Beweren en bewijzen/de zuilen/Zekerheid/3. Waarheid

spraakverwarring In de natuurwetenschappen en de informatica noemt men een wiskundige structuur (rechts onder in de vier werelden) die inzicht verschaft of een stukje realiteit (links onder in de vier werelden) een model van dit stukje realiteit. Soms noemt men ook een stel formules (rechts boven in de vier werelden) een model. Logici gebruiken het woord „model” voor iets anders: Een „model” van een formule (die in de vier werelden rechts boven leeft) is een verzameling van situaties (die in de vier werelden rechts onder leven) waarvoor deze formule waar is.

literatuur Boek Logica voor alfa's en informatici

In de propositielogica is alles eindig. Een formule of een verzameling van formules waarin n verschillende proposities voorkomen, kent 2ⁿ verschillende situaties.

Wat heeft dit met de realiteit te maken?

Soms lukt het, alle situaties uit een stukje realiteit waarover we willen redeneren te beschrijven met een domeinmodel voor eindig veel fenomenen die al dan niet het geval kunnen zijn. Bij elk zulk fenomeen hoort een propositie die al dan niet waar kan zijn. Een formule waarin deze proposities voorkomen is waar voor bepaalde situaties en onwaar voor andere.

Een situatie die een gegeven formule waar maakt noemen logici ietwat misleidend een model van deze formule. Wij zullen dit gebruik van het woord model vermijden en het altijd hebben over situaties die een formule waar maken.

Een waarheidstabel is een overzicht van de waarheidswaarden van een formule voor elke mogelijke situatie. Een waarheidstabel voor een formule in de propositielogica is daarom eindig. Waarheidstabellen kun wel erg groot worden; dat heeft met 2ⁿ te maken. Daarom moet men ze systematisch opschrijven, anders vergeet men situaties.

Een formule is een tautologie als ze waar is in elke situatie. Met een waarheidstabel kan men dus nagaan of een formule een tautologie is.

Met een waarheidstabel nagaan of een formule B een geldig gevolg van een aantal formules b₁, b₂, ..., b_n is, is iets ingewikkelder, en je moet de definitie van geldig gevolg goed kennen: B is een geldig gevolg van b₁, b₂, ..., b_n als B waar is voor elke situatie waarvoor alle b_i waar zijn.

Voorbeeld: A → B ∨ C , (B → D) ∨ (C → D) ⊨ A → D (met dank aan Rogier Lommers) – zie ook: Implicatie.

A  B  C  D   A → (B ∨ C)     (B → D) ∨ (C → D)   (A→(B ∨ C)) ∧ ((B→D)∨(C→D))       A → D 
0  0  0  0     1    0           1    1    1                  1                       1
0  0  0  1     1    0           1    1    1                  1                       1
0  0  1  0     1    1           1    1    0                  1                       1
0  0  1  1     1    1           1    1    1                  1                       1
0  1  0  0     1    1           0    1    1                  1                       1
0  1  0  1     1    1           1    1    1                  1                       1
0  1  1  0     1    1           0    0    0                  0                       1
0  1  1  1     1    1           1    1    1                  1                       1
1  0  0  0     0    0           1    1    1                  0                       0
1  0  0  1     0    0           1    1    1                  0                       1
1  0  1  0     1    1           1    1    0                  1                       0
1  0  1  1     1    1           1    1    1                  1                       1
1  1  0  0     1    1           0    1    1                  1                       0
1  1  0  1     1    1           1    1    1                  1                       1
1  1  1  0     1    1           0    0    0                  0                       0
1  1  1  1     1    1           1    1    1                  1                       1

Rogier: „Een geldig gevolg P ⊨ Q houdt in dat overal waar P een situatie een model is, dit ook bij Q het geval moet zijn. Als ik de waarheidstabel uitlees zie ik dat dit bij de volgende situaties niet klopt: 1010 en 1100. Conclusie: A → D is geen logisch gevolg van A → B ∨ C , (B → D) ∨ (C → D).”

Het aantonen van de juistheid van een gevolgtrekking met een waarheidstabel is een uitputtingsslag. Als de stelling een beetje interessant is, is de tabel zo groot dat men bij het opstellen al fouten maakt en niemand het resultaat wil controleren. Daarom zullen veel logici en wiskundigen een waarheidstabel niet als „bewijs” accepteren.

een waarheidstabel voor een realistisch probleem

* A | B | C | D | E | F | G | H | I | (A & B => C) & (C => D) & (E & D => F) & (F => G) & (H & G => I) => (E & H & A & B => I) ---+---+---+---+---+---+---+---+---+------------------------------------------------------------------------------------------ 0) 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 1 1 1 1 0 1 1 1 1 0 1 1 0 0 0 1 1) 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 1 1 1 1 0 1 1 1 1 0 1 1 0 0 0 1 2) 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 1 1 1 1 0 1 1 1 1 0 1 1 0 0 0 1 3) 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 1 1 1 1 0 1 1 1 1 0 1 1 0 0 0 1 4) 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 1 1 1 1 0 1 1 1 1 0 1 1 0 0 0 1 5) 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 1 1 1 1 0 1 1 1 1 0 1 1 0 0 0 1 6) 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 1 7) 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 1 1 1 1 0 1 1 1 1 1 1 1 0 0 0 1 8) 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 1 1 1 1 0 1 0 0 0 0 1 1 0 0 0 1 9) 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 1 1 1 1 0 1 0 0 0 0 1 1 0 0 0 1 10) 0 | 0 | 0 | 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