LOGICA & FORMELE SYSTEMEN
23 december 2025
Ma.sse van Wassenhove
Wetenschappen en Bio-ingenieurswetenschappen
,Inhoudsopgave
1 Proposi)elogica ....................................................................................................................................... 3
1.1 Inleiding................................................................................................................................................ 3
1.2 Syntaxis................................................................................................................................................. 3
1.2.1 Alfabet ........................................................................................................................................ 3
1.2.2 Formules ..................................................................................................................................... 3
1.2.3 Terminologie ............................................................................................................................... 3
1.2.4 Formuleschema en instan)e ...................................................................................................... 3
1.2.5 Subs)tu)e ................................................................................................................................... 4
1.2.6 Construc)eboom ........................................................................................................................ 4
1.3 Seman)ek............................................................................................................................................. 4
1.3.1 Waarheidswaarden ..................................................................................................................... 4
1.3.2 Waarheidstabellen en waardering .............................................................................................. 5
1.3.3 Model ......................................................................................................................................... 5
1.3.4 Tautologie en contradic)e .......................................................................................................... 6
1.3.5 Logisch equivalent ...................................................................................................................... 6
1.3.6 Func)oneel volledig .................................................................................................................... 7
1.3.7 Disjunc)eve normaalvorm .......................................................................................................... 7
1.4 Geldig gevolg ........................................................................................................................................ 7
1.4.1 Geldig gevolg .............................................................................................................................. 7
1.4.2 Seman)sche tableaus ................................................................................................................. 8
1.5 Afleidingen ........................................................................................................................................... 9
1.5.1 Afleidbaar ................................................................................................................................... 9
1.5.2 Syntac)sch consistent ............................................................................................................... 10
1.5.3 Stellingen .................................................................................................................................. 10
1.5.4 Overzicht afleidingsregels ......................................................................................................... 10
2 Predikaatlogica ...................................................................................................................................... 11
2.1 Inleiding.............................................................................................................................................. 11
2.2 Syntaxis............................................................................................................................................... 11
2.2.1 Alfabet ...................................................................................................................................... 11
2.2.2 Plaatsigheid............................................................................................................................... 11
2.2.3 Term .......................................................................................................................................... 11
2.2.4 Formule..................................................................................................................................... 11
2.2.5 Bereik van kwantoren ............................................................................................................... 12
2.2.6 Vrije of gebonden variabelen.................................................................................................... 12
2.2.7 Gesloten formule ...................................................................................................................... 12
2.2.8 Open formule............................................................................................................................ 12
2.2.9 Subs)tu)e ................................................................................................................................. 12
2.2.10 Veilige subs)tu)es ............................................................................................................... 13
2.2.11 Aflabe)sche variant ............................................................................................................. 13
2.3 Seman)ek........................................................................................................................................... 13
2.3.1 Structuur ................................................................................................................................... 13
2.3.2 Interpreta)efunc)e .................................................................................................................. 13
2.3.3 Model ....................................................................................................................................... 14
2.3.4 Bedeling .................................................................................................................................... 14
2.3.5 Waardering van termen ............................................................................................................ 14
2.3.6 Waardering van formules ......................................................................................................... 15
2.3.7 Gelijkheid van termen .............................................................................................................. 15
2.3.8 Eigenschap waarheidsfunc)e ................................................................................................... 15
2.3.9 Eigenschap subs)tu)e .............................................................................................................. 15
2.3.10 Geldig gevolg ...................................................................................................................... 15
2.3.11 Universeel geldig .................................................................................................................. 16
1
, 2.3.12 Logisch equivalent................................................................................................................ 16
2.3.13 Theorie // ............................................................................................................................. 16
2.3.14 Axiomaverzameling .............................................................................................................. 16
2.3.15 Modelverzameling ............................................................................................................... 16
2.4 Geldig gevolg ...................................................................................................................................... 17
2.4.1 Proposi)elogica vs predikaatlogica ........................................................................................... 17
2.5 Afleidingen ......................................................................................................................................... 17
2.6 Metatheorie ....................................................................................................................................... 17
2.6.1 Subs)tu)e ................................................................................................................................. 17
2.6.2 Lemma ...................................................................................................................................... 18
2.6.3 Prenexvorm .............................................................................................................................. 19
2.6.4 Fragmenten van predikaatlogica ???? ...................................................................................... 19
2.6.5 Adequaatheid van tableaus ???? .............................................................................................. 19
2.6.6 Volledigheidsstelling ???? ......................................................................................................... 19
3 Lambda calculus..................................................................................................................................... 20
3.1.1 Basis begrippen......................................................................................................................... 20
3.1.2 Lambda expressies (syntax) ...................................................................................................... 20
3.1.3 Currying .................................................................................................................................... 20
3.1.4 Binding en bereik ...................................................................................................................... 20
3.1.5 Vrije variabelen ......................................................................................................................... 21
3.1.6 Gesloten lambda-expressie ...................................................................................................... 21
3.1.7 Subs)tu)e: defini)e ................................................................................................................ 21
3.2 Rekenen met lambda experssies ........................................................................................................ 21
3.2.1 Beta-gelijkheid ........................................................................................................................ 21
3.2.2 Church getallen ......................................................................................................................... 22
3.2.3 Lemma (“+”) ............................................................................................................................. 22
3.2.4 Lambda-definieerbaar .............................................................................................................. 22
3.2.5 Optelling ................................................................................................................................... 23
3.2.6 Lemma (“*”) ............................................................................................................................. 23
3.2.7 True, false, if… ........................................................................................................................... 24
3.3 Fixpunten ........................................................................................................................................... 24
3.3.1 Fixpunten defini)e .................................................................................................................... 24
3.3.2 Fixpunten stelling...................................................................................................................... 24
3.3.3 Recursie .................................................................................................................................... 25
3.3.4 Gevolg ....................................................................................................................................... 25
2
23 december 2025
Ma.sse van Wassenhove
Wetenschappen en Bio-ingenieurswetenschappen
,Inhoudsopgave
1 Proposi)elogica ....................................................................................................................................... 3
1.1 Inleiding................................................................................................................................................ 3
1.2 Syntaxis................................................................................................................................................. 3
1.2.1 Alfabet ........................................................................................................................................ 3
1.2.2 Formules ..................................................................................................................................... 3
1.2.3 Terminologie ............................................................................................................................... 3
1.2.4 Formuleschema en instan)e ...................................................................................................... 3
1.2.5 Subs)tu)e ................................................................................................................................... 4
1.2.6 Construc)eboom ........................................................................................................................ 4
1.3 Seman)ek............................................................................................................................................. 4
1.3.1 Waarheidswaarden ..................................................................................................................... 4
1.3.2 Waarheidstabellen en waardering .............................................................................................. 5
1.3.3 Model ......................................................................................................................................... 5
1.3.4 Tautologie en contradic)e .......................................................................................................... 6
1.3.5 Logisch equivalent ...................................................................................................................... 6
1.3.6 Func)oneel volledig .................................................................................................................... 7
1.3.7 Disjunc)eve normaalvorm .......................................................................................................... 7
1.4 Geldig gevolg ........................................................................................................................................ 7
1.4.1 Geldig gevolg .............................................................................................................................. 7
1.4.2 Seman)sche tableaus ................................................................................................................. 8
1.5 Afleidingen ........................................................................................................................................... 9
1.5.1 Afleidbaar ................................................................................................................................... 9
1.5.2 Syntac)sch consistent ............................................................................................................... 10
1.5.3 Stellingen .................................................................................................................................. 10
1.5.4 Overzicht afleidingsregels ......................................................................................................... 10
2 Predikaatlogica ...................................................................................................................................... 11
2.1 Inleiding.............................................................................................................................................. 11
2.2 Syntaxis............................................................................................................................................... 11
2.2.1 Alfabet ...................................................................................................................................... 11
2.2.2 Plaatsigheid............................................................................................................................... 11
2.2.3 Term .......................................................................................................................................... 11
2.2.4 Formule..................................................................................................................................... 11
2.2.5 Bereik van kwantoren ............................................................................................................... 12
2.2.6 Vrije of gebonden variabelen.................................................................................................... 12
2.2.7 Gesloten formule ...................................................................................................................... 12
2.2.8 Open formule............................................................................................................................ 12
2.2.9 Subs)tu)e ................................................................................................................................. 12
2.2.10 Veilige subs)tu)es ............................................................................................................... 13
2.2.11 Aflabe)sche variant ............................................................................................................. 13
2.3 Seman)ek........................................................................................................................................... 13
2.3.1 Structuur ................................................................................................................................... 13
2.3.2 Interpreta)efunc)e .................................................................................................................. 13
2.3.3 Model ....................................................................................................................................... 14
2.3.4 Bedeling .................................................................................................................................... 14
2.3.5 Waardering van termen ............................................................................................................ 14
2.3.6 Waardering van formules ......................................................................................................... 15
2.3.7 Gelijkheid van termen .............................................................................................................. 15
2.3.8 Eigenschap waarheidsfunc)e ................................................................................................... 15
2.3.9 Eigenschap subs)tu)e .............................................................................................................. 15
2.3.10 Geldig gevolg ...................................................................................................................... 15
2.3.11 Universeel geldig .................................................................................................................. 16
1
, 2.3.12 Logisch equivalent................................................................................................................ 16
2.3.13 Theorie // ............................................................................................................................. 16
2.3.14 Axiomaverzameling .............................................................................................................. 16
2.3.15 Modelverzameling ............................................................................................................... 16
2.4 Geldig gevolg ...................................................................................................................................... 17
2.4.1 Proposi)elogica vs predikaatlogica ........................................................................................... 17
2.5 Afleidingen ......................................................................................................................................... 17
2.6 Metatheorie ....................................................................................................................................... 17
2.6.1 Subs)tu)e ................................................................................................................................. 17
2.6.2 Lemma ...................................................................................................................................... 18
2.6.3 Prenexvorm .............................................................................................................................. 19
2.6.4 Fragmenten van predikaatlogica ???? ...................................................................................... 19
2.6.5 Adequaatheid van tableaus ???? .............................................................................................. 19
2.6.6 Volledigheidsstelling ???? ......................................................................................................... 19
3 Lambda calculus..................................................................................................................................... 20
3.1.1 Basis begrippen......................................................................................................................... 20
3.1.2 Lambda expressies (syntax) ...................................................................................................... 20
3.1.3 Currying .................................................................................................................................... 20
3.1.4 Binding en bereik ...................................................................................................................... 20
3.1.5 Vrije variabelen ......................................................................................................................... 21
3.1.6 Gesloten lambda-expressie ...................................................................................................... 21
3.1.7 Subs)tu)e: defini)e ................................................................................................................ 21
3.2 Rekenen met lambda experssies ........................................................................................................ 21
3.2.1 Beta-gelijkheid ........................................................................................................................ 21
3.2.2 Church getallen ......................................................................................................................... 22
3.2.3 Lemma (“+”) ............................................................................................................................. 22
3.2.4 Lambda-definieerbaar .............................................................................................................. 22
3.2.5 Optelling ................................................................................................................................... 23
3.2.6 Lemma (“*”) ............................................................................................................................. 23
3.2.7 True, false, if… ........................................................................................................................... 24
3.3 Fixpunten ........................................................................................................................................... 24
3.3.1 Fixpunten defini)e .................................................................................................................... 24
3.3.2 Fixpunten stelling...................................................................................................................... 24
3.3.3 Recursie .................................................................................................................................... 25
3.3.4 Gevolg ....................................................................................................................................... 25
2
