Mathematical Logic - 2015 Propositional Logic: exercises Fausto - - PowerPoint PPT Presentation
Mathematical Logic - 2015 Propositional Logic: exercises Fausto Giunchiglia and Mattia Fumagalli (ref. Chiara Ghidini slides PL Formalization and Enzo Maltese sleides PL Exercises) Index q From NL to PL q Truth tables q
Fausto Giunchiglia and Mattia Fumagalli (ref. Chiara Ghidini slides “PL Formalization” and Enzo Maltese sleides “PL Exercises”)
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q (∧ P Q)
q (P¬¬)
q (P+P)
q ¬¬¬¬¬¬¬¬¬P
q ((P→Q) → (Q→P))
q ¬¬¬¬¬¬P¬→Q
q ((P∧Q) → P)
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q It is the case that P:
q It is not the case that P:
q P and Q. P but Q. Although P, Q:
q P or Q:
q P if and only if Q:
q If P, then Q:
q P if Q:
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q Q only if P:
q P just in case Q:
q Not P or not Q:
q It is not the case that both P and Q:
q Both not P and not Q:
q P is necessary for Q:
q P is sufficient for Q:
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q P is both necessary and sufficient for Q:
q P unless Q:
q Among P and Q, only P:
q Among P and Q, not P:
q Only one among P and Q:
q At most one among P and Q:
q At least one among P and Q:
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q "if Paola is happy and paints a picture then Renzo isn't happy”
q "if Paola is happy, then she paints a picture”
q "Paola is happy only if she paints a picture”
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q A truth valuation on a propositional language L is a mapping ν
q ν(A)
q ν(¬A)
q ν(A∧B)
q ν(A∨B)
q ν(⊥)
q ν(⊤)
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q Two formulas F and G are logically equivalent (denoted with F ↔ G )
q Let F and G be formulas. G is a logical consequence of F
q Let F be a formula: q F is valid if every interpretation satisfies F q F is satisfiable if F is satisfied by some interpretation q F is unsatisfiable if there isn't any interpretation satisfying F
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q A truth valuation on a PL language L is a mapping ν that assigns to
q A truth table is composed of one column for each input variable and
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POSSIBLE ASSIGNEMENTS VARIABLES LOGICAL OPERATION
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POSSIBLE ASSIGNEMENTS VARIABLES (1) (2) (3)
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MODEL
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