See [[Formal Language]]. # Properties > [!NOTE] Definition (truth functional form) > A sentence is truth functional form if... ^efade4 A complex sentence is given different names dependent on its truth value in every possible situation: > [!NOTE] Definition (logical truth) > A sentence is a ***logical truth*** or ***tautology*** iff it is true in every possible situation. ^e8f9c8 > [!Example] > Given a sentence $P$ of [[L1P (Propositional Logic)|propositional logic]] then $(P \lor \lnot P)$ is a tautology as shown: > > | P | $(P \lor \lnot P )$| >| --- | --- | >| T |T | >|T | T| > [!NOTE] Definition (contradiction) > 1. A sentence is a ***contradiction*** iff it is false in every possible situation. > 2. A sentence is a ***contradiction in virtue of its truth functional form*** iff it has a truth functional form that can be captured by an $L_{1p}$ sentence which is a contradiction. (We can call it TT-contradiction.) > [!Example] > Suppose $P$ is an $L_{1p}$ sentence then $(P \land \lnot P)$ is a contradiction: > >|$P$ | $(P \land \lnot P)$| | --- | --- | | T |F | | F | F| > [!NOTE] Definition (logical possibility) > A sentence is a ***logical possibility*** iff there is at least one possible situation in which it is true. > [!NOTE] Definition (logical equivalence) > 1. Two sentences are ***logically equivalent*** iff they have the same truth value in every possible situation. If $P$ and $Q$ are logically equivalent, we may write $P \iff Q$. > 2. $P \iff Q$ iff the arguments $P\vdash Q$ and $Q\vdash P$ are [[Logically Valid Argument|logically valid]]. > [!Example] > Two [[L1P (Propositional Logic)|propositional logic]] sentences are logically equivalent iff in a joint truth table for the two sentences there is no row in which one of them is $T$ and the other $F$. E.g. > 1. Double Negation: $P \iff \lnot \lnot P$. > 2. [[De Morgan's Laws for Union]]: $(\lnot P \lor \lnot Q)\iff \lnot(P \land Q)$ & $\lnot(P \lor Q)\iff (\lnot P \land \lnot Q)$. > [!NOTE] Substitution principle > If $P$ and $Q$ are logically equivalent sentences of $L_{1}$, then the result of substituting one for the other in an $L_{1}$ sentence is logically equivalent to the original sentence. > [!Example] > We know that $\lnot(P \land Q)\iff(\lnot P \lor \lnot Q)$ > If we substitute for $\lnot(P\land Q)$ in $\lnot(R\lor \lnot(P\land Q))$ we get $\lnot(R\lor(\lnot P\lor \lnot Q))$. > Substitution principle tell us that $\lnot(R\lor(\lnot P\lor \lnot Q)) \iff \lnot(R\lor \lnot(P\land Q))$.