Key Concepts Representation Inference Semantics Discourse Pragmatics Computation

Key Concepts Representation a formula of some logic (FOL) Inference –constency checking task; –informativity checking Semantics –language  world

Key Concepts Discourse –anaphora; –time; –rhetorical structure Pragmatics –presupposition Computation –computability –implementability

First Order Logic from a Model-Theoretic Perspective

Fundamental Syntactic Concepts Vocabularies First Order Languages

Vocabulary Example { (love,2), (customer,1), (robber,1), (mia,0), (vincent,0), (honey-bunny,0), (yolanda,0) }

Vocabulary Vocabularies specify the basic elements of te language we are going to use. Items in a vocabulary associated with the number 0 are called constants. For example Items in a vocabulary associated with a number greater than 0 are called relation symbols, (or, sometimes, predicate symbols) For example, love is a relation symbol. In fact, it is a 2-place relation symbol of arity 2.

First Order Languages All symbols in the vocabulary (non-logical symbols). An infinite collection of variables x, y, z, w,... Boolean operators ¬ (negation), → (implication), ∨ (disjunction), and ∧ (conjunction). The quantifiers ∀ (the universal quantifier) and ∃ (the existential quantifier). The equality symbol =. The round brackets ) and ( and the comma.

Terms Terms are the noun phrases of first- order languages. All constants and variables are terms. Nothing else is a term.

Atomic Formula We combine our noun phrases with our predicates to form atomic formulas: If R is a relation symbol of arity n, and τ1,..., τn are terms, then R(τ1,..., τn ) is an atomic (or basic) formula. One special case of this definition is worth drawing attention to: as = is a two place relation symbol, then τ1 = τ2 is an atomic formula.

Well formed formulas All atomic formulas are wffs If φ and ψ are wffs then so are  φ, (φ → ψ), (φ ∨ ψ), and (φ ∧ ψ) If φ is a wff, and x is a variable, then both ∃ xφ and ∀ xφ are wffs (We call φ the matrix of such wffs) Nothing else is a wff

Free and Bound Variables ( customer(x) ∨ ∀ x(robber(x) ∧ ∀ y person(y))) It is useful to think of a free variable as deictic pronoun. A sentence of first-order logic is a formula that contains no free variables.

Proof Theory We now have a formal language We might start by exploring syntactical patterns of inference. In fact, the pioneers of formal logic did precisely this.

Examples Butch is boxer and Butch is happy therefore Butch is happy φ ∧ ψ therefore φ All boxers are brutal, Butch is boxer, therefore Butch is brutal ∀ x(φ(x) → ψ(x)) φ[c/x] therefore ψ[c/x]

Model Theory But around 1930 this syntactic view was broadened with the development of model theory, and the semantic turn The key figure here is Alfred Tarski, a Polish logician who emigrated to the US in He gave his celebrated satisfaction definition and thereby founded the subject that is now known as model theory.

Semantic Perspective Language  World LOGIC  MODEL

Models In a nutshell, models are like little worlds They are mathematical entities that first-order language talk about. The crucial idea is to say exactly how first- order languages talk about models, via Tarski's famous satisfaction definition That is, we are giving a precise model concerning how first-order languages can be about something. Tarski added aboutness.

Model: definition A model is a pair (D, F) where D is a domain containing the set of entities we want to talk about. It must be non-empty. F is the interpretation function. It specifies what each symbol in the vocabulary stands for. F associates each symbol in the vocabulary with an appropriate entity built from items in D.

Example Model D = {d1, d2, d3, d4, d5 } F (mia) = d2 F (honey-bunny) = d1 F (vincent) = d4 F (yolanda) = d1 F (customer) = {d1, d2, d4 } F (robber) = {d3, d5 } F (love) = {(d3, d4 )} NB. not every entity has a name and one entity can have 2 names.

Variable Assignment Functions Suppose we are working with a model M = (D, F ) Then an assignment g of values to variables in M is a function from the set of variables to D Think of g as a context which specifies values for our pronouns (free variables) Assignment functions allows us to define satisfaction for any formula.

Interpretation of Terms Let M = (D, F ) be a model g be an assignment of values to variables in M τ be a term Then by the interpretation of τ with respect to M and g is meant: F (τ ) if τ is a constant g(τ ) if τ is a variable We denote the interpretation of τ by I g F (τ ).

Satisfaction Definition M, g |= R(τ1, · · ·, τn ) iff (I(τ1 ), · · ·, I(τn )) ∈ F(R) M, g |= τ1 = τ2 iff I(τ1 ) = I(τ2 ) M, g |= ¬φ iff not M, g |= φ M, g |= φ ∧ ψ iff M, g |= φ and M, g |= ψ M, g |= φ ∨ ψ iff M, g |= φ or M, g |= ψ M, g |= φ → ψ iff not M, g |= φ or M, g |= ψ M, g |= ∃ xφ iff M, gφ, for some x-variant gM, g |= ∀ xφ iff M, gφ, for all x-variants g

What Does this Buy Us? Fundamental inferential concepts –Consistency –Validity (or Noninformativity) Framework for natural language semantics Precise and intuitively appealing accounts of what these concepts actually are.

Consistency A formula is consistent if it is satisfied in at least one model. So consistent formulas describe conceivable or possible states of affairs. For example, robber(mia) is consistent. A formula that is not consistent is called inconsistent. So inconsistent describes inconceivable or impossible states of affairs: robber(mia) ∧ ¬ robber(mia) is inconsistent. ･

Consistency of a Set of Formulas A finite set of formulas {φ1,..., φn } is consistent if φ1 ∧... ∧ φn is consistent. A finite set of formulas that is not consistent is called inconsistent.

Validity A valid (or uninformative) formula is a formula that is satisfied in all models (under any variable assignment) for example: robber(mia) ∨ ¬robber(mia)). A formula that is not valid is called invalid (or informative).

Validity The notation |= φ means that φ is a valid formula. The notation |/=φ means that φ is not a valid formula. That is, it means that there is at least one model in which φ is false, or to put it another way, at least one model in which ¬φ is true.

Validity of an Argument Suppose φ1,..., φn, and ψ are a finite collection of first-order sentences. The argument with premises φ1,..., φn and conclusion ψ is a valid argument if whenever all the premises are true in some model, the conclusion is true in that model also. The notation φ1,..., φn |= ψ is used to indicate a valid argument. The notation φ1,..., φn |/= ψ is used to indicate a invalid argument.

Deduction Theorem Suppose φ1,..., φn, and ψ is a finite collection of first-order sentences. Then: φ1,..., φn |= ψ iff |= φ1 ∧... ∧ φn → ψ DT shows that the notions of formula validity and argument validity are intimately interrelated. DT says that an argument is valid iff the given sentence is valid.

Informativity/Validity φ is informative (that is, not valid) if and only if ¬φ is consistent. That is, informativity means the opposite really was an option. φ is uninformative (that is, valid) if and only if ¬φ is inconsistent. That is, uninformativity means that the opposite simply was not an option.

The Programme LOGIC MODEL LANGUAGE WORLD

Conclusion We have a map and a method Much work in formal semantics can be seen as filling in aspects of this diagram Computational semantics can be seen as giving the diagram computational content.