Explain the different between proportional calculus and predicate calculus with exampl
Answers
Answer:Predicate logic is an extension of propositional logic.
In propositional logic, a statement that can either be true or false is called a proposition. For example, the statement “it’s raining outside” is either true or false. This statement would be translated into propositional logic’s language as a capital letter like P. If you have one or more propositions, you can connect them to make more complex sentences using logical connectives like “not,” “and,” “or,” “if…then,” and “if and only if.” In symbols these connectives look like this
not:¬
and:∧
or:∨
if,then:⟹
if and only if:⟺
In predicate logic, you have everything that exists in propositional logic, but now you have the ability to attribute properties and relationships on things or variables. A 1-place predicate is a statement that says something about an object. An example of this would be “two is an even number.” This statement is saying the number two has the property of being even. We can also use variables that range over objects, but aren’t names of specific things themselves. An example of that would be “x is an even number.” Now the first statement was true about two, but the second statement is only true if x stands in for an even number. In the predicate language we can represent these as:
P1x= x is an even number
Let a= two, then P1a= two is an even number
These 1-place predicates can be connected with the connectives from propositional logic.
Now, let’s say you wanted to say that everything is blue. To indicate the idea of everything we use the symbol ∀. A statement that uses this symbol is called a universally quantified statement. Also, what if you wanted to say that there are just some things that are blue. We usually say, “there exists a thing that is blue.” The symbol we use for there exist is ∃. A statement with that symbol is called an existentially quantified statement.
When we use them in a logical sentence, we put the variable to the right of these symbols and then a predicate with that variable next to it, like:
∀xP1x
∃xP1x
You can also relate one object with another (or itself) using a multi-place predicate. Like “x respects y,” or “x is between y and z.” The difference is the superscript used and how many variables you place on the right side of the capital letter (2-place predicate R2xy or a 3-place predicate B3xyz).
With this new formal system you can have more complicated logical arguments or use it in mathematical definitions and proofs.
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