Questions tagged [propositional-calculus]

Appropriate for questions about truth tables, conjunctive and disjunctive normal forms, negation, and implication of unquantified propositions. Also for general questions about the propositional calculus itself, including its semantics and proof theory. Questions about other kinds of logic should use a different tag, such as (logic), (predicate-logic), or (first-order-logic).

Propositional logic is a branch of logic dealing with logical connectives and statements involving them. A logical connective connects finitely many sentences and forms a compound sentence, in a way that the truth value of the compound sentence depends only on the truth value of its constituents. The most common connectives are the binary connectives conjunction ($\land$), disjunction ($\lor$) and implication ($\rightarrow$), the unary connective negation ($\neg$), and the nullary connectives true ($\top$) and false ($\bot$).

Any proposition is considered to be either atomic (in which case it has no constituents) or compound (in which case it's formed by mean a connective using simpler propositions). A propositional model is a function assigning to each atomic proposition a truth value $0$ or $1$. The truth values of compound propositions are then determined by the truth values of their constituents. For example, if $I$ is a function assigning truth values to propositions, one would have $I(\top)=1$, $I(\bot)=0$, $I(\neg A)=1-I(A)$, $I(A\land B)=\min\big(I(A),I(B)\big)$, $I(A\lor B)=\max\big(I(A),I(B)\big)$ and $I(A\rightarrow B)=\max\big(1-I(A),I(B)\big)$. The propositions having the value $1$ for every model, are called tautologies, and those having the value $0$ for every model, are called absurdities. A central task of propositional logic is characterizing tautologies and absurdities.

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Can every proof by contradiction also be shown without contradiction?

Are there some proofs that can only be shown by contradiction or can everything that can be shown by contradiction also be shown without contradiction? What are the advantages/disadvantages of proving by contradiction? As an aside, how is proving by…
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Using proof by contradiction vs proof of the contrapositive

What is the difference between a "proof by contradiction" and "proving the contrapositive"? Intuitive, it feels like doing the exact same thing. And when I compare an exercise, one person proves by contradiction, and the other proves the…
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In classical logic, why is $(p\Rightarrow q)$ True if both $p$ and $q$ are False?

I am studying entailment in classical first-order logic. The Truth Table we have been presented with for the statement $(p \Rightarrow q)\;$ (a.k.a. '$p$ implies $q$') is: $$\begin{array}{|c|c|c|} \hline p&q&p\Rightarrow q\\…
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In classical logic, why is $(p\Rightarrow q)$ True if $p$ is False and $q$ is True?

Provided we have this truth table where "$p\implies q$" means "if $p$ then $q$": $$\begin{array}{|c|c|c|} \hline p&q&p\implies q\\ \hline T&T&T\\ T&F&F\\ F&T&T\\ F&F&T\\\hline \end{array}$$ My understanding is that "$p\implies q$" means "when there…
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"Modus moron" rule of inference?

This is an exercise I got from the book "First Order Mathematical Logic" by Angelo Margaris (1967). I have never heard of this rule before, the question is whether what Margaris calls the modus moron rule of inference is correct or not and to…
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What would happen if we just made vacuous truths false instead?

It's well known that vacuous truths are a concept, i.e. an implication being true even if the premise is false. What would be the problem with simply redefining this to be evaluated to false? Would we still be able to make systems work with this…
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Still struggling to understand vacuous truths

I know, I know, there are tons of questions on this -- I've read them all, it feels like. I don't understand why $(F \implies F) \equiv T$ and $(F \implies T) \equiv T$. One of the best examples I saw was showing how if you start out with a false…
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How do I make proofs with long formulae more readable without sacrificing clarity?

Question A lot of things I'm trying to prove just now are turning into "notational hell", which I think makes them very hard to read. I've tried to cut down on this by assuming my reader will understand what definitions are in play, modularising my…
Ten O'Four
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Why isn’t ‘because’ a logical connective in propositional logic?

In simple terms, could someone explain why there is not a logical connective for ‘because’ in propositional logic like there is for ‘and’ and ‘or’? Is this because the equivalent of ‘because’ is the argument of the form ‘if p, then q’, or am I…
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Why aren't vacuous truths just undefined?

I am struggling to understand this. According to truth tables, if $P$ is false, it doesn't matter whether $Q$ is true or not: Either way, $P \implies Q$ is true. Usually when I see examples of this people make up some crazy premise for $P$ as a way…
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Associativity of logical connectives

According to the precedence of logical connectives, operator $\rightarrow$ gets higher precedence than $\leftrightarrow$ operator. But what about associativity of $\rightarrow$ operator? The implies operator ($\rightarrow$) does not have the…
Amber Jain
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Do De Morgan's laws hold in propositional intuitionistic logic?

In Wikipedia page on intuitionistic logic, it is stated that excluded middle and double negation elimination are not axioms. Does this mean that De Morgan's laws, stated $$ \lnot (p \land q) \iff \lnot p \lor \lnot q \\ \lnot (p \lor q) \iff \lnot p…
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Associativity of $\iff$

In this answer, user18921 wrote that the $\iff$ operation is associative, in the sense that $(A\iff B)\iff C$ $A\iff (B\iff C)$ are equivalent statements. One can brute-force a proof fairly easily (just write down the truth table). But is there…
Willie Wong
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Assumed True until proven False. The Curious Case of the Vacuous Truth

Given two statements, $P$ and $Q$, and the logical connective, $\implies$, the truth table for $P \implies Q$ is: $$\begin{array}{ c | c || c | } P & Q & P\Rightarrow Q \\ \hline \text T & \text T & \text T \\ \text T & \text F & \text F \\ …
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Prove XOR is commutative and associative?

Through the use of Boolean algebra, show that the XOR operator ⊕ is both commutative and associative. I know I can show using a truth table. But using boolean algebra? How do I show? I totally have no clue. Any help please?
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