Meaning of Atoms Models assign truth values A model assigns truth - PowerPoint PPT Presentation

Atoms and Propositions Motivation Semantics of Propositional Logic Propositional Atoms Proof Theory Constructing Propositions Soundness and Completeness (preview) Syntax of Propositional Logic Meaning of Atoms Models assign truth values A model assigns truth values ( F or T ) to each atom. 03b—Propositional Logic

Atoms and Propositions Motivation Semantics of Propositional Logic Propositional Atoms Proof Theory Constructing Propositions Soundness and Completeness (preview) Syntax of Propositional Logic Meaning of Atoms Models assign truth values A model assigns truth values ( F or T ) to each atom. More formally A model for a propositional logic for the set A of atoms is a mapping from A to { T , F } . 03b—Propositional Logic

Atoms and Propositions Motivation Semantics of Propositional Logic Propositional Atoms Proof Theory Constructing Propositions Soundness and Completeness (preview) Syntax of Propositional Logic Meaning of Atoms Models assign truth values A model assigns truth values ( F or T ) to each atom. More formally A model for a propositional logic for the set A of atoms is a mapping from A to { T , F } . How do you call them? Models for propositional logic are called valuations . 03b—Propositional Logic

Atoms and Propositions Motivation Semantics of Propositional Logic Propositional Atoms Proof Theory Constructing Propositions Soundness and Completeness (preview) Syntax of Propositional Logic Examples Example Some valuation Let A = { p , q , r } . Then a valuation v 1 might assign p to T, q to F and r to T. 03b—Propositional Logic

Atoms and Propositions Motivation Semantics of Propositional Logic Propositional Atoms Proof Theory Constructing Propositions Soundness and Completeness (preview) Syntax of Propositional Logic Examples Example Some valuation Let A = { p , q , r } . Then a valuation v 1 might assign p to T, q to F and r to T. More formally p v 1 = T , q v 1 = F , r v 1 = T . 03b—Propositional Logic

Atoms and Propositions Motivation Semantics of Propositional Logic Propositional Atoms Proof Theory Constructing Propositions Soundness and Completeness (preview) Syntax of Propositional Logic Examples Example Some valuation Let A = { p , q , r } . Then a valuation v 1 might assign p to T, q to F and r to T. More formally p v 1 = T , q v 1 = F , r v 1 = T . write v 1 ( p ) instead of p v 1 03b—Propositional Logic

Atoms and Propositions Motivation Semantics of Propositional Logic Propositional Atoms Proof Theory Constructing Propositions Soundness and Completeness (preview) Syntax of Propositional Logic Building Propositions We would like to build larger propositions, such as arguments, out of smaller ones, such as propositional atoms. We do this using operators that can be applied to propositions, and yield propositions. 03b—Propositional Logic

Atoms and Propositions Motivation Semantics of Propositional Logic Propositional Atoms Proof Theory Constructing Propositions Soundness and Completeness (preview) Syntax of Propositional Logic Unary Operators Let p be an atom. All possibilities The following options exist: p v = F : ( op ( p )) v = F . 1 03b—Propositional Logic

Atoms and Propositions Motivation Semantics of Propositional Logic Propositional Atoms Proof Theory Constructing Propositions Soundness and Completeness (preview) Syntax of Propositional Logic Unary Operators Let p be an atom. All possibilities The following options exist: p v = F : ( op ( p )) v = F . p v = T : ( op ( p )) v = F . 1 03b—Propositional Logic

Atoms and Propositions Motivation Semantics of Propositional Logic Propositional Atoms Proof Theory Constructing Propositions Soundness and Completeness (preview) Syntax of Propositional Logic Unary Operators Let p be an atom. All possibilities The following options exist: p v = F : ( op ( p )) v = F . p v = T : ( op ( p )) v = F . 1 p v = F : ( op ( p )) v = T . p v = T : ( op ( p )) v = T . 2 03b—Propositional Logic

Atoms and Propositions Motivation Semantics of Propositional Logic Propositional Atoms Proof Theory Constructing Propositions Soundness and Completeness (preview) Syntax of Propositional Logic Unary Operators Let p be an atom. All possibilities The following options exist: p v = F : ( op ( p )) v = F . p v = T : ( op ( p )) v = F . 1 p v = F : ( op ( p )) v = T . p v = T : ( op ( p )) v = T . 2 p v = F : ( op ( p )) v = F . p v = T : ( op ( p )) v = T . 3 03b—Propositional Logic

Atoms and Propositions Motivation Semantics of Propositional Logic Propositional Atoms Proof Theory Constructing Propositions Soundness and Completeness (preview) Syntax of Propositional Logic Unary Operators Let p be an atom. All possibilities The following options exist: p v = F : ( op ( p )) v = F . p v = T : ( op ( p )) v = F . 1 p v = F : ( op ( p )) v = T . p v = T : ( op ( p )) v = T . 2 p v = F : ( op ( p )) v = F . p v = T : ( op ( p )) v = T . 3 p v = F : ( op ( p )) v = T . p v = T : ( op ( p )) v = F . 4 03b—Propositional Logic

Atoms and Propositions Motivation Semantics of Propositional Logic Propositional Atoms Proof Theory Constructing Propositions Soundness and Completeness (preview) Syntax of Propositional Logic Unary Operators Let p be an atom. All possibilities The following options exist: p v = F : ( op ( p )) v = F . p v = T : ( op ( p )) v = F . 1 p v = F : ( op ( p )) v = T . p v = T : ( op ( p )) v = T . 2 p v = F : ( op ( p )) v = F . p v = T : ( op ( p )) v = T . 3 p v = F : ( op ( p )) v = T . p v = T : ( op ( p )) v = F . 4 03b—Propositional Logic

Atoms and Propositions Motivation Semantics of Propositional Logic Propositional Atoms Proof Theory Constructing Propositions Soundness and Completeness (preview) Syntax of Propositional Logic Unary Operators Let p be an atom. All possibilities The following options exist: p v = F : ( op ( p )) v = F . p v = T : ( op ( p )) v = F . 1 p v = F : ( op ( p )) v = T . p v = T : ( op ( p )) v = T . 2 p v = F : ( op ( p )) v = F . p v = T : ( op ( p )) v = T . 3 p v = F : ( op ( p )) v = T . p v = T : ( op ( p )) v = F . 4 The fourth operator negates its argument, T becomes F and F becomes T . We call this operator negation , and write ¬ p (pronounced “not p”). 03b—Propositional Logic

Atoms and Propositions Motivation Semantics of Propositional Logic Propositional Atoms Proof Theory Constructing Propositions Soundness and Completeness (preview) Syntax of Propositional Logic Nullary Operators are Constants The constant ⊤ The constant ⊤ always evaluates to T , regardless of the valuation. 03b—Propositional Logic

Atoms and Propositions Motivation Semantics of Propositional Logic Propositional Atoms Proof Theory Constructing Propositions Soundness and Completeness (preview) Syntax of Propositional Logic Nullary Operators are Constants The constant ⊤ The constant ⊤ always evaluates to T , regardless of the valuation. The constant ⊥ The constant ⊥ always evaluates to F , regardless of the valuation. 03b—Propositional Logic

Atoms and Propositions Motivation Semantics of Propositional Logic Propositional Atoms Proof Theory Constructing Propositions Soundness and Completeness (preview) Syntax of Propositional Logic Binary Operators: 16 choices op 1 ( p , q ) op 2 ( p , q ) op 3 ( p , q ) op 4 ( p , q ) p q F F F F F F F T F F F F T F F F T T T T F T F T 03b—Propositional Logic

Atoms and Propositions Motivation Semantics of Propositional Logic Propositional Atoms Proof Theory Constructing Propositions Soundness and Completeness (preview) Syntax of Propositional Logic Binary Operators: 16 choices (continued) op 5 ( p , q ) op 6 ( p , q ) op 7 ( p , q ) op 8 ( p , q ) p q F F F F F F F T T T T T T F F F T T T T F T F T 03b—Propositional Logic

Atoms and Propositions Motivation Semantics of Propositional Logic Propositional Atoms Proof Theory Constructing Propositions Soundness and Completeness (preview) Syntax of Propositional Logic Binary Operators: 16 choices (continued) op 9 ( p , q ) op 10 ( p , q ) op 11 ( p , q ) op 12 ( p , q ) p q F F T T T T F T F F F F T F F F T T T T F T F T 03b—Propositional Logic

Atoms and Propositions Motivation Semantics of Propositional Logic Propositional Atoms Proof Theory Constructing Propositions Soundness and Completeness (preview) Syntax of Propositional Logic Binary Operators: 16 choices (continued) op 13 ( p , q ) op 14 ( p , q ) op 15 ( p , q ) op 16 ( p , q ) p q F F T T T T F T T T T T T F F F T T T T F T F T 03b—Propositional Logic

Atoms and Propositions Motivation Semantics of Propositional Logic Propositional Atoms Proof Theory Constructing Propositions Soundness and Completeness (preview) Syntax of Propositional Logic Three Famous Ones op 2 : op 2 ( p , q ) is T when p is T and q is T , and F otherwise. 03b—Propositional Logic

Atoms and Propositions Motivation Semantics of Propositional Logic Propositional Atoms Proof Theory Constructing Propositions Soundness and Completeness (preview) Syntax of Propositional Logic Three Famous Ones op 2 : op 2 ( p , q ) is T when p is T and q is T , and F otherwise. Called conjunction , denoted p ∧ q . 03b—Propositional Logic