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Blanchette, Patricia, , . Logical Consequence
2001, In Lou Goble (Ed). Blackwell Guide to Philosophical Logic. Wiley-Blackwell: 115-135.
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Added by: Berta Grimau, Contributed by: Patricia Blanchette

Description: This article is a short overview of philosophical and formal issues in the treatment and analysis of logical consequence. The purpose of the paper is to provide a brief introduction to the central issues surrounding two questions: (1) that of the nature of logical consequence and (2) that of the extension of logical consequence. It puts special emphasis in the role played by formal systems in the investigation of logical consequence.

Comment: This article can be used as background or overview reading in a course on the notion of logical consequence. It could also be used in a general course on philosophy of logic having a section on this topic. It makes very little use of technical notation, even though familiarity with first-order logic is required. It closes with a useful list of suggested further readings.

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Blanchette, Patricia, , . Models and Modality
2000, Synthese 124(1): 45-72.
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Added by: Berta Grimau, Contributed by: Patricia Blanchette

Abstract: This paper examines the connection between model-theoretic truth and necessary truth. It is argued that though the model-theoretic truths of some standard languages are demonstrably “necessary” (in a precise sense), the widespread view of model-theoretic truth as providing a general guarantee of necessity is mistaken. Several arguments to the contrary are criticized.

Comment: This text would be best used as secondary reading in an intermediate or an advanced philosophy of logic course. For example, it can be used as a secondary reading in a section on the connection between model-theoretic truth and necessary truth.

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Kouri Kissel, Teresa, , Stewart Shapiro. Classical Logic
2018, The Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.)
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Summary: This article provides the basics of a typical logic, sometimes called ‘classical elementary logic’ or ‘classical first-order logic’, in a rigorous yet accessible manner. Section 2 develops a formal language, with a syntax and grammar. Section 3 sets up a deductive system for the language, in the spirit of natural deduction. Section 4 provides a model-theoretic semantics. Section 5 turns to the relationships between the deductive system and the semantics, and in particular, the relationship between derivability and validity. The authors show that an argument is derivable only if it is valid (soundness). Then they establish a converse: that an argument is valid only if it is derivable (completeness). They also briefly indicate other features of the logic, some of which are corollaries to soundness and completeness. The final section, Section 6, is devoted to a brief examination of the philosophical position that classical logic is ‘the one right logic’.

Comment: This article introduces all the necessary tools in order to understand both the proof-theoretic and the model-theoretic aspects of first-order classical logical consequence. As such it can be used as a main reading in an introductory logic course covering classical first-order logic (assuming the students will have already looked at classical propositional logic). Moreover, the article covers some metatheoretic results (soundness, completeness, compactness, upward and downward Löwenheim-Skolem), which makes it suitable as a reading for a slightly more advanced course in logic. Finally, the article includes a brief incursion into the topic of logical pluralism. This makes it suitable to be used in a course on non-classical logics with an introduction module on classical logic.

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Sagi, Gil, , . Models and Logical Consequence
2014, Journal of Philosophical Logic 43(5): 943-964.
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Abstract: This paper deals with the adequacy of the model-theoretic definition of logical consequence. Logical consequence is commonly described as a necessary relation that can be determined by the form of the sentences involved. In this paper, necessity is assumed to be a metaphysical notion, and formality is viewed as a means to avoid dealing with complex metaphysical questions in logical investigations. Logical terms are an essential part of the form of sentences and thus have a crucial role in determining logical consequence.

Gila Sher and Stewart Shapiro each propose a formal criterion for logical terms within a model-theoretic framework, based on the idea of invariance under isomorphism. The two criteria are formally equivalent, and thus we have a common ground for evaluating and comparing Sher and Shapiro philosophical justification of their criteria. It is argued that Shapiro’s blended approach, by which models represent possible worlds under interpretations of the language, is preferable to Sher’s formal-structural view, according to which models represent formal structures. The advantages and disadvantages of both views’ reliance on isomorphism are discussed.

Comment: This paper provides an original view on the debate on the adequacy of the model-theoretic notion of logical consequence as well as a good overview of the relevant part of the debate. It can be used as standing on its own, but it can also serve as a complement to Sher (1996), also written by a female logician, and Shapiro (1998). Adequate for a general course on philosophy of logic or in a more specialized course on logical consequence. The paper is not technical, although students should’ve have taken at least an introductory logic course.

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