علم الدلالة (المنطق)

في علم المنطق ، يُعنى علم الدلالة، أو الدلالة الصورية، بدراسة معنى وتفسير اللغات الصورية ، والأنظمة الصورية ، واللغات الطبيعية (بصورها المثالية) . ويسعى هذا المجال إلى توفير نماذج رياضية دقيقة تُجسّد المفاهيم ما قبل النظرية للصدق ، والصحة ، والنتيجة المنطقية . وبينما يهتم بناء الجملة المنطقي بالقواعد الصورية لبناء تعابير سليمة، فإن الدلالة المنطقية تُرسّخ أُطرًا لتحديد متى تكون هذه التعابير صحيحة، وما يترتب عليها.

أدى تطور الدلالات الرسمية إلى ظهور عدة مناهج مؤثرة، منها الدلالات النموذجية (التي رائدها ألفريد تارسكي )، والدلالات البرهانية (المرتبطة بجيرهارد جنتزن ومايكل دوميت )، ودلالات العوالم الممكنة (التي طورها شاول كريبكي وآخرون للمنطق الموجه والأنظمة ذات الصلة)، والدلالات الجبرية (التي تربط المنطق بالجبر المجرد )، ودلالات الألعاب (التي تفسر الصلاحية المنطقية من خلال مفاهيم نظرية الألعاب ). وتعكس هذه المناهج المتنوعة وجهات نظر فلسفية مختلفة حول طبيعة المعنى والحقيقة في الأنظمة المنطقية.

تعتمد شروط صدق الجمل المختلفة التي قد نصادفها في الحجج على معناها، ولذا لا يستطيع علماء المنطق تجنب الحاجة إلى معالجة معنى هذه الجمل. يشير علم دلالات المنطق إلى المناهج التي وضعها علماء المنطق لفهم وتحديد ذلك الجزء من المعنى الذي يهمهم؛ فالعالم المنطقي تقليديًا لا يهتم بالجملة كما هي، بل بالقضية ، وهي جملة مثالية قابلة للمعالجة المنطقية.

Before modern logic, interpretations of logic were based on Aristotle's Organon, especially De Interpretatione. The problem of multiple generality required quantifications to be introduced, and that made it impossible to perform the kind of subject–predicate analysis in Aristotle's logic. Term logic is an attempt to modernize Aristotle's logic: find deductive systems in the spirit of Aristotle's syllogisms, but with the generality of modern logics based on the quantifier.

The main modern approaches to semantics for formal languages are the following:

• The archetype of model-theoretic semantics is Alfred Tarski's semantic theory of truth, based on his T-schema, and is one of the founding concepts of model theory. This is the most widespread approach, and is based on the idea that the meaning of the various parts of the propositions are given by the possible ways we can give a recursively specified group of interpretation functions from them to some predefined mathematical domains: an interpretation of first-order predicate logic is given by a mapping from terms to a universe of individuals, and a mapping from propositions to the truth values "true" and "false". Model-theoretic semantics provides the foundations for an approach to the theory of meaning known as truth-conditional semantics, which was pioneered by Donald Davidson. Kripke semantics introduces innovations, but is broadly in the Tarskian mold.
• Proof-theoretic semantics associates the meaning of propositions with the roles that they can play in inferences. Gerhard Gentzen, Dag Prawitz and Michael Dummett are generally seen as the founders of this approach; it is heavily influenced by Ludwig Wittgenstein's later philosophy, especially his aphorism "meaning is use".
• Truth-value semantics (also commonly referred to as substitutional quantification) was advocated by Ruth Barcan Marcus for modal logics in the early 1960s and later championed by J. Michael Dunn, Nuel Belnap, and Hugues Leblanc for standard first-order logic. James Garson has given some results in the areas of adequacy for intensional logics outfitted with such a semantics. The truth conditions for quantified formulas are given purely in terms of truth with no appeal to domains whatsoever (and hence its name truth-value semantics).
• Game semantics or game-theoretical semantics made a resurgence mainly due to Jaakko Hintikka for logics of (finite) partially ordered quantification, which were originally investigated by Leon Henkin, who studied Henkin quantifiers.
• Probabilistic semantics originated from Hartry Field and has been shown equivalent to and a natural generalization of truth-value semantics. Like truth-value semantics, it is also non-referential in nature.

• Jaakko Hintikka (2007), Socratic Epistemology: Explorations of Knowledge-Seeking by Questioning, Cambridge: Cambridge University Press.
• Ilkka Niiniluoto (1999), Critical Scientific Realism, Oxford: Oxford University Press.
• John N. Martin (2019), The Cartesian Semantics of the Port Royal Logic , Routledge.