Logic, Language, and Calculus

The difference between object-language and metalanguage is crucial for logical analysis, but has yet not been examined for the field of computer science. In this paper the difference is examined with regard to inferential relations. It is argued that inferential relations in a metalanguage (like a calculus for propositional logic) cannot represent conceptual relations of natural language. Inferential relations govern our concept use and understanding. Several approaches in the field of Natural Language Understanding (NLU) and Natural Language Inference (NLI) take this insight in account, but do not consider, how an inference can be assessed as a good inference. I present a logical analysis that can assesss the normative dimension of inferences, which is a crucial part of logical understanding and goes beyond formal understanding of metalanguages.

[1]  John R. Searle,et al.  Minds, brains, and programs , 1980, Behavioral and Brain Sciences.

[2]  Robert Brandom,et al.  Reason in Philosophy: Animating Ideas , 2009 .

[3]  G. Gentzen.  Untersuchungen über das logische Schließen. I , 1935 .

[4]  Drew McDermott,et al.  Introduction to artificial intelligence , 1986, Addison-Wesley series in computer science.

[5]  Phil Blunsom,et al.  Teaching Machines to Read and Comprehend , 2015, NIPS.

[6]  Christopher Potts,et al.  Posing Fair Generalization Tasks for Natural Language Inference , 2019, EMNLP.

[7]  Saul A. Kripke,et al.  Semantical Analysis of Modal Logic I Normal Modal Propositional Calculi , 1963 .

[8]  T. Zoglauer,et al.  Einführung in die formale Logik für Philosophen , 2008 .

[9]  T. Williamson.  Review of Robert Brandom, Reason in Philosophy: Animating Ideas , 2014 .

[10]  Christopher Potts,et al.  Modular Representation Underlies Systematic Generalization in Neural Natural Language Inference Models , 2020, ArXiv.

[11]  Richard Evans,et al.  Can Neural Networks Understand Logical Entailment? , 2018, ICLR.

[12]  W. Quine,et al.  On What There Will Be , 1948, Realisms Interlinked.

[13]  Monika Gruber.  Alfred Tarski and the "Concept of Truth in Formalized Languages" , 2016 .

[14]  Leonard Linsky,et al.  Reference and modality , 1971 .

[15]  Christopher Potts,et al.  A large annotated corpus for learning natural language inference , 2015, EMNLP.

[16]  K. Gödel.  Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I , 1931 .

[17]  Christopher Potts,et al.  On the Effective Use of Pretraining for Natural Language Inference , 2017, ArXiv.

[18]  Christopher Potts,et al.  Learning Distributed Word Representations for Natural Logic Reasoning , 2014, AAAI Spring Symposia.

[19]  G. Ryle.  I.—Knowing How and Knowing that: The Presidential Address , 1946 .

[20]  Robert Brandom,et al.  Making it explicit : reasoning, representing, and discursive commitment , 1996 .

[21]  Paul Hoyningen-Huene,et al.  Formal Logic: A Philosophical Approach , 2004 .

[22]  R. Carnap.  Logical Syntax of Language , 1937 .

[23]  Mohit Bansal,et al.  Adversarial NLI: A New Benchmark for Natural Language Understanding , 2020, ACL.

[24]  B. Russell.  The Principles of Mathematics , 1938 .

[25]  Patrick Pantel,et al.  From Frequency to Meaning: Vector Space Models of Semantics , 2010, J. Artif. Intell. Res..

[26]  J. Bessant,et al.  Saying and Doing: Child Protective Service and participation in decision-making , 2014 .

[27]  Ido Dagan,et al.  The Third PASCAL Recognizing Textual Entailment Challenge , 2007, ACL-PASCAL@ACL.

[28]  Christopher Potts,et al.  Stress-Testing Neural Models of Natural Language Inference with Multiply-Quantified Sentences , 2018, ArXiv.

[29]  Sanjay Kaul,et al.  What the Tortoise said to Achilles. , 2010, The American journal of cardiology.

[30]  Phil Blunsom,et al.  Reasoning about Entailment with Neural Attention , 2015, ICLR.