Hardegree, Formal Semantics, Handout of 8

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1 Hardegree, Formal Semantics, Handout of 8 1. Ambiguity According to the traditional view, there are two kinds of ambiguity lexical-ambiguity, and structural-ambiguity. 1. Lexical-Ambiguity Lexical-ambiguity occurs when a single morpheme has two or more entries in the lexicon; alternatively, the same surface form (spelling/pronunciation) corresponds to two or more morphemes. A prominent logical example is and which seems to have different meanings in the following examples. Jay respects Kay and Elle Jay is between Kay and Elle = Jay respects Kay, and Jay respects Elle Jay is between Kay, and Jay is between Elle Jay and Kay are the only students Jay is the only student, and Kay is the only student Other logical examples of lexical-ambiguity are the following. ed passive Jay is respected by Kay past tense Jay presented Kay with an award er comparative Jay is taller than Champ agentive Jay is the owner of Champ 2. Structural-Ambiguity When three or more phrases are combined, the order of combination may affect the content of the resulting compound phrase. For example, the following phrase two plus three times four admits two different analyses. two plus three times four two plus three times four (2+3) (3 4) 14 Sometimes an instance of structural-ambiguity is semantically INsignificant, in the sense that the content is unaffected by the order of computation. For example, two plus three plus four computes to the same value regardless of how parentheses are inserted. Sometimes an instance of structural-ambiguity is phonetically hidden, as in Jay respects Kay more than Elle which is ambiguous between the following. Jay respects Kay more than [Jay respects] Elle Jay respects Kay more than Elle [respects Kay] In instances such as this, different underlying forms give rise to the same overt form.

2 Hardegree, Formal Semantics, Handout of 8 3. Lexical and Structural Ambiguity Some phrases are both lexically-ambiguous and structurally-ambiguous, including the following one. poor violin player On the one hand, poor has three relevant lexical entries. poor versus rich poor versus good poor(ly) versus well. On the other hand, it is also structurally ambiguous according to whether the violin player is poor or the violin is poor or the playing is poor. 2. A More Complex Example The following example from elementary logic is a key example of a sentence that involves structural/scope ambiguity. every man respects some woman This most plausibly has a standard SVO form, which is analyzed as follows [using our original account of quantifiers]. every man [+1] respects some woman [+2] λy 2 λx 1 Rxy λq 2 y{wy&qy} λq 1 x{mx Qx} λx 1 y{wy&rxy} x{ Mx y{wy&rxy} } However, in addition to this reading according to which every man has wide scope, there is another (less obvious) reading according to which some woman has wide scope, in which case the following is its translation. y { Wy & x { Mx Rxy } } In this connection, notice the following series. every man respects some woman some woman is respected by every man there is some woman who is respected by every man In order to construct this reading we structurally-parse the sentence in the following way. every man [+1] respects some woman [+2] λq 1 x{mx Qx} λy 2 λx 1 Rxy λy 2 x{mx Rxy} λq 2 y{wy&qy} y { Wy & x{mx Rxy } } According to this analysis, we first combine every man [+1] and respects, which results in an accusative predicate [D2S], and then combine the resulting expression with some woman [+2], which takes an accusative predicate and delivers a sentence.

3 Hardegree, Formal Semantics, Handout of 8 3. Compositional-Ambiguity We propose that, in addition to lexical-ambiguity and structural-ambiguity, there is also compositional-ambiguity, which is can be more clearly seen when we redo our previous example using out new account of quantification. Of course, we first need an official account of existential-quantification, which we do as follows, using infinitary-disjunction. type(some) = CD trans(some) = λp 0 {x Px} every man [+1] respects some woman [+2] λp 0 {x Px} M 0 λp 0 {y Py} W 0 { x Px } λx.x 1 { y Wy } λx.x 2 λy 2 λx 1 Rxy { y 2 Wy } { x 1 Px } { λx 1 Rxy Wy }?? Here we employ the following further composition rules. -Composition α {β Φ} α ; β γ α, β, γ are any expressions; Φ is any formula any sub-derivation of γ from {α,β} {γ Φ} -Simplification Φ, Ψ are formulas {Ψ Φ } ν{φ&ψ} ν are the variables c-free in Φ ν 1 ν k Φ ν 1 ν k Φ Notice that the compositions proceed very easily until the last one. In order to compute the proper entry for??, we must compose the following two expressions. { x 1 Px } { λx 1 Rxy Wy }

4 Hardegree, Formal Semantics, Handout of 8 The key to the composition is noting that both -composition and -composition allow the minor-premise to be any expression even a "junction". 1 So we can do either of the following. 1. treat as major, and as minor {x 1 Mx} major { λx 1 Rxy Wy } minor x 1 ; { λx 1 Rxy Wy }??? {??? Mx } All we need to do is replace???, but this can be solved using -comp, as follows. { λx 1 Rxy Wy } x 1 x 1 ; λx 1 Rxy Rxy a simple derivation provides this { Rxy Wy } So plugging the resulting expression back into the previous incomplete inference, we have: {x 1 Mx} major { λx 1 Rxy Wy } minor x 1 ; { λx 1 Rxy Wy } { Rxy Wy } { { Rxy Wy } Mx } The output expression in turn may be transformed as follows. { { Rxy Wy } Mx } { y(rxy & Wy) Mx } -simplification x{ Mx y(rxy & Wy) } -simplification 1 We propose the term junction for all infinitary operators. Before the dust settles, we will have a total of seven such operators, including our current two [conjunction and disjunction].

5 Hardegree, Formal Semantics, Handout of 8 2. treat as major, and as minor. { λx 1 Rxy Wy } major {x 1 Mx} minor λx 1 Rxy ; {x 1 Mx} { Rxy Mx } -comp { { Rxy Mx } Wy } -comp { x{mx Rxy} Wy } -simp y{ Wy & x{mx Rxy} } -simp Notice that the tree structure is the same for both analyses, but they produce different products. Thus, this is a case, not of lexical-ambiguity, or structural-ambiguity, but rather compositionalambiguity. 4. An Example of Insignificant Ambiguity As noted earlier, structural-ambiguity is sometimes insignificant; the different readings are all equivalent. The same holds for compositional-ambiguity, as we see in the following example. 1. every man respects every woman every man [+1] respects every woman [+2] λy 2 λx 1 Rxy {y 2 Wy} {x 1 Mx} { λx 1 Rxy Wy }??? Once again, we can treat either junction as major and the other one as minor, in which case we obtain the following derivations. every man [+1] respects every woman [+2] λy 2 λx 1 Rxy {y 2 Wy} {x 1 Mx} { λx 1 Rxy Wy } { { Rxy Wy } Mx } { y {Wy Rxy} Mx } x { Mx y {Wy Rxy} } every man [+1] respects every woman [+2] λy 2 λx 1 Rxy {y 2 Wy} {x 1 Mx} { λx 1 Rxy Wy } { { Rxy Mx } Wy } { x {Mx Rxy} Wy } y { Wy x {Mx Rxy} } In the first computation, we accord wide-scope to every man ; in the second one, we accord wide-scope to respects every woman. Notice that the two resulting formulas are logically equivalent.

6 Hardegree, Formal Semantics, Handout of 8 5. Parallel-Composition The two compositions above are serial one QP has wider scope than the other. There is another composition that treats the two QPs as having "equal scope" or "parallel scope", which is under-written by the following further general rule, is any junction. 2 α Φ β Ψ } α ; β γ α, β, γ are any expressions; Φ, Ψ are any formulas; any sub-derivation of γ from γ Φ&Ψ } When applied to our two universals, we obtain the following. {x 1 Mx} { λx 1 Rxy Wy } x 1 ; λx 1 Rxy Rxy { Rxy Mx & Wy } With this composition in mind, we can construct the following semantic-tree. every man [+1] respects every woman [+2] λy 2 λx 1 Rxy {y 2 Wy} {x 1 Mx} { λx 1 Rxy Wy } { Rxy Mx & Wy } x y { (Mx & Wy) Rxy } Notice that simplification produces two universal quantifiers, since the abstracts have two free variables. Also notice that the resulting formula is logically-equivalent to the two previous formulas. 2 The Cyrillic (Russian phonogram for soft j sound) is short for junction, which is the term we propose for the various infinitary-operators, including conjunction and disjunction, but also several others we propose later.

7 Hardegree, Formal Semantics, Handout of 8 Finally, we propose the following general schema, is any junction. 3 Serial-Composition β Φ } α ; β γ α, β, γ are any expressions; Φ is any formula; any sub-derivation of γ from γ Φ } 6. The No Quantifier We have given an account of every and some in terms of junctions, so it is natural next to consider the corresponding account of no, which we do as follows. type(no) = C D trans(no) = λp 0 {x Px} The new junction is called nunction, 4 and satisfies the general composition rules from the previous section, plus the following special simplification-rule. -Simplification Φ, Ψ are formulas {Ψ Φ} ν{φ&ψ} ν are the variables c-free in Φ ν 1 ν k Φ ν 1 ν k Φ 3 This is not the final form; an adjustment is needed to account for scope-restrictions. See later chapter. 4 Read the nun in nunction as none.

8 Hardegree, Formal Semantics, Handout of 8 7. Three-Fold Ambiguity The following may come as a surprise: unlike every and some, combinations of no with itself are subject to significant scope-ambiguities, as seen in the following three examples. 1. no man respects no woman [granting wide scope to no man ] no man 1 respects no woman 2 λy 2 λx 1 Rxy { y 2 Wy } { x 1 Mx } { λx 1 Rxy Wy } { { Rxy Wy } Mx } { ~ y{ Wy & Rxy } Mx } ~ x { Mx & ~ y{ Wy & Rxy } } 2. no man respects no woman [granting wide scope to no woman ] no man 1 respects no woman 2 λy 2 λx 1 Rxy { y 2 Wy } { x 1 Mx } { λx 1 Rxy Wy } { { Rxy Mx } Wy } { ~ x{ Mx & Rxy } Wy } ~ y { Wy & ~ x{ Mx & Rxy }} 3. no man respects no woman [granting equal-scope to no man and no woman ] no man 1 respects no woman 2 λy 2 λx 1 Rxy { y 2 Wy } { x 1 Mx } { λx 1 Rxy Wy } { Rxy Mx & Wy} ~ xy { (Mx & Wy) & Rxy } Notice that no two of 1-3 are logically-equivalent, which is to say that scope-ambiguity for nono is significant.

Hardegree, Formal Semantics, Handout of 8

Hardegree, Formal Semantics, Handout 2015-03-24 1 of 8 1. Number-Words By a number-word or numeral 1 we mean a word (or word-like compound 2 ) that denotes a number. 3 In English, number-words appear to