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PHIL 1010 INTRODUCTION TO LOGIC (3) LEC. 3. Humanities Core. Basic logical principles and applications: definition, informal fallacies, categorical logic, elementary propositional logic, analogy, and selected inductive inferences.

PHIL 1017 HONORS LOGIC (3) LEC. 3. Pr. Honors College. Humanities Core. Basic logical principles and applications: definition, informal fallacies, categorical logic, elementary propositional logic, analogy, and selected inductive inferences.

In [61] a large collection of over 100 different systems is discussed which are all captured by pattern (Fig. 7). In this pattern, the training phase of a learning system is guided by information that is obtained from a symbolic inference system (pattern Fig. 2b). For this purpose, the training step from elementary pattern (Fig. 1a) is extended with a further input to allow for this guidance by inferred symbolic information. A particular example is where domain knowledge (such as captured in modern large knowledge graphs) is used as a symbolic prior to constrain the search space of the training phase [8]. In general, this patterns also captures all systems with a so-called semantic loss function [66], where (part of) the loss-function is formulated in terms of the degree to which the symbolic background knowledge is violated. Such a semantic loss-function is also used in [38], where the semantic loss is calculated by weighted model counting. In [20] and [41] the semantic loss-function is realised through approximate constraint satisfaction. Another example is [19] where logical rules are used as background knowledge for a gradient descent learning task in a high-dimensional real-valued vector space. In the same spirit, [52] exploits a type-hierarchy to inform an embedding in hyperbolic space.

An entirely different category of systems that is captured by the same pattern are constrained reinforcement learners (e.g. [28]), where the exploration behaviour of a reinforcement learning agent is constrained through symbolic constraints that enforce safety conditions. Similarly, [33] uses high-level symbolic plans to guide a reinforcement learner towards efficiently learning a policy. Silvestri et al. [57] shows how adding domain knowledge in the form of symbolic constraints greatly improves the sampling-frequency of a neural network trained to solve a combinatorial problem. The LYRICS system [42] proposes a generic interface layer that allows to define arbitrary first order logic background knowledge, allowing a learning system to learn its weights under the constraints imposed by the prior knowledge.

Whereas pattern (Fig. 9) provides representation learning by embedding vector/tensor representations of logical structures in a neural network, there are also attempts to learn the reasoning process itself in neural networks. This is motivated by the ability of neural networks to provide higher scalability and better robustness when dealing with noise in the input data (incomplete, contradictory, and erroneous). The focus of pattern (Fig. 10) is on reasoning with first-order logic on knowledge graphs. This pattern learns specific reasoning tasks based on symbolic input tuples and the inferencing results from the symbolic reasoner. Pattern (Fig. 10) is a combination of our basic patterns for symbolic reasoning (Fig. 2b) and training to produce a statistical model (Fig. 1a).

This article discusses the process of creating and using visual maps to guide legal reasoning because law and the complex facts legal professionals confront are not intuitive, and visuals are underutilized in legal writing. Part of the reason for this is that we have inherited guides to legal reasoning imbedded in judicial decisions based on literal descriptions of logic that have limited application to legal reasoning. This article seeks to overcome these literal bounds by illustrating how formal and informal logic can be described more figuratively or visually.

Section II describes how facts and law material to legal reasoning have been traditionally described in terms of formal logic by leading jurists and scholars. They provide a verbal description of the power and limits of formal logic in legal reasoning. Section II makes the important distinction between formal logic and informal logic and their continuing role in reaching legal conclusions. Section II then begins to describe how formal logic in legal reasoning can be described figuratively or visually in a more accessible manner.

Lawyers and judges often disparage an opposing position as illogical.[33] But for such an argument to be persuasive, lawyers should know how logic strengthens legal conclusions or reveals fallacies in them. To help clarify the use of logic in legal reasoning, this section starts by discussing the two fundamental formal types of logic: deductive reasoning and inductive reasoning, including the inductive forms of reasoning by analogy and generalization. It is important to understand what formal logic actually is because the application of such forms to legal reasoning can be very powerful and very limited.

Once applicable rules and principles are inferred, legal reasoning becomes deductive in an informal sense based on the probability that a conclusion is the best one.[64] One may also use analogies to reach a conclusion. In other words, legal reasoning may involve an interplay of analogy, generalization, and informal deduction to reach a conclusion.[65] This interplay allows judges and lawyers to address complex issues involving long chains of propositions.[66] And in this way, formal logic can strengthen legal conclusions or reveal fallacies in them.

Generalization is the other form of inductive reasoning that is such an instinctive thought process that it can be performed intuitively without thinking of it as a form of logic. Generalization involves the process of forming a conclusion from a variety of particular pieces of information.[74] Generalizations serve to sharpen our understanding of concepts by concentrating our minds on a few essentials.[75] In legal reasoning, such information may take the form of facts or precedents. For example, the scene of a naked couple making love without showing anything below the neck, except for backs, arms, and legs has stood for over 50 years as a standard for what is not obscenity.[76] Like analogy, the strength of a generalization may be formally analyzed based on a number of statistical criteria.[77] A generalization based on particular facts may be stronger depending on the following: a) the larger the size of the sample or amount of evidence relative to the size of the subject about which one is generalizing, and b) the similarity between the sample and the object of generalization.[78] Such criteria may even be codified into standards, as they are in the actuarial profession.[79]

To try to describe the logical relationship between two concepts without formal logic, judges have begun using or referring to Venn diagrams.[88] This section gives examples of Venn diagrams in judicial decisions to introduce this important form of informal logic.

Instead of using formal terms of logic, judges schooled in symbolic logic began to conceptualize legal relationships by visualizing and drawing them in cases to describe the same fundamental relationships of inclusion and exclusion described by Venn. Such cases involve statutory schemes common to administrative hearings, such as those that appear on the docket of this author. They also involve complex law and subject matter, as shown in the examples that follow.

Such diagrams are a symbolic form of logic using sets to express the same relationships of containment described in deductive reasoning or, more simply, the relationships between concepts and the parts they contain.

Visuals are a realistic tool to help visualize the inherent logic of legal reasoning in its informal or broadest sense in a way that makes the law come alive.[101] Visuals can perform this function by representing the fundamental elements of reasoning separately and together. This section gives examples of how these qualitative elements can be visualized separately and together to draw conclusions. A separate subsection is provided below to discuss how visuals can be used to summarize statistical inferences used to find facts or to support legal conclusions.

The pictures worth a thousand legal words in legal writing are those that simply summarize complex facts, law, or combinations of both accompanied by text. Judges and lawyers can use visuals to form mental pictures in their minds needed to aid the process of inferring the applicable law, mapping what the law is, and envisioning the circumstances that are contemplated by it in ways that formal and informal logic described by text alone cannot. Lawyers can insert visuals in legal briefs and judges can incorporate electronic versions of them in their decisions or create their own images as instructed above.

What distinguishes the text is its graded step-by-step approach to the subject, with informal logic forming the basis and Symbolic logic and Inductive logic forming the more advanced steps. The book also uses a hands-on approach to teaching of logic to induce self-learning, as shown in sections such as on how to create a truth table or a truth tree, on providing strategic tips for formal derivations, and on how to approach symbolization in predicate logic.

\r \tWhat distinguishes the text is its graded step-by-step approach to the subject, with informal logic forming the basis and Symbolic logic and Inductive logic forming the more advanced steps. The book also uses a hands-on approach to teaching of logic to induce self-learning, as shown in sections such as on how to create a truth table or a truth tree, on providing strategic tips for formal derivations, and on how to approach symbolization in predicate logic.

Frameworks always involve logical consequence relation(s), theirdefinition(s) of what follows from what logically. But this need notalways mean that frameworks have purely logical object languages thatare then provided with empirical interpretations. Other modes ofinference employed in existing scientific disciplines, includingconceptual inference, experimental procedures, and the reasoning fromthese, can also be reconstructed in or by frameworks. Carnap continuedto hope that all of them could, eventually, be understood in morestraightforwardly logical terms, but he realized that this was along-term program and not immediately on the horizon. In particular,the framework could also come equipped with an inductive logic, wherethe deductive consequence relation for the object language isaugmented by a numerical degree-of-confirmation assignment thatsatisfies the axioms of probability. In any case, Carnapian languageor framework go beyond what we now mean by a formal language: they donot just involve a syntactic vocabulary and syntactic formation rules.From the modern logical point of view, a Carnapian framework is closeto a logic or a formal theory (but perhaps with an interpretation),while from the philosophy of science point of view, a Carnapianframework is meant to reconstruct the conceptual and inferentialpresuppositions of a scientific theory rather than a scientific theoryitself. 2b1af7f3a8