History of Formal Semantics, Lecture 6 History of Formal Semantics, Lecture 6 B. H. Partee, RGGU, March 28, 2012 p.1 B. H. Partee, RGGU, March 28, 2012 p.2 Lecture 6. History of Semantics in logic and philosophy, including the With the rise of mathematics in the 19th century, George Boole (1815-64) had an "Ordinary Language vs. Formal Languages" wars. Montague. algebraic conception for a system governing the “Laws of Thought”, a kind of calculus ratiocinator independent from the vagaries of natural language. (Boolean algebra turns 1. Philosophical and logical advances that made formal semantics possible................................... 1! out to have widespread application to natural language semantics, whether Boole would 1.1 Early philosophers ................................................................................................................. 1! like that or not.) 1.2 Frege ...................................................................................................................................... 2! 1.3 Russell, Carnap, Tarski.......................................................................................................... 3! 1.2 Frege 1.4 The Ordinary Language – Formal Language war.................................................................. 4! 1.5 Responses to the OL-FL war ................................................................................................. 5! 1.6 Possible Worlds Semantics.................................................................................................... 5! The greatest foundational figure for formal semantics is Gottlob Frege (1848-1925). He is 2. Montague’s work ......................................................................................................................... 6! credited with a number of ideas that have been crucial for logic and for semantics. One of 2.1 Why did Montague turn to natural language work? .............................................................. 7! his central contributions is the idea that function-argument structure is the key to 2.1.1 A note on the Kalish and Montague textbook................................................................ 7! semantic compositionality. Without the idea that some expressions denote functions that 2.1.2. A New Clue about Montague’s Motivations................................................................. 8! can apply to the denotations of other expressions, it was a mystery how compositionality 2.2 Central ideas in Montague’s work ......................................................................................... 9! should work – what kinds of things could the meanings of phrases be such that they could References ...................................................................................................................................... 11! be combined – by what kind of a “calculus”? -- to give meanings of larger phrases? Readings. To illustrate with a simple example, consider the sentence John is happy. Following (1) (Partee 2011) “Formal semantics: Origins, issues, early impact”, Sections 3 and 4. Frege, we say that the predicate is happy denotes the characteristic function of the set of Optional: happy entities; that function applies to the individual denoted by the name John to give (1) (Stanley 2008) Philosophy of language in the twentieth century ‘true’ or ‘false’, the denotation (extension) of the sentence. Schematically: (2) (Stokhof 2006) The development of Montague grammar (3) (Abbott 1999) The formal approach to meaning: Formal semantics and its recent (1) ||Happy|| (||John||) = T (or F) developments. A very readable brief overview of formal semantics, its linguistic and philosophical history, and current controversies. (4) (Janssen 2011) Montague semantics. Well-written history and overview. Frege proposed that there were two categories of expressions whose denotations were basic: sentences, denoting truth values, and names (simple DPs), denoting entities. These 1. Philosophical and logical advances that made formal semantics two kinds of denotations he described as “saturated”. All other kinds of expressions, possible1 according to Frege, have denotations that are “unsaturated”; they have to combine with other denotations to become saturated. So Frege had a kind of type theory, though it was 1.1 Early philosophers not restrictive enough to block Russell’s paradox – see below. Aristotle (384–322 B.C.E.) is widely regarded as the inventor of logic, and he focused on quantification. (Operators like and and or were added by the Stoics.) We’ll say more In contemporary formal semantics starting from Montague, we say basically the same when we focus on the history of quantification (Week 8). thing when we decide that we will take t and e as our two basic types2, corresponding to truth values and entities respectively, serving as the denotations of sentences and e-type Leibniz (1646-1716) dreamed of a characteristica univeralis, based on an ars DPs. All other expressions take their semantic value in domains corresponding to combinatoria, a system of symbolization that would have simple forms for simple functional types. Then by function-argument application, the meanings of various concepts, and unambiguous logical forms displaying the logical structure of all complex expressions can combine, leading eventually to sentence meanings of type t. expressions, together with a calculus ratiocinator, a complete system of deduction that would allow new knowledge to be derived from old. Leibniz’s program aimed to Another of Frege’s great contributions was the logical structure of quantified sentences. encompass the three relationships between language and reality, language and thought, Frege was the first to figure out a systematic semantics for variables and variable- and language and knowledge. Leibniz also had a notion of possible worlds. But his work binding, different from what Tarski did 50 years later, which is what we have inherited. on these topics had little impact. That was part of the design of a “concept-script” (Begriffschrift), a “logically perfect language” to satisfy Leibniz’s goals; he did not see himself as offering an analysis of And Leibniz may have been the first to use bound variables, but in mathematics, not natural language, but a tool to augment it, as the microscope augments the eye. in his logic. (More on that when we discuss history of anaphora and of quantification.) 1 Much of this handout is drawn from (Partee 2011). See that paper for acknowledgements and sources, 2 There are other choices. Some add additional basic types for times, or for possible situations, or for which include (Cocchiarella 1997, Soames 2010, Stanley 2008) and conversations with many people. “degrees”. See (Partee 2006) for some speculation about doing everything with just one basic type. RGGU126.doc - 1 - ! RGGU126.doc - 2 - ! History of Formal Semantics, Lecture 6 History of Formal Semantics, Lecture 6 B. H. Partee, RGGU, March 28, 2012 p.3 B. H. Partee, RGGU, March 28, 2012 p.4 either analytic – true by virtue of their meaning and logic – or knowable on the basis of Frege is also credited with the Principle of Compositionality3, a cornerstone of formal experience – ‘verifiable.’ (Quine admired Carnap greatly, but found a deep circularity in semantics, and a principle quite universally followed in the design of the formal the notions of ‘analytic’ and ‘meaning’ and related notions.) languages of logic, with a few interesting exceptions4. Later Carnap read and appreciated Tarski’s work, saw the need for previously excluded The Principle of Compositionality: The meaning of a complex expression is a non-extensional language, and developed a semantic approach, where meaning = truth function of the meanings of its parts and of the way they are syntactically combined. conditions (Carnap 1956). (Quine, for one, distrusted all non-extensional language.) Carnap introduced possible worlds as state-descriptions; see Section 1.5 below. And Frege introduced the distinction between sense and reference (Sinn and Bedeutung), Later still, Carnap recognized the importance of adding pragmatics to his theorizing, with which later philosophers of language tried to formalize in various ways, e.g. as the issues of gaining and communicating knowledge: not everything important about distinction between intension and extension. language could be expressed with pure logical syntax and semantics. 1.3 Russell, Carnap, Tarski Tarski (1902-1983) developed model theory based in set theory and with it made major Russell discovered a paradox in Frege’s Begriffschrift. advances in providing a semantics for logical languages, including a semantical definition of truth (Hodges 2010), still with an extensional metalanguage. Model theory leads to a From his definition of the basic notions of arithmetic, the logic of the Begriffsschrift, clear and fruitful distinction between syntax and semantics in logic, and that distinction and a single axiom which governs extensions, Frege derives the Peano axioms for has been central in linguistics as well, although what linguists mean by semantics isn’t number theory. always model-theoretic. Basic Law V: !F!G (ext(G) = ext(F) ! !x(Fx ! Gx)) Note: To get a taste of what “syntax” vs “semantics” mean in logic, see But Basic Law V is inconsistent. In particular, it is subject to a version of Russell’s http://en.wikipedia.org/wiki/Syntax_%28logic%29 . For instance, in logic, syntax Paradox. includes proof theory. [For possible discussion