Download Logic of Attitudes

Logic of Attitudes
Natural language processing
Lecture 7
Logic of attitudes
1) ‘propositional’ attitudes
• Tom Att1 (believes, knows) that P
a) Att1/(): relation-in-intension of an individual to a proposition
b) Att1*/(n): relation-in-intension of an individual to a ; hyper-proposition
2) ‘notional’ attitudes
• Tom Att2 (seeks, finds, is solving, wishing, wanting to, …) P
a) Att2/(): relation-in-intension of an individual to an intension
b) Att2*/(n): relation-in-intension of an individual to a hyper-intension
Moreover, both kinds of attitudes come in two variants; de dicto and de re
Propositional attitudes
1) doxastic (ancient Greek δόξα; from verb δοκεῖν dokein, "to appear",
"to seem", "to think" and "to accept")
• “a believes that P”
2) epistemic (ancient Greek; ἐπίσταμαι, meaning "to know, to
understand, or to be acquainted with“)
• “a knows that P”
• Epistemic attitudes represent factiva; what is known must be true
• Doxastic attitudes may be false beliefs
Propositional attitudes
a) The embedded clause P is mathematical or logical  hyper-propositional
• “Tom believes that all prime numbers are odd”
b) The embedded clause P is analytically true/false and contains empirical
terms  hyper-propositional
• “Tom does not believe that whales are mammals“
c) The embedded clause P is empirical and contains mathematical terms 
hyper-propositional
• “Tom thinks that the number of Prague citizens is 1048576“
d) The embedded clause P is empirical and does not contain mathematical
terms  propositional / hyper-propositional
• “Tom believes that Prague is larger than London“
a) Attitudes to mathematical propositions
• “Tom believes that all prime numbers are odd”
• Believe* must be a relation to a construction;
otherwise  the paradox of an idiot; Tom would believe every false
mathematical sentence
• “Tom knows that some prime numbers are even”
• Know* must be a relation to a construction;
otherwise  the paradox of logical/mathematical omniscience; Tom
would know every true mathematical sentence
a) Attitudes to mathematical propositions
• “Tom believes that all prime numbers are odd”
1. Types. Believe*/(n); Tom/; All/(()()): restricted
quantifier; Prime, Odd/()
2. Synthesis. wt [0Believe*wt 0Tom 0[[0All 0Prime] 0Odd]]
3. Type-checking … (yourself)
If the analysis were not hyperintensional, i.e., as an attitude to a
construction, then Tom would believe every analytic False, e.g. that
1+1=3; the paradox of an idiot
Similarly, the paradox of logical/mathematical omniscience would arise
the paradox of logical/mathematical omniscience
• Tom knows that 1+1=2
• 1+1=2 iff arithmetic is undecidable
• ------------------------------------------------------• Tom knows that arithmetic is undecidable
Iff/(): the identity of truth-values
wt [0Know*wt 0Tom 0[0= [0+ 01 01] 02]]
0[0= [0+ 01 01] 02]  0[0Undecidable 0Arithmetic]
The paradox is blocked; /(nn): the non-identity of constructions
All true (false) mathematical sentences denote the truth-value T (F); yet
not in the same way. They construct a truth-value in different ways
the paradox of logical/mathematical omniscience
Similarly, an attitude to an analytically true (false) sentence must be
hyperintensional; otherwise – the paradox of logical omniscience
(idiocy)
Analytically true sentence denotes TRUE: the proposition that takes the
truth-value T in all worlds w and times t
Analytically false sentence denotes FALSE: the proposition that takes the
truth-value F in all worlds w and times t
Example. Whales are mammals denotes TRUE;
Read in de dicto way; the property being a mammal is a requisite of the
property of being a whale
Requisite/(()()); Whale, Mammal/()
[0Requisite 0Mammal 0Whale]
the paradox of logical/mathematical omniscience
b) The embedded clause P is analytically true/false and contains empirical
terms  hyper-propositional
• “Tom does not believe that whales are mammals“
wt [0Believe*wt 0Tom 0[0Requisite 0Mammal 0Whale]]
• “Tom knows that no bachelor is married“
• “No bachelor is married” iff “Whales are mammals”
• Iff/(): the identity of propositions
• “Tom knows that whales are mammals“ ??? No, not necessarily
wt [0Know*wt 0Tom 0[0Requisite 0Unmarried 0Bachelor]]
0[0Requisite 0Unmarried 0Bachelor]  0[0Requisite 0Mammal 0Whale]
The paradox is blocked; /(nn): the non-identity of constructions
properties of propositions True, False, Undef/()
• [0Truewt P] iff Pwt v-constructs T, otherwise F
• [0Falsewt P] iff Pwt v-constructs F, otherwise T
• [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.
[0Requisite 0Mammal 0Whale] =
wt x [[0Truewt wt [0Whalewt x]]  [0Truewt wt [0Mammalwt x]]
Hyper-propositional attitudes
c) The embedded clause P is empirical and contains mathematical
terms  hyper-propositional
• “Tom thinks that the number of Prague citizens is 1048576“
• 1048576(dec) = 100000(hexa)
• “Tom does not have to think that the number of Prague citizens is
100000(hexa)“
• Note that 1048576(dec), 100000(hexa) denote one and the same number
constructed in two different ways:
• 1048576(dec) = 1.106 + 0.105 + 4.104 + 8.103 + 5.102 + 7.101 + 6.100
• 100000(hexa) = 1.165 + 0.164 + 0.163 + 0.162 + 0.161 + 0.160
Hyper-propositional attitudes
• “Tom thinks that the number of Prague citizens is 1048576“
• Think*/(n); Tom, Prague/; Number_of/(());
Citizen_of/(());
• wt [0Think*wt 0Tom
0[wt [0Number_of [0Citizen_of 0Prague]] = 01048576]]
wt
• Type-checking …. yourself
Propositional attitudes
d) The embedded clause P is empirical and does not contain
mathematical terms  propositional / hyper-propositional
• “Tom knows that London is larger than Prague“ iff
• “Tom knows that Prague is smaller than London“ iff
• “Tom knows that (London is larger than Prague and whales are
mammals)“
• Implicit Know/(): the relation-in-intension of an individual to a
proposition
• Explicit Know*/(n): the relation-in-intension of an individual to a
hyper-proposition
Implicit knowledge
•
•
•
•
wt [0Knowwt 0Tom wt [0Largerwt 0London 0Prague]]
--------------------------------------------------------------------------wt [0Knowwt 0Tom wt [0Smallerwt 0Prague 0London]]
Additional types. Larger, Smaller/()
Proof. In all worlds w and times t the following steps are truth-preserving:
1. [0Knowwt 0Tom wt [0Largerwt 0London 0Prague]]
assumption
2. wt xy [[0Largerwt x y] =o [0Smallerwt y x]]
axiom
3. [[0Largerwt 0London 0Prague] =o [0Smallerwt 0Prague 0London]]
2) Elimination of , 0London/x, 0Prague/y
4. wt [[0Largerwt 0London 0Prague] =o [0Smallerwt 0Prague 0London]]
3) Introduction of 
5. wt [[0Largerwt 0London 0Prague] =o wt [0Smallerwt 0Prague 0London]]
4) Introduction of 
6. [0Knowwt 0Tom wt [0Smallerwt 0Prague 0London]]
5) substitution of id.
Knowing is factivum
• What is known must be true
• Agent a knows that P  P is true
• Agent a does not know that P  P is true
• P being true is a presupposition of knowing
• Do you know that Earth is flat? Futile question, because the Earth is
not flat! (Unless you are in a Terry Pratchett’s Discworld )
()[0Knowwt a P]
()[0Know*wt a C]
----------------------------------------------[0Truewt P]
[0Truewt 2C]
Types. P  ; 2C  ; C  n.
Computational, inferable knowledge
Knowexp(a)wt

Knowinf(a)wt

Knowimp(a)wt
idiot a
rational a
omniscient a
How to compute inferable knowledge?
• K0(a)wt = Knowexp(a)wt
• K1(a)wt = [Inf(R) Knowexp(a)wt]
• K2(a)wt = [Inf(R) K1(a)wt]
• …
• Non-descending sequence of known hyper-propositions
• There is a fixed point – computational, inferable knowledge of a rational agent who
masters the set of rules R
• Inf(R)/((n)(n)) is a function that associates a given set S of constructions (hyperpropositions) with the set S’ of those constructions that are derivable from S by means of
the rules R