1. Transparent Intensional Logic TIL
Marie Duží

2. Natural Language Processing
The most important applications
Logical analysis of natural language
Multi-agent systems; agent’s communication and reasoning
The TIL-Script functional programming language
Materials to study:
Duží M., Jespersen B., Materna P. (2010): Procedural Semantics for Hyperintensional Logic. Springer

3. If you oversleep, you will be late
Logical semantics
Logic
is about reasoning, argumentation
is going from premises to a conclusion
is the analysis and appraisal of arguments
When you do logic, you try to clarify reasoning and separate good from bad reasoning, i.e.,
Separate valid arguments from invalid ones
Valid argument:
If you oversleep, you will be late
You are not late

You didn’t oversleep
The conclusion is logically entailed by the premises

4. Logical semantics
An argument is valid if it would be contradictory (impossible) to have the premises all true and conclusion false.
When you do reasoning, you use natural language
When you analyse arguments, you must analyse premises in a fine-grained way so that not to infer something that is not entailed by the premises.
Our goal: to build up an inference machine that neither over-infers ( paradoxes), nor under-infers ( lack of knowledge)
The more fine-grained the analysis of the meaning of premises is, the better inference machine can we build up

5. Logical semantics Propositional logic 1st-order predicate logic (FOL)
Is very limited; semantics reduced to assigning T(rue), F(alse) to atomic propositions and composing these propositions by means of truth-value functions  (Boolean) algebra of truth-values
1st-order predicate logic (FOL)
Limited analysis of the structure of atomic propositions – up to assigning properties and relations to individuals
Apt for mathematics, problems with natural language

6. Coarse-grained analysis
Some prime numbers are even
Some odd numbers are even
Some students are lazy
Formalization in FOL: x [P(x)  Q(x)]
Questions:
How comes that the sentences (1), (2), (3) have the same analysis?
is analytically true sentence
is analytically false sentence
is empirical sentence, maybe true, maybe false
How comes that the formula x [P(x)  Q(x)] has interpretations in which it is true and other interpretations in which it is false if it is the analysis of (1) or (2)?
How does the translation of natural language sentences into the language of FOL contribute to understanding their meaning ?

7. Coarse-grained analysis
No bachelor is married
No bachelor is rich
FOL: x [P(x)  Q(x)] or x [P(x)  Q(x)]
Questions:
How comes that both the sentences have to analyses, and which of them is the ‘correct’ one? ?
After all, (1) is analytically true (true at all state-of-affairs), while (2) is empirical statement, which might be true at some state-of-affairs and false in others

8. Coarse-grained analysis of premises
Does it matter?
If we always could validly derive conclusions that are entailed by the premises, the coarse-grained analysis would be OK
Does the analysis in FOL make it possible to validly derive consequences entailed by the premises?
Unfortunately, it does not.
Coarse-grained analysis of premises yields paradoxes  inferring something that is not entailed by the premises (over-inferring) and under-inferring

9. Paradoxes Necessarily, 8  5 The number of planets = 8
––––––––––––––––––––––––––––––– ???
Necessarily, the number of planets  5
Quine’s paradox
Solution: Modal logic, introducing an operator □
It is ordered to deliver a letter
If the letter is delivered, then it is delivered or burnt out
––––––––––––––––––––––––––––––––––––––––––– ???
It is ordered to deliver the letter or burn out
Ross’s paradox
Deontic logic, introducing operator O
Substitution of identicals is not allowed within the scope of operators  opaque, intensional contexts

10. Paradoxes John believes that Prague has 1.048.576 citizens
= (16)
––––––––––––––––––––––––––––––––– ???
John believes that Prague has (16) citizens
Doxastic / Epistemic logics, introduce operators B, K
John knows that 1+1=2
1+1=2  Sin() = 0
––––––––––––––––––––––– ???
John knows that Sin() = 0
Doxastic / Epistemic logics, introduce operators B(elieve) and K(now)
The paradox of logical/mathematical omniscience

11. Paradoxes John calculates 2 + 5 2 + 5 = 7 –––––––––––––– ???
–––––––––––––– ???
John calculates 7
Oidipus seeks the murderer of his father
Oidipus is the murderer of his father
––––––––––––––––––––––––– ???
Oidipus seeks Oidipus
Attitude logics, …

12. Paradoxes The US President is the husband of Melania
Hillary wanted to become the US president
––––––––––––––––––––––––––––––––––––––––
Hillary wanted to become the husband of Melania
Tom believes that the King of France is wise
––––––––––––––––––––––––––––––––––––––– ???
The King of France exists
Tom is seeking an abominable snowman (yeti)
––––––––––––––––––––––––––––––––––––––– ???
Abominable snowman exists
Logics – magic ???

13. Extensional vs. intensional (opaque) context
When is the context extensional?
The context is extensional if the extensional rules like substitution identities a existential generalization are valid
And when are these rules valid?
In an extensional context
Hmmm
We stir clear of this circle by
Defining three kinds of context first
Defining universally valid rules

14. TIL There is a spreading and still growing tree of particular logics
It has been growing bottom up
Is it OK?
Shouldn’t here be just one universal logic?
Aren’t logical rules valid universally?
TIL – universal logical framework
„top down“ approach
Logical rules are valid universally, only they have to be properly applied

15. Procedural semantics of TIL
Expression
Sense (procedure, construction)
denotation
Ontology of TIL: ramified hierarchy of types 15

16. TIL: three kinds of context
Hyperintensional; construction of the denoted function is an object of predication
Tom computes Sin()
Tom believes that the Pope is wise but does not believe that the Bishop of Rome is wise
Intensional; the denoted function itself is an object of predication
Sine is a periodic function
Tom wants to become the Pope
Extensional; value of the denoted function is an object of predication
Sin() = 0
The Pope is wise.

17. TIL Ontology (types of order 1)
(non-procedural objects)
Basic types
truth-values {T, F} ()
universe of discourse {individuals} ()
times or real numbers ()
possible worlds ()
Functional types ( 1…n) partial functions (1  …  n)  
PWS Intensions – entities of type (());  17

18. Possible worlds no sci-phi ! No multiple universes
Universe of discourse: the collection of bare individuals – abstract hangers (determined just by an ID) to hang particular traits and relations on
Possible world: chronology of maximal consistent distributions of these basic traits among individuals
PWS-intensions / (()); or  for short

19. Examples of PWS-intensions
propositions of type 
properties of individuals of type ()
binary relations-in-intension between individuals of type ()
individual offices (or roles) of type 

20. Examples of extensions (not functions with the domain )
Logical objects like truth-functions and quantifiers are extensional
 (conjunction),  (disjunction) and  (implication) are of type (), and  (Boolean negation) of type ().
Quantifiers ,  are type-theoretically polymorphic total functions of type (()), for an arbitrary type , defined as follows.
The universal quantifier  is a function that associates a class A of -elements with T if A contains all elements of the type , otherwise with F.
The existential quantifier  is a function that associates a class A of -elements with T if A is a non-empty class, otherwise with F.

21. Constructions Variables x, y, p, w, t, … v-construct
Trivialization 0C constructs C (of any type)
a fixed pointer to C and the dereference of the pointer.
In order to operate on C, C needs to be grabbed, or ‘called’, first. Trivialization is such a grabbing mechanism.
Closure [x1…xn X]  ( 1…n)
 n 
Composition [F X1 … Xn]  
( 1…n)  n
Execution 1X, Double Execution 2X 21

22. TIL Ontology (higher-order types)
Constructions of order 1 (1)
 construct entities belonging to a type of order 1
/ belong to 1 : type of order 2
Constructions of order 2 (2)
 construct entities belonging to a type of order 2 or 1
/ belong to 2 : type of order 3
Constructions of order n (n)
 construct entities belonging to a type of order n  1
/ belong to n : type of order n + 1
Functional entities: ( 1…n) / belong to n
(n: the highest of the types to which , 1, …, n belong)
And so on, ad infinitum 22

23. explicit intensionalization and temporalization
constructions of possible-world intensions directly encoded in the logical syntax:
w t […w….t…]
w  ; t  ; 0Happy  (); 0Pope  
w t [0Happywt 0Popewt]  
In any possible world (w) at any time (t):
Take the property of being happy (0Happy)
Take the papal office (0Pope)
Extensoinalize both of them (0Happywt, 0Popewt)
Check whether the holder of the Papal office is happy at that w, t of evaluation ([0Happywt 0Popewt])

24. Method of analysis
Assing types to objects that are mentioned by the expression E, i.e. to the objects denoted by some subexpression of E.
Compose constructions of objects ad 1) to construct the object denoted by E.
Semantically simple expressions (including idioms) are furnished with Trivialization of the denoted object as their meaning
Type checking.

25. Example: „The Mayor of Ostrava“
Types: Mayor_of/((())) – abbr. ():attribute; Ostrava/, Mayor_of_Ostrava/(()) – abbr. 
Synthesis: wt [0Mayor_ofwt 0Ostrava]
Type checking:
w t [[[0Mayor_of w] t] 0Ostrava]
((())) 
(()) 
()  
()
(()) abbreviated as  (individual office)

26. „The Mayor of Ostrava is rich“
Additional type: Rich/()
Synthesis: wt [0Richwt wt [0Mayor_ofwt 0Ostrava]]wt]
Typechecking (shortened):
w t [[[0Richwt wt [0Mayor_ofwt 0Ostrava]]wt]
()  
()
(()) abbr.  (proposition)

27. TIL vs. Montague’s IL
IL is an extensional logic, since the axiom of extensionality is valid: x (Ax = Bx)  A = B.
This is a good thing. However, the price exacted for the simplification of the language (due to ghost variables) is too high;
the law of universal instantiation, lambda conversion and Leibniz’s Law do not generally hold, all of which is rather unattractive.
Worse, IL does not validate the Church-Rosser ‘diamond’. It is a well-known fact that an ordinary typed -calculus will have this property. Given a term x(A)B (the redex), we can simplify the term to the form [B/x]A, and the order in which we reduce particular redexes does not matter. The resulting term is uniquely determined up to -renaming variables.
TIL does not have this defect; it validates the Church-Rosser property though it works with n-ary partial functions
the functions of TY2 are restricted to unary total functions (Schönfinkel)

28. TIL: logical core constructions + type hierarchy (simple and ramified)
The ramified type hierarchy organizes all higher-order objects: constructions (types n), as well as functions with domain or range in constructions.
The simple type hierarchy organizes first-order objects: non-constructions like extensions (individuals, numbers, sets, etc.), possible-world intensions (functions from possible worlds) and their arguments and values.

29. Hyperintensionality Which contexts are intensional / hyperintensional?
was born out of a negative need, to block invalid inferences
Carnap (1947, §§13ff); there are contexts that are neither extensional nor intensional (attitudes)
Cresswell; any context in which substitution of necessary equivalent terms fails is hyperintensional
Yet, which inferences are valid in hyperintensional contexts?
How hyper are hyperintensions?
Which contexts are intensional / hyperintensional?
TIL definition is positive: a context is hyperintensional if the very meaning procedure is an object of predication 29

30. Three kinds of context
hyperintensional context: a meaning construction occurs displayed
so that the very construction is an object of predication
though a construction at least one order higher need to be executed in order to produce the displayed construction
intensional context: a meaning construction occurs executed in order to produce a function f
so that the whole function f is an object of predication
moreover, the executed construction does not occur within another displayed construction
extensional context: the meaning construction is executed in order to produce a particular value of the so-constructed function f at its argument
so that the value of the function f is an object of predication
moreover, the executed construction does not occur within another intensional or hyperintensional context.

31. Hyperintensionality Extensional logic of hyperintensions
Transparency: no context is opaque
The same (extensional) logical rules are valid in all kinds of context;
Leibniz’s substitution of identicals, existential quantification even into hyperintensional contexts, …
Only the types of objects these rules are applied at differ according to a context
Anti-contextualism: constructions are assigned to expressions as their context-invariant meanings