Natural language processing Lecture 7 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” Types. Believe*/(n); Tom/; All/(()()): restricted quantifier; Prime, Odd/() Synthesis. wt [0Believe*wt 0Tom 0[[0All 0Prime] 0Odd]] 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_ofwt 0Prague]] = 01048576]] 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: [0Knowwt 0Tom wt [0Largerwt 0London 0Prague]] assumption wt xy [[0Largerwt x y] =o [0Smallerwt y x]] axiom [[0Largerwt 0London 0Prague] =o [0Smallerwt 0Prague 0London]] 2) Elimination of , 0London/x, 0Prague/y wt [[0Largerwt 0London 0Prague] =o [0Smallerwt 0Prague 0London]] 3) Introduction of  wt [[0Largerwt 0London 0Prague] =o wt [0Smallerwt 0Prague 0London]] 4) Introduction of  [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 (hyper- propositions) with the set S’ of those constructions that are derivable from S by means of the rules R