About the difference between the propositional logic and the (first order) predicate logic->
- can you give me one or more remarkable examples which underly the differences and the similarities between the two?
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1$\begingroup$ Is this a test question? Learn about them and the similarities and differences will be obvious. "Predicate logic" (or "first order logic"), not "predicative logic" (which sort-of means something technical, but, given your question, clearly is not what you mean). $\endgroup$– BrianOFeb 24, 2016 at 17:10
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$\begingroup$ In short, The main difference is that in propositional logic you cannot express stuff like "there is some element (in some set) such that..." or "for all elements (in some set) we have...", But in predicative logic (or first-order logic) we can express those statements. The main similarity is that the tautologies and contradictions of propositional logic stay true in first-order logic. $\endgroup$– MathNerdFeb 24, 2016 at 17:12
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$\begingroup$ @BrianO thanks for the suggestion, I edited the question. $\endgroup$– franz1Feb 24, 2016 at 17:17
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$\begingroup$ @MathNerd thank you. Does what you mentioned have anything to do with the substitutional interpretation of quantifiers? (I mean, is it correct to say that in propositional logic I subsitute universal and existential quantifiers with conjonctions and disjunctions?) $\endgroup$– franz1Feb 24, 2016 at 17:25
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$\begingroup$ Re the substitutional interpretation of quantifiers: not a very credible approach when the intended model of a theory is uncountable as with the theory of real closed fields, whose intended model is $\Bbb R$. The substitutional interpretation claims that $\forall x$ would really mean "for whichever names we can substitute for $x$, ..." This anticipates uncountably many names, all of which are treated as finite things (terms of the language). That seems an unnatural point of view. We truly don't have more than countably many names for reals; there really are that many more transcendentals. $\endgroup$– BrianOFeb 24, 2016 at 21:13
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2 Answers
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The obvious difference is that predicate logic allows for quantifiers. E.g.
- Propositional: $p\implies p $
- predicate: $\forall x:p(x)\implies p(x) $
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In propositional logic the "atoms" are propositions, whith them you can always build new ones using the logical conectives, for example you could have a proposition $p$ which means "there is a dog in Luca's house" and another propositio $q$ which means "Luca is Sandra's boyfriend", now you can say "there is a dog in Sandra's boyfriend's house" in the language of propositional logic in the following way: $p \land q$.
In predicate logic you can "break" those "atoms" and work with the "subatomic particles", and so this form of logic allows us to analize the internal structure of the propositions. Now you can use quantifiers, terms, relations and functions. You can define, for example: $d(x)$ means "x is a dog" $h(x,y)$ means "x is in y's house" $b(x)$ means "the boyfriend of x" and finally define Sandra as the constant $s$, now your proposition would be: $$\exists x (d(x) \land h(x,b(s))$$
