Dr. Rogelio Dávila Pérez Formal Semantics for Natural language

Theory of Meaning A theory of meaning is understood as providing a detailed specification of the knowledge that a native speaker has about his/her own language. [Dummett, 91] In doing this, a theory of meaning has to provide a way to assign meaning to all the different words in the language and then a mechanism by means of which all these meanings can be combined into larger expressions to form the meaning of phrases, sentences, discourses, and so on. Formal Semantics for Natural language

Two philosophical views: Realistic (classical) School: The proper job of a semantic theory consists of stating the way in which the language is related to the world. This means to establish for every sentence in the language what conditions must hold in the world for the sentence to be true. This is generally known as the truth-conditional semantics program. Formal Semantics for Natural language

Antirealistic (constructive) school A theory of meaning is a theory of understanding. The knowledge that a native speaker has about a sentence of his/her own language is reflected in the use he/she makes of that sentence. In order to understand a sentence, to use a sentence, it is not enough to know its truth conditions, but actually to be able to recognize when those conditions hold in the world. The framework endows what is called a verificationist theory of meaning. Formal Semantics for Natural language

Truth-conditional semantics program To state the meaning of a sentence we should state which conditions must be true in the world for this sentence to be true. e.g. Every man loves a woman. Truth-conditions: For each member “x” of the set of men, there should be at least one member “y” of the set of women, in such a way that the pair is in the relation loves. Logic: x.(man(x)  y.(woman(y)  loves(x,y))) Classic approach to semantics

Frege’s Compositional Semantics The meaning of the sentence is determined by the meaning of the words of which it is composed, and the way in which these are put together. The linear order of the words in a sentence hide the role that different kinds of words play in the building of the meaning of the whole. Classic approach to semantics

Syntactic structure John likes Mary PN VtVt NP VP S DetPNVtVt NP VP S Every man likes Mary Noun

Quantification problem John likes Mary PN VtVt NP VP S DetPNVtVt NP VP S Every man likes Mary Noun x.(man(x)  likes(x, mary)) likes(john, mary)

The Lambda Calulus The lambda calculus is a theory of functions, or, more accurately, a theory of operations. People familiar with elementary set theory will have assimilated the idea that functions are special kinds of set; sets of ordered pairs. e.g. The identity function on natural numbers: {(x,y)| xN & yN & x=y}

The Lambda Calulus Some examples of lambda expressions: (i) ( x.x) (ii) ( x.( y.(xy)) (iii) (( x.x)( x.x)) (iv) ( x.( y.x)) (v) ( x.( y.( z.((xz)(yz)))))

-operator - abstraction t x.t - conversion ( x.t)a = t [a/x]

The Lambda Calulus Reduce the following examples as far as possible: (i) ( x.x(xy))( z.zx) (ii) ( x.xy)( z.zx)( z.zx) (iii) ( x.( y.yx)z)(( z.zx)( z.zx)) (iv) ( x.xxy)( x.xxy)( x.( y.yx)z)

Semantic structure Every man likes Mary Det PNVtVt NP Noun NP VP S

R1. If S then S´= ´(  ´)R8. If NP then NP’ = ´ R2. If NP then NP’ = ´(  ´)R9. man’ =w.man(w) R3. If VP then VP’ = z. ´(  ´(z))R10. likes’ =x.y.likes(x,y) R4. If Det then Det’ = ´R11. mary’ =P.P(mary) R5. If Noun then Noun’ = ´ R12. every’=P.Q.x.(P(x)Q(x)) R6. If Vt then Vt’ = ´ R7. If PN then PN’ = ´ Semantics rules       

Semantic structure Det PNVtVt NP P.Q.x.(P(x)  Q(x)) w.man(w) x.y.likes(x,y) P.P(mary) Noun (R1) x.(man(x)  likes(x, mary)) (R3) z.likes(z,mary) (R2) Q.x.(man(x)  Q(x)) Every man likes Mary