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07-predsem

# 07-predsem - The semantics of predicate logic Readings...

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The semantics of predicate logic Readings: Section 2.4, 2.5, 2.6. In this module, we will precisely define the semantic interpretation of formulas in our predicate logic. In propositional logic, every formula had a fixed, finite number of models (interpretations); this is not the case in predicate logic. As a consequence, we must take more care in defining notions such as satisfiability and validity, and we will see that there cannot be algorithms to decide if these properties hold or not for a given formula. 1

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Models (2.4.1) In the semantics of propositional logic, we assigned a truth value to each atom. In predicate logic, the smallest unit to which we can assign a truth value is a predicate P ( t 1 ,t 2 ,...,t n ) applied to terms. But we cannot arbitrarily assign a truth value, as we did for propositional atoms. There needs to be some consistency. We need to assign values to variables in appropriate contexts, and meanings to functions and predicates. Intuitively, this is straightforward, but we must define such things precisely in order to ensure consistency of interpretation. 2
Example In Module 5, we considered the formula x ( P ( x ) ∧ ¬ Q ( x ) R ( x )) . Our interpretation of this statement was, “Every student who took CS245, but did not pass CS245, failed CS245.” Under this interpretation, x ranges over all students (say, at UW). So, since x is a placeholder for a term, terms t denote UW students. P , Q , and R , then are properties of students. We can think of them as B -valued functions on UW students: P ( x ) = x took CS245”, Q ( x ) = x passed CS245”, R ( x ) = x failed CS245” 3

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More abstractly, P , Q , and R are sets: P = { students who took CS245 } Q = { students who passed CS245 } R = { students who failed CS245 } Then P ( x ) is shorthand for x P , and similarly for Q and R . 4
We could also, however, interpret the predicate symbols P , Q , and R as follows: P = { natural numbers } Q = { even numbers } R = { odd numbers } Then terms t range over some numeric domains, say the integers or the real numbers, and the formula says that within that domain, all natural numbers that are not even are odd. As we see, there is no requirement that our interpretation be about UW students, regardless of our initial motivation! 5

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Finally, we could also apply the following interpretation: P = { natural numbers } Q = { even numbers } R = { prime numbers } Then the formula says that all natural numbers that are not even are prime. This is clearly a false statement, but a possible interpretation of the formula. 6
Consider the following formula: x ( yL ( x,y ) L ( x,c )) We can take our domain of concrete values, in this case, to include students and courses. Then c might denote the constant CS245, and L is a B -valued function on two variables that we might define as follows: L ( x,y ) = x is a student, y is a course, and x loves y Then the formula says that every student who loves a course must love CS245. 7

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07-predsem - The semantics of predicate logic Readings...

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