Bits and pieces: understanding information decomposition from part-whole relationships and formal logic

Partial information decomposition (PID) seeks to decompose the multivariate mutual information that a set of source variables contains about a target variable into basic pieces, the so-called ‘atoms of information’. Each atom describes a distinct way in which the sources may contain information about the target. For instance, some information may be contained uniquely in a particular source, some information may be shared by multiple sources and some information may only become accessible synergistically if multiple sources are combined. In this paper, we show that the entire theory of PID can be derived, firstly, from considerations of part-whole relationships between information atoms and mutual information terms, and secondly, based on a hierarchy of logical constraints describing how a given information atom can be accessed. In this way, the idea of a PID is developed on the basis of two of the most elementary relationships in nature: the part-whole relationship and the relation of logical implication. This unifying perspective provides insights into pressing questions in the field such as the possibility of constructing a PID based on concepts other than redundant information in the general n-sources case. Additionally, it admits of a particularly accessible exposition of PID theory.

On the action of the implicative closure operator on the set of partial functions of the multivalued logic

On the set P k ∗ $\begin{array}{} \displaystyle P_k^* \end{array}$ of partial functions of the k-valued logic, we consider the implicative closure operator, which is the extension of the parametric closure operator via the logical implication. It is proved that, for any k ⩾ 2, the number of implicative closed classes in P k ∗ $\begin{array}{} \displaystyle P_k^* \end{array}$ is finite. For any k ⩾ 2, in P k ∗ $\begin{array}{} \displaystyle P_k^* \end{array}$ two series of implicative closed classes are defined. We show that these two series exhaust all implicative precomplete classes. We also identify all 8 atoms of the lattice of implicative closed classes in P 3 ∗ $\begin{array}{} \displaystyle P_3^* \end{array}$ .

Semantic Relativism and Logical Implication

Semantic relativism is the view that the truth-value of some types of statements can vary depending on factors besides possible worlds and times, without any change in their propositional content. It has grown increasingly popular as a semantic theory of several types of statements, including statements that attribute knowledge of a proposition to a subject (knowledge attributions). The ways of knowing claim is the view that perception logically implies knowledge. In my “Semantic Relativism and Ways of Knowing” (2019) I argued that a relativist semantics for knowledge attributions is incompatible with the ways of knowing claim. I suggested that this incompatibility depends on some basic features of the logic of relativist semantics, and therefore can be shown to generalise beyond the discussion of knowledge attributions to semantic relativism more broadly. Here I make this generalisation. I demonstrate that for any proposition p expressed by a statement that does not have a relativist semantics, and for any proposition q expressed by a statement that does have a relativist semantics, p fails to logically imply q. I explain why this happens, discuss some of its philosophical consequences, and consider a way to modify relativist semantics to avoid it. I conclude that semantic relativism raises interesting philosophical questions that have gone largely unnoticed in discussions of this view until now.

Kasih Sejati: Memahami Misi Secara Teks, Konteks, dan Komunitas Berdasarkan Yohanes 17:18 Pada PTT/AK

The existence of God is in essence impossible to shift, but, in fact seems to experience a shift that appears in human efforts to respond to the mission call. Efforts to "narrow" the meaning, purpose and impact of understanding the "true love and mission" for His people here are in contrast to God's love that cannot be separated from His essence. God himself wants His love to be part of His ransom people, so this is where "mission" becomes the logical implication of God's existence as a mission initiator that has been, is being, and continues to be done through His chosen people in every age in (community). The author used qualitative method with text and language analysis approaches. Based on research results in the "text" it can be understood that God's love enables Him to send His only begotten Son to fulfill His mission. Then in His grace; The Son continues the Father's mandate to disciples to become a "community" sharing the love of God the Father. This task is then passed on to the present generation specifically for graduates of Theological/Christian Religious College.Key Words: True Love, Text, Context, CommunityAbstrak:Eksistensi Allah pada hakikatnya tidak mungkin bergeser, namun, pada kenyataannya terkesan mengalami pergeseran yang nampak dalam upaya manusia meresponi panggilan missions. Upaya “menyempitkan” makna, tujuan dan dampak dari pemahaman akan “kasih sejati dan misi” bagi umat-Nya disini bertolak belakang dengan kasih Allah yang tidak dapat dipisahkan dengan hakikat-Nya. Allah sendiri menginginkan agar kasih-Nya menjadi bagian dari umat tebusan-Nya”, maka disinilah hal “misi” menjadi implikasi logis dari eksistensi Allah sebagai Inisiator misi yang telah, sedang, dan terus dikerjakan melalui orang-orang pilihan-Nya pada setiap zamannya dalam (komunitas). Penulis menggunakan metode kualitatif (Qualitative Research) dengan pendekatatan analisis teks dan Bahasa. Berdasarkan hasil kajian secara “teks” dapat dipahami bahwa kasih Allah memungkinkan Dia mengutus Anak-Nya yang tunggal untuk menggenapi misi-Nya. Kemudian dalam anugerah-Nya; Sang Anak meneruskan mandat Allah Bapa itu kepada para murid untuk menjadi satu “komunitas” pembagi kasih Allah Bapa. Tugas inilah yang kemudian diteruskan pada generasi masa kini secara khusus bagi lulusan Perguruan Tinggi Teologi / Agama Kristen.

Kasih Sejati: Memahami Misi Secara Teks, Konteks, dan Komunitas Berdasarkan Yohanes 17:18 Pada PTT/AK

The existence of God is in essence impossible to shift, but, in fact seems to experience a shift that appears in human efforts to respond to the mission call. Efforts to "narrow" the meaning, purpose and impact of understanding the "true love and mission" for His people here are in contrast to God's love that cannot be separated from His essence. God himself wants His love to be part of His ransom people, so this is where "mission" becomes the logical implication of God's existence as a mission initiator that has been, is being, and continues to be done through His chosen people in every age in (community). The author used qualitative method with text and language analysis approaches. Based on research results in the "text" it can be understood that God's love enables Him to send His only begotten Son to fulfill His mission. Then in His grace; The Son continues the Father's mandate to disciples to become a "community" sharing the love of God the Father. This task is then passed on to the present generation specifically for graduates of Theological/Christian Religious College.

Contextualism and closure

Skepticism confronts us with a paradox (sometimes known as “the skeptical trilemma”), a version of which follows: (1) I know that I am working on a computer right now; (2) I know that knowing that I am working on a computer right now logically implies that I am not being deceived or manipulated in the way that skeptical hypotheses imagine. (This implication is called “closure under known logical implication”); (3) I do not or cannot know that I am not being deceived or manipulated in the way skeptical hypotheses imagine. The paradox of skepticism is that these three statements are logically incompatible. A relatively new movement in epistemology called contextualism proposes that we can accept all three of the claims in the trilemma, by recognizing that they are not all true within the same epistemic context. Briefly, contextualists claim that we can know in ordinary contexts, but cannot know that we are not being deceived or manipulated in a skeptical scenario, but the latter fact is true in a different epistemic context than the ordinary knowledge that we might have. Closure under known logical implication will remain true, but only insofar as the implications involved are alternatives that belong to the same epistemic context as the original knowledge claim. In this paper, I claim that contextualism’s account of how epistemological contexts change, together with its acceptance of closure, is implausible.

The Scholastic Logic of Statistical Hypotheses

Among the important conceptual innovations introduced in the second scholasticism era and motivated by theological debates following the Council of Trent were the theories of moral necessity and moral implication. As they were centred upon a view of moral necessity as a form of necessity weaker than physical (and, ipso facto, metaphysical and logical) necessity, and moral implication as weaker than physical (and, ipso facto, metaphysical and logical) implication, some interpretations of moral necessity encouraged the logic of statistical hypotheses and probability. Three branches of this debate are studied in this paper: the explanation of moral necessity in terms of suppositio (Vega, Molina, Hurtado, Sforza Pallavicino), the confrontation over the interpretation of moral necessity (Quirós, Herrera), and the theory of statistical quantification (Elizalde, Terill, de Benedictis).

Deductive closure principle

It seems that one can expand one’s body of knowledge by making deductive inferences from propositions one knows. The ‘deductive closure principle’ captures this idea: if S knows that P, and S correctly deduces Q from P, then S knows that Q. A closely related principle is that knowledge is closed under known logical implication: if S knows that P and S knows that P logically implies Q, then S knows that Q. These principles, if they hold, are guaranteed by general features of the concept of knowledge. They would form part of a logic of knowledge. An influential argument for scepticism about knowledge of the external world employs the deductive closure principle. The sceptic begins by sketching a logically possible hypothesis, or counter-possibility (for example, that one is a brain in a vat, with computer-induced sense experience) which is logically incompatible with various things one claims to know (such as that one has hands). The proposition that one has hands logically implies the falsity of the sceptical hypothesis. Supposing that one is aware of this implication, the deductive closure principle yields the consequence that if one knows that one has hands, then one knows that one is not a brain in a vat. The sceptic argues that one does not know this: if one were in a vat, then one would have just the sensory evidence one actually has. It follows that one does not know that one has hands. Some philosophers have sought to block this argument by denying the deductive closure principle.