We introduce a new formal Logic, called equality Propositional Logic. It has two basic connectives, ^ (conjunction) and ≡ (equivalence). Moreover, the ⇒ (implication) connective can be derived as A ⇒ B: = (A ^ B) ≡ A. We formulate the equality Propositional Logic and demonstrate that the resulting Logic has reasonable properties such as Modus Ponens(MP) rule, Hypothetical Syllogism(HS) rule and completeness, etc. Especially, we provide two ways to prove the completeness of this Logic system. We also introduce two extensions of equality Propositional Logic. The first one is involutive equality Propositional Logic, which is equality Propositional Logic with double negation. The second one adds prelinearity which is rich enough to enjoy the strong completeness property. Finally, we introduce additional connective Δ (delta) in equality Propositional Logic and demonstrate that the resulting Logic holds soundness and completeness.

The first chapter of the paper discusses the concept of “Propositionality” attributed to Logical arguments, and by adopting the criterion introduced by Christopher Martin, the “Propositionality” of Logical arguments is elaborated based on the Principle of Substitution in compound propositions. The second chapter discusses the viewpoints of Kamran Karimullah on the “non-Propositionality” of Al-Farabi’s conditional arguments and “Propositionality” of Avicenna to reach a judgment. In this approach, the “non-Propositionality” of Al-Farabi’s Logical teaching is confirmed not only in its dialectical context, but is also confirmed beyond it. The chapter continues by showing that Avicenna’s viewpoints on the conditional arguments and propositions maintain differs somehow from modern “Propositional Logic” despite satisfying the criterion of Propositionality.

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Usually two-dimensional substitution is used for the production of integrated circuits. Due to heavy use of communications, two-dimensional substitution has high losses as well as, density of elements is low in it. To resolve this problem, three-dimensional substitution method was proposed. Rather than the two-dimensional arrangement of elements in a row, elements are layout in three dimensions in this substitution. In this paper, three-dimensional substitution algorithms using order used in two-dimensional substitution, three-dimensional substitution analysis by mPL and three-dimensional substitution simultaneous with two-dimensional substitution by mPL have been studied in terms of structure and function, and a butterfly processing element (PE) and an Advanced Encryption Standard (AES) block and a wireless MIMO decoder to assess them have been implemented with the mentioned methods. . Applying these methods shows that the use of face to face integration by microbuses in communications of substitution algorithm, on average, improves the maximum clock and block speed of AES encryption to 15. 3% and the maximum clock and block speed of PE module to 22. 6% as well as the maximum clock and speed of MIMO modules to 17. 1%, while the use of these methods has led to the average reduction in power consumption of 2. 6% for the AES module and the average reduction in power of 12. 9% for the PE module and the average reduction in power consumption of 5. 1% for MIMO module.

The present paper is devoted to the phenomenon of "presupposition" and its variants. This Phenomenon has alluded generations of linguists. In the same vein, it has drawn the attention of Logicians in view of their interest in the expansion of the first-rank predicate Logic and the development of the second-rank predicate Logic. This paper is purported to address the concept of presupposition and to analyze its nature from both Logical and linguistic vantage-points.

In Classical Logic, it is not possible to conclude from "If P then Q" that "It is not the case that if P then ∼Q". This argument, whose conditional counterpart is known as Boethius' thesis, is abundantly attested in the realm of causal, conceptual, and Logical relations. Aristotle's thesis "It is not the case that if P then P" is not a theorem in this Logic. Furthermore, in Classical Logic, each of the two propositions P and Q is derived from "It is not the case that if P then ∼Q", against which there is a lot of evidence. The Non-Truth Functional System of Propositional Logic is an answer to these problems, in which causal, conceptual, and Logical relations are analyzed, formulated, and evaluated in accordance with natural intuition without exception. In article "Hajhosseini's Non-Truth Functional Logic", Asadollah Fallahi makes three specific criticisms of this system: "The number of inference rules are reducible", "Every Propositional variable is a theorem, and this system and its extension are trivial" and "The extension of the Non-Truth Functional System reduces to classical Logic". In this article, we show that the first criticism is based on some incorrect proofs. Also, the second criticism arises from the incorrect definition of some non-truth functional combinations or the incorrect proof of some arguments. Finally, the third criticism is solved by reducing the rules of distributivity. For our answer to his repeated criticisms we refer to the article "Critical Review of a Criticism on the Theory of Truth-Functional System".

The nature of Propositional knowledge is of important issues in contemporary epistemology. The common theory of epistemologists, knows Propositional knowledge as "justified true belief", but this view encountered the problem called “Gettier problem ". The contemporary epistemologists caused a fundamental rethink with the concept of knowledge and different approaches, based on Gettier criticism on the tripartite analysis of knowledge. This essay seeks to extract theory of Propositional knowledge sadr-ol-mote’allehin to correspond to contemporary epistemology approaches through review of 'knowledge' issue in transcendent theosophy (Hikmah Muta’liyah). Sadra's theory of Propositional knowledge is multifaceted theory; an aspect of epistemic and cognitive function. He not only does not reduce knowledge to Propositional knowledge alone, but also he considers it as an aspect of cognitive function. Knowledge of recent aspect affecting on subject. He does not limit to a way achieving knowledge by one method, rather with high attitude, knows effective in achieving knowledge of all faculties, knowledge and moral virtues, divine faculty even God.