1. Lecture 3 Natural Language Processing
Empirical vs. analytical expressions
Partiality

2. Empirical expressions
Empirical expressions denote non-trivial (i.e. non-constant) intensions
„King of France“, „the prezident of ČR“, „the highest mountain“ denote offices (roles) of type , and actually refer (in this w and t) to nothing, Miloš Zeman and Mount Everest, respectively.
Yet the referred object must be found out empirically
Predicates like „to be a student“, „to be red“, „to be happy“, „to be of the age 60“ denote properties of type () and refer to their population, i.e. to a set of individuals who happen to be students, red, happy and of the age 60, respectively.
Sentences like „Prague is greater than Brno“ denote propositions of type  and refer to a truth-value (this sentence to T).
Hence these expressions are empirical, because the object to which they happen to refer in a given state of the world is beyond logical analysis; it can be determined only by empirical investigation of the state of the world at a given time moment.

3. Analytic expressions
Analytic expressions denote extensions or trivial (i.e. constant) intensions
The objects they denote can be determined by mere understanding a given expression, without empirically investigating the state of the world.
Mathematical and logical expressions are analytical, they denote extensions
Expressions that contain empirical subexpressions are analytical, if they denote constant intensions. Hence the referred object is necessarily, in all worlds w and times t one and the same.
„No bachelor is married“
„Whales are mammals“
Denote the constant proposition True, that takes value T in all w and t

4. Restricted quantifiers
All students are clever.
Some students are lazy.
wt x [[0Studentwt x]  [0Cleverwt x]]
wt x [[0Studentwt x]  [0Lazywt x]]
These analyses are not literal ones, because they are not in harmony with the so- called „Parmenides principle“ and with our method of literal analysis: the sentences do not mention , .
Thus we introduce restricted quantifiers All, Some, No, Most, … of type ((())())
[0All M], M v (), v-constructs the set of all supersets of M
[0Some M], M v (), v-constructs the set of those sets that have a non-empty intersection with M
[0No M], M v (), v-constructs the set of all sets that have an empty intersection with M
wt [[0All 0Studentwt] 0Cleverwt]
wt [[0Some 0Studentwt] 0Lazywt]

5. Analytic expressions “No bachelor is married”  True
Bachelor, Married/(); No/((())())
wt [[0No 0Bachelorwt] 0Marriedwt]
Let us define and refine:
m, n/1 v (), x v : 0No = m n x [[m x]  [n x]], [[0No m] n] = x [[m x]  [n x]].
0Bachelor = wt x [[0Marriedwt x]  [0Manwt x]].
(to be unmarried and man are requisites of the property of being a bachelor)
[[0No 0Bachelorwt] 0Marriedwt] = x [[0Bachelorwt x]  [0Marriedwt x]] = x [[0Marriedwt x]  [0Manwt x]  [0Marriedwt x]].
For any valuation v-constructs T, thus we can generalize:
wt x [[0Marriedwt x]  [0Manwt x]  [0Marriedwt x]].

6. Analytic vs. empirical expressions
Necessarily, 8 > 5
The number of planets = 8
 ???
Necessarily, the number of planets > 5
wt [0> 08 05] (analytical necessity)
w t [[0Number_of 0Planetwt] = 08] (empirical fact)

w t [0> [0Number_of 0Planetwt] 05] (empirical fact)
Types. Number_of/(()): the number of elements of a set, Planet/(), >, =/()
Proof:
[0> 08 05] assumption, E
[[0Number_of 0Planetwt] = 08] 2. assumption, E
[0> [0Number_of 0Planetwt] 05] 1, 2 Leibniz
wt [0> [0Number_of 0Planetwt] 05] 3, I

7. Partiality, v-improper constructions
Incoherent typing
If X is not a construction of order n (n  1), then 1X is improper; (a non-procedural object cannot be executed)
If X is not a construction of order n (n  2), then 2X is improper; (a non-procedural object cannot be executed)
If X, X1, …, Xn are not constructions typed to produce objects of types according to Def., then [X X1…Xn] is improper by failing to v-construct anything for any valuation v.
Example. Tom/; 5/; Student/() 1Tom, 15, wt [0Studentwt 05], 2[wt [0Studentwt 0Tom]]
Application of a function f to an argument a such that f is not defined at a
[0: x 00] is v-improper for any valuation v
x [0: x 00] is not v-improper, as it constructs a degenerate function
0[0: x 00] is not v-improper, as it constructs the Composition [0: x 00]
[0Improper 0[0: x 00]] constructs T
Improper/(1): the class of constructions v-improper for any v.

8. Partiality and Compositionality
„If five divided by zero equals five, then Tom is the Pope“
denotes the degenerate proposition undefined at all w,t !
wt [[[0: 05 00] = 05]  [0Tom = 0Popewt]]  .
0, 5/; :/(); Tom/; Pope/; [0: 05 00]  ; [[0: 05 00] = 05]  ; 0Tom  ; 0Popewt v ; [0Tom = 0Popewt] v ; [[[0: ] = 05]  [0Tom = 0Popewt]] v .
The relation = does not obtain the first argument, hence the implication truth-function also does not obtain the first argument; therefore, the whole Composition is v-improper by failing to construct anything.
Partiality is propagated up.
The Closure  degenerate proposition

9. Partiality and Compositionality
If you intuitively feel that the sentence could be true, because the speaker wanted to say that it is not true that five divided by zero is five, we must analyse this sentence:
„If it is true that five divided by zero equals five, then Tom is the Pope“.
True*/(n): the class of those constructions that v- construct T for every valuation v.
[0True* 0C] v-constructs T, iff C v-constructs T for any valuation v, otherwise F.
wt [[0True* 0[[0: 05 00] = 05]]  [0Tom = 0Papežwt]]  
Now the Closure constructs the proposition True.
The sentence is analytically true

10. Partiality and Compositionality
False*/(n) and Improper*/(n) are classes of constructions that v-construct F or are v- improper for all valuations v, respectively:
[0True* 0C] = [0False* 0C]  [0Improper* 0C]
[0False* 0C] = [0True* 0C]  [0Improper* 0C]
[0Improper* 0C] = [0True* 0C]  [0False* 0C]
Similarly, we often need these properties of propositions: True, False, Undef/()

11. properties of propositions True, False, Undef/()
Tom never smoked
Tom stop smoking
Can the second sentence be true or false under the assumption of the first sentence?
If Tom never smoked, he couldn’t stop smoking  the sentence (2) is false (?)
But then it is true that Tom did not stop smoking  he still smokes (?)
The sentence (2), or rather the proposition denoted by (2) has no truth-value
Being an ex-smoker is a requisite of the property Stop_smoking
The presupposition of (2) is the proposition that Tom previously smoked

12. properties of propositions True, False, Undef/()
[0Truewt P] iff Pwt
[0Falsewt P] iff Pwt
[0Undefwt P] = [0Truewt P]  [0Falsewt P]
P,Q  
Requisites.
[0Req F G] =
wt x [[0Truewt wt [Gwt x]]  [0Truewt wt [Fwt x]]
F, G  ()
Gloss. The property F is a requisite of the property G iff necessarily, for all x holds: if it is true that x is a G then it is true that is x an F
Example. If it is true that Tom stopped smoking then it is true that Tom previously smoked.