# propositional logic Definition and Topics - 9 Discussions

Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions.
Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic.

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1. ### B Simple probability question: Suppose P(B|A)=1. Does that mean that P(A|B)=1?

Suppose P(B|A)=1. Does that mean that P(A|B)=1?
2. ### A How are incompatible ##\hat p\hat q\neq\hat q\hat p## related to ##p\wedge q## ?

If propositions ##p,q\in{\mathscr L}_{\mathcal H}## (i.e., the lattice of subspaces of ##\mathcal H##) are incompatible, then ##\hat p\hat q\neq\hat q\hat p##. But since it's a lattice, there exists a unique glb ##p\wedge q=q\wedge p##. How are they mathematically related? In particular, I...
3. ### Logic Proposition Proof

Homework Statement Need to demonstrate this proposition: (P→Q)↔[(P ∨ Q)↔Q] . My textbook use truth tables, but I'd like to do without it. It asks me if it's always truth The Attempt at a Solution Im unable to demonstrate the Tautology and obtain (¬Q) as solution. I start by facing the right...
4. ### Question about propositional logic

Homework Statement I have to prove that ##(p \equiv q) \equiv ((p ∧ q) ∨ (¬p ∧ ¬q))## With no premisses In order to prove this, I first need to prove that: ##(p \equiv q) \to ((p ∧ q) ∨ (¬p ∧ ¬q))## And: ##((p ∧ q) ∨ (¬p ∧ ¬q)) \to (p \equiv q)## I was able to find the second implication...
5. ### I The truth value of ##P(x)## → ##Q(x)##

I'm reading Velleman's book titled "How to Prove it" and I'm very confused when I'm reading about conditional statements. I understand that there exists some issue with the conditional connective and I accept that because that's the cost of espousing a truth-functional view. I came here to ask...
6. M

### Propositional function problems

1. Suppose P(x) and Q(x) are propositional functions and D is their domain. Let A = {x ∈ D: P(x) is true}, B = {x ∈ D: Q(x) is true} (a) Give an example for a domain D and functions P(x) and Q(x) such that A∩B = {} (b) Give an example for a domain D and functions P(x) and Q(x) such that A ⊆ B...
7. ### Truth table interpretation

Homework Statement A person can either be a knight (always tells the truth) or a knave (always tells a lie). On an island with three persons (A, B and C), A tells "If I am a knight, then at least one of us is a knave". Homework Equations Truth tables, logic rules. The Attempt at a Solution...
8. ### I Weird statement of conditions in propositional logic

So I am studying conditionals in proposition logic, and I have discovered that there are a variety of ways to phrase a conditional "if p, then q" in English. Some of the harder ones are... p is sufficient for q a necessary condition for p is q q unless ~p (where ~ is the not operator) p only...
9. ### Symbolize propositions using predicate logic

Homework Statement Suppose that predicates and individuals are dened as follows: S: should be shunned, U: is prone to unruly behaviour, P: is a friend of Peter's, M: is a friend of mine, a: Ann, d: David. Symbolize the following: i. Ann is a friend of Peter's and David is a friend of mine...