It's possible that considering propositional logic is making this too simplified, and that, at the cost of some additional complexity, the distinctions would be clearer in predicate logic. For example, you could identify a truth function $\mathsf {Prop}\to\mathbf {2}$ with a subset of $\mathsf {Prop}$, the set of propositions. Doing this for first-order logic, though, doesn't make as much ...
Propositional logic (also called sentential logic) is logic that includes sentence letters (A,B,C) and logical connectives, but not quantifiers. The semantics of propositional logic uses truth assignments to the letters to determine whether a compound propositional sentence is true. Predicate logic is usually used as a synonym for first-order logic, but sometimes it is used to refer to other ...
In some contexts, though, people don't make this distinction between material implication (the connective) and logical implication (the $\implies$ arrow). But they are not the same thing in every context of propositional logic.
Based on Wolf's definition, 'x+3=7' is a statement and a propositional function? This doesn't make sense at all coming from my understanding of things. Is a propositional function like 'x is prime' considered a proposition in propositional logic? What is truth-functional form? How does truth-functional form relate to propositions?
The suggestions given are fine, but there is not always a direct read over from natural language to formal logic: when could mean "whenever" but there could, in natural language, be an implied "only" as "I only buy food when I get paid" and that is just one of the slippery ambiguities which formal language is explicitly designed to avoid.
The most misunderstood element of standard propositional logic is the $\to$ symbol. It is often read as "implies," but that has human meaning that one statement follows from the other in some direct way.
For details and proof of soundness/completeness, see e.g. Mordechai Ben-Ari, Mathematical Logic for Computer Science (Springer, 3rd ed 2012), Chapter 4 Propositional Logic: Resolution, page 82.
For resolution in propositional logic, the order in which you resolve the literals does not matter for the end result, if that was your question. Resolution can be applied across any two conjuncts of a CNF; the rule implicitly incorporates commutativity.
I am trying to clear my doubts about various terms: tautology, contradiction, contingent, satisifiable, unsatisfiable, valid and invalid. I have read on them from various sources, and am putting al...
I'm attempting to solve a proof my professor asked. We are able to use any of the rules of inference, Indirect Proof or Conditional Proof. Every time I think am making progress I run into a brick w...
