an equivalent digital circuit. Provided that the logical values ‚true‘ and ‚false‘ correspond to the ‚high‘ and ‚low‘ voltage levels, the output of a circuit equivalent with contradiction is always low-level for every input state, while the output of a circuit corresponding to a tautology is always high-level, irrespective of input state. On the other hand, the remaining propositions correspond to circuits the output of which is high level if and only if some of the atomic components of the proposition are true, or rather, the inputs equivalent with the atomic propositions are high-level. But what is the equivalent of a circular statement? The propositions are true or false irrespective of time, whereas the voltage level of the circuits can change in time. More accurately, one can say that the input levels of the circuits are high or low depending on whether we evaluate the atomic formulae of the formula which expresses the logical structure of the proposition to be true or false, for the voltage level of the output of the circuit and truth-value of the formula result from these formulae. I call the digital circuits ‚combinatorial automata‘, which thus may serve to model the formulae of propositional calculus. Formulae connected with truth functions yield further formulae. Although there are always corresponding combinatorial automata for these, the situation is not quite so simple in every case, for we do not find combinatorial automata joined to each other in each case; it is also possible that we will not find an automaton – an operating machine or circuit – there at all. There are digital circuits the output of which is not a function of the input. The range of automata is wider than that of the combinational automata. It includes machines the input states of which do not determine unambiguously their output states, i.e. the output is not a function of the input. This is because the circuit has a feedback. Most digital circuits belong to this latter group, which I call the ‚sequential automata‘. The question arises whether there is a logical structure of circulating statement which corresponds to such a sequential automaton (or sequential circuit). In my view the logical structure of the Liar Paradox coincides with the operation of a sequential automaton irrespective of the logical correctness of the paradox itself. The analysis also examines possibilities of how the model could be further developed.

The study does not claim to offer the absolute explanation that makes all other explanations superfluous. It merely states that the model it offers is worth thinking about and developing.

`My spreadsheets work on PC and Mac, to the best of my knowledge equally well with either MS Excel or Gnumeric software.`
## Spreadsheets: |

Modal
Logic - K system

Modal
Logic - T system

Tarski's style truth model Kripke's counterexample

Arithmetic Translation of Truth Functions

Everything the kings says is true.

Liar paradox Strengthened Liar

Curry paradox - simplified model Curry paradox - in Tarski's style

Buridan's proof of God's existence

simple Yablo's paradox model Yablo's paradox finite version