Logic semantics

Having defined the syntax of propositional logic, we can proceed to the definition of its semantics. Semantics dictates the rules for revealing the truth of a statement in relation to a particular model. In propositional logic, any model simply fixes a truth value (true or false) for each propositional symbol. For example, if the propositional symbols P _1,2 , P _2,2 and P _3,1 are used in statements of some knowledge base, then one of the possible models is as follows:

m ₁ = {P _1,2 = false, P _2,2 = false, P _3,1 = true}

If there are three propositional characters, there are 2 ³ = 8 possible models; just as many models are shown. However, it should be noted that since we defined the syntax , models have become purely mathematical objects that do not have to be associated with the world of vampus . For example, P _1,2 is just a symbol; it may mean that "there is a pit in the square [1,2]" or "I will be in Paris today and tomorrow."

The semantics of propositional logic should determine how to calculate the truth value of any statement in the presence of a model. This procedure is performed recursively. All statements are formed from atomic statements and five bundles, so you need to specify how to calculate the truth of atomic statements, and then how to calculate the truth of statements formed using each of these five bundles. The task of calculating the truth of atomic utterances, as shown below, is simple.

• The statement True is true in any model, and the statement False is false in any model.
• The truth value of any other propositional symbol must be indicated directly in the model. For example, in the model m ₁ , the above statement P _1,2 is false.

To determine the truth of complex sentences, rules like the one below apply.

• For any statement s and any model m, the statement ¬s is true in model m if and only if s is false in model m.

These rules allow us to reduce the problem of determining the truth of complex sentences to the problem of determining the truth of simpler sentences. The rules for determining truth for each bundle can be summarized as a truth table , which determines the truth value of a complex statement for each possible assignment of truth values ​​to its components. The truth tables for the five logical connectives in question are listed in the table. Using these tables, the truth value of any statement s as applied to any model m can be calculated using a simple recursive evaluation process. For example, the statement ¬P _1,2 ˆ (P _2,2 v P _3,1 ), estimated in the model m ₁ results in true ˆ (false v true) = true ˆ true = true.

It was said above that any knowledge base consists of a multitude of statements. Now it can be shown that the logical knowledge base is a conjunction of these statements. This means that, starting with the empty KB knowledge base and applying the Tell operations (KB, S ₁ ), ..., Tell (KB, S _n ), we get: KB = S ₁ ˆ ... ˆS _n . Thus, knowledge bases and utterances can be considered interchangeable concepts.

Truth tables for five logical connectives. To use this table, for example, to calculate the value of P v Q, when P is true and Q is false, you first need to find a row in the left part of the table where P is true and Q is false (third row). Then you need to look for an entry in this row that corresponds to the column P v O to determine the result - true. Another way to search for truth values ​​is that each row can be considered as a model, and the entries under each column for this row are asserted if the corresponding statement is true in this model.

Truth tables for bundles of l ("and"), v ("or") and -. ("no") almost completely correspond to the intuitive notions about the meaning of the same words of a natural language. The main source of confusion is that the statement P v Q is true if p, or Q, or both, are true. There is also another bundle, called the "exclusive OR" (abbreviated "XOR" - exclusive OR), which takes a false value if both clauses are true. There is no general agreement as to what designation should be used for the bundle “exclusive OR”; two options are v and ©.

The truth table for the link => at first glance may seem puzzling, since it does not quite correspond to the intuitive understanding of the expressions "P entails Q", or "if P, then Q". But first of all it should be noted that the propositional logic does not require that between the statements of P and Q there should be established any causal relations or relations determining their relevance with respect to each other. Saying “the fact that 5 is an odd number implies that Tokyo is the capital of Japan” is a true statement of propositional logic (with normal interpretation), even though it certainly appears strange in natural language. Another source of confusion is that any implication is true if its antecedent is false. For example, the statement “that 5 is an even number implies that that Sam is smart” is true, regardless of whether Sam is smart. This may seem strange, but it makes sense if we consider the statement in the form "P => Q" as a statement: "If P is true, then I claim that 0 is true. Otherwise, I do not make any statements." The only way in which this statement can take the value false is to make the statement Rbylo true, and Q - false.

The truth table for two-sided implication, P <=> 0, shows that this statement is true if true and P => O, and?) => P. In verbal form, the corresponding statement is often written as follows: "P then and only when Q ", or abbreviated" RTG Q "(then - not a typo). The rules for the world of vampus are best written using a bunch of <=>. For example, a breeze is felt in a certain square, if there is a hole in the adjacent square, and a breeze can be felt in a certain square only if there is a hole in one of the neighboring squares. Therefore, we will need bilateral implications, such as

B _1,1 <=> (P _1,2 v P _2,1 )
where B _1,1 means that a breeze is felt in the square [1/1]. Note that the following one-way implication:
B _1,1 => (P _1,2 v P _2,1 )
in the world of vampus is true, but incomplete. It does not allow to exclude from consideration models in which the statement B _{1.1 is} false and P _{1.2 is} true and which violate the rules of the world of vampus. This idea can be expressed in a different way in such a way that the implication requires the presence of holes if the wind is felt, and a two-sided implication also requires the absence of holes if the wind is not felt.