(22nd-July-2020)
• Semantics specifies how to put symbols of the language into correspondence with the world. Semantics can be used to understand sentences of the language. The semantics of propositional calculus is defined below.
• An interpretation consists of a function π that maps atoms to {true, false}. If π(a)=true, we say atom a is true in the interpretation, or that the interpretation assigns true to a. If π(a)=false, we say a is false in the interpretation. Sometimes it is useful to think of π as the set of atoms that map to true, and that the rest of the atoms map to false.
• The interpretation maps each proposition to a truth value. Each proposition is either true in the interpretation or false in the interpretation. An atomic proposition a is true in the interpretation if π(a)=true; otherwise, it is false in the interpretation. The truth value of a compound proposition is built using the truth table of Figure 5.1.
• we only talk about the truth value in an interpretation. Propositions may have different truth values in different interpretations.
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