Some experimental theories of quantum gravity, such as loop quantum gravity, propose a discrete or ``quantized'' structure for space-time at very small scales. These theories hypothesize that space-time is fundamentally made up of discrete units or ``atoms'' of space, in a similar way to how matter is fundamentally made up of discrete particles. In the context of space-time, the term ``atomic structure'' is used metaphorically to suggest a discrete or granular nature at extremely small scales. In Einstein's special theory of (...) relativity, there is a maximum limit of speed, beyond which no point of mass in an inertial reference frame can travel. In the following, I will demonstrate that in a given inertial reference frame, in addition to the existence of an upper limit velocity of the motion of a point of mass, there are also logical reasons to think that space and time have an atomic structure. The basic idea of this argument was suggested by Zeno's well-known aporias. (shrink)

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 being equivalent with contradiction is always low level for every input state, whereas the output of a circuit corresponding to a tautology is always high level, irrespective of the input (...) states. On the other hand, the remaining propositions correspond to circuits whose output is high-level if and only if certain atomic components of the proposition are true, and the rest of the atomic components are false. The inputs equivalent to the atomic propositions are either high or low level. However, what logical circuits are equivalent to a circulating statement or argumentation? -/- The propositions are true or false irrespective of time, whereas the voltage level of the circuits can change over time. To be more precise, we can say that the input levels of the circuits are high or low depending on whether we evaluate the atomic formulae of the formula expressing the logical structure of the proposition as true or false. The voltage level of the output of the circuit corresponds to the truth value of the formula. I will call `{combinational automaton}' the digital circuit that may model the formulae of propositional logic. -/- Formulae connected with truth functions yield formulae again. Although there are always corresponding automata for them, the situation is not simple in this case. We do not always get combined automata joined to each other, and in some cases, it is also possible that we do not get an automaton--an operating machine or circuit--at all. -/- There are digital circuits whose output is not a function of their input. The range of automata is wider than that of combinational automata. It includes machines whose input states do not unambiguously determine the output states; that is, the output is not a function of the input. This is because the circuit has feedback. Most digital circuits belong to this latter group. I will call them "sequential automata". -/- The question arises whether there is a logical structure of a circulating statement that 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 for further developing the model. (shrink)

Many logic handbooks allude to the obvious connection between propositional logic and logic circuits. Truth functions in logic can be represented by logic circuits in which the high or low voltage levels of the circuits correspond to the true and false logic values, respectively. At the propositional logic level, the logical connectives of propositions can be simulated by logic circuits as follows: the true or false logical evaluation of atomic propositions corresponds to the high or low level of the circuit (...) input and the truth value of compound propositions corresponds to the circuit output state. A high circuit output signifies that the compound sentence is evaluated as true, whereas a low output indicates that the compound sentence is false. It is well known that in the world of logic circuits, the AND connective in logic corresponds to the AND gate, the OR connective to the OR gate and the negation operation to the inverter. The output of a circuit equivalent to contradiction is always low and that of a circuit corresponding to tautology is always high irrespective of the input state. The remaining compound formulas correspond to logic circuits with a high output level for some inputs and low output level for other inputs. However, what logical circuit can model a circular sentence? (shrink)

The Russell's paradox concerns the foundations of naive set theory. This short short paper is about how it can be interpreted in other contexts and has significance in the world of commands. Understanding the paper assumes that the reader is broadly familiar with the foundations of set theory and its history. The text contains many formulas and therefore the reader should be comfortable in the world of logical formulas. My example is somewhat similar to the barber paradox. There, too, we (...) are puzzled by the feasibility of a task. In the case of the barber paradox there is a solution: the barber is a woman, in my example there is no such escape. (shrink)

The Truth problem is one of the central problems of philosophy. Nowadays, every major theory of truth that applies to formal languages utilizes devices referring to formulae. Such devices include name-forming functions. The theory of truth discussed in this paper applies to strict formal logic languages, the critique of which must, therefore, also obey mathematical rigour. This is why I have used formal logic derivations below rather than the argumentation of ordinary language.

To solve the ship of Theseus puzzle, you need several keys. First of all, we need to fix the application condition of the physical object—it is a vehicle or an object memory; then the identification criterion—the ship was named The Ship of Theseus when he boarded it; and finally the re-application criterion—a ship (or an object memory) when it is the same ship (or object memory) as before. The degree of change from the previous and reference states must be given. (...) The identity of physical objects over time can be expressed by the uniformity of the time slices of the persisting objects. Uniformity exists from the point of view that these instances are temporal cross-sections of the same object; they may differ in other characteristics; there are no general a priori principles independent of practice. The definition of identity is decided by the community of language users. The two states of an object are the same as long as the rate of change of state remains below the threshold value relative to the reference. The two states of an object are similar as long as the change is below a threshold value with respect to each other. Similarity can be described by a tolerance relationship. The mathematical relation describing uniformity is formally an equivalence relation, just as the relation describing identity is an equivalence relation. A physical object is identical to an equivalence class when we speak of the object's identity over time and identical or not identical to an element of the equivalence class when we speak of the object having changed or ceased to exist. (shrink)

Sometimes it is worth using Tarski’s solution rather than merely mentioning it.

The book the reader is holding in his/her hands present the life and work of painter Tibor András focusing on his works made during the II. World War and the 1956 Hungarian revolution. His paintings and drawings would be hard to comprehend without outlining the context of his personal life and the historic events he got entangled in. This is why this book has three parts not counting the appendix. The first part presents the revolution as a visual reality of (...) the historical course of events; the second compares the drawings and paintings to themselves and to other contemporary works; and the final part showcases personal recollections. These three presentations were written by different authors. Beside the logical coherence we didn’t mean to harmonize these approaches stylistically or otherwise. Rather we thought that different views and different aspects better highlight and emphasize the personal qualities that are very important considering the characteristics of Tibor András’ artistic activity that extended to half of a century. Tibor András had significantly changed his subject matters, style and technique of applied graphics and paintings several times during his career. It’s hard to separate his artistic periods since he often went back to a work that seemed finished or to a style he used to paint. “A painter can just simply finish a painting; he/she must stop working on it.” - was his motto and he either was right or not that approach was the leading guide through his career and in spite of the flaws in his oeuvre his style is clearly visible behind every change. (shrink)