A [[Formal Language|formal language]] is defined by its symbols, [[Syntax|syntax]] and [[Semantics|semantics]]. Note that [[L1P (Propositional Logic)]] is a sublanguage of $L_{1}$. The advantage of $L_{1}$ are constants and predicates. # Symbols > [!Definition] Constants > Constants or individual constants in $L_{1}$ are: lower-case letters from the beginning of the alphabet, potentially followed by a numerical index e.g. $a,a_{1},a_{2},\dots,b,b_{1},b_{2},\dots$ > > >Constants are analogous to names in [[Natural language]] (e.g., John, Jane, London, France, Ben Nevis, etc.). Like names their role is to refer to particular objects >[!note] Variables >We use lower case letters from the end of the alphabet as variables e.g. x, y, z. > [!Definition] Predicates > A *predicate* is that which is said of a (grammatical) subject, or rather, the part of a sentence or clause in which something is said about the subject. > > In $L_{1}$ predicates are written as strings of letters and numerals, always starting with a capital letter, followed by a number of variables surrounded by brackets, and separated from each other by commas. E.g. $F(x),\text{Smiling}(x), \text{TallerThan}(x,y),\dots$ > >In addition, we have a special predicate '$=. > > > >The *arity* of a predicate is the number of different constants it can be combined with. (Each predicate has a fixed arity). > > >[!note] Sentence letters > - For atomic sentences, we use upper case letters A-E and P-S, potentially followed by a numerical index. E.g. $P, Q,R_{1},\dots$ > - For any sentence (atomic or complex), we use $\varphi,\psi,\chi$ as sentence variables. > [!Definition] Symbols for [[Connective|connectives]] >- Negation: $\neg$; >- Conjunction: $\land$; >- Disjunction: $\lor$; >- Material conditional: $\to$; >- Biconditional symbol: $\leftrightarrow$. > # Syntax >[!note] Defintion (atomic sentence) >Any predicate of arity $n$ combined with $n$ occurrences of constants is an atomic sentence of $L_{1}$. >[!note] Definition (sentence) >1. All atomic sentences are sentences of $L_{1}$. >2. If $\varphi$ is a sentence of $L_{1}$, then $\lnot \varphi$ is a sentence of $L_{1}$. >3. If $\varphi$ and $\psi$ are sentences of $L_{1}$ then $(\phi \land \psi)$, $(\phi \lor \psi)$, $(\phi \to \psi)$, $(\phi \leftrightarrow \psi)$ are sentences (called complex sentences) of $L_{1}$. > [!NOTE] Syntax for identity symbol '=' > - Like other 2-place predicates, ‘=’ combines with two constants to form an atomic sentence. > - Instead of preceding a pair of constants (which are enclosed within brackets and separated from each other by a comma), ‘=’ is flanked by the constants (without brackets). > - So $a=b, a=a$, etc. are atomic sentences, and thus sentences, of $L_{1}$ > > > [!info] Bracketing Convetions > 1. Cojoining any number of sentences with using bracket since order doesn't matter - and similarly for disjunctions > 2. Omitting the outermost brackets of sentences > 3. Adding extra brackets around any sub-sentence. # Semantics The meaning of a sentence in $L_{1}$ is its ***truth conditions***. E.g., the truth conditions of ‘John is smiling’ can be stated thus. ‘John is smiling’ is true in a given situation iff John is smiling in the situation in question. #### Semantics for the identity symbol For any pair of constants $c_{1}$ and $c_{2}$, $c_{1}=c_{2}$ is true iff the object assigned to $c_{1}$ and the object assigned to $c_{2}$ are one and the same object. #### Semantics for complex sentences Connectives of $L_{1}$ are [[Connective#^e0da44|truth functional]]. In any given situation, - $\lnot \varphi$ is true iff $\varphi$ is false. - $(\varphi \land \psi)$ is true iff $\varphi$ is true and $\psi$ is true. - $(\varphi \lor \psi)$ is true iff $\varphi$ is true or/and $\psi$ is true. - $(\varphi \to \psi)$ is false iff $\psi$ is false and $\varphi$ are true. - $(\varphi \leftrightarrow \psi)$ is true iff $\varphi$ and $\psi$ have the same truth value. This can be expressed by means of a truth table: | $\varphi$ | $\psi$ | $\lnot \varphi$ | $(\varphi \land \psi)$ | $(\varphi \lor \psi)$ | $(\varphi \to \psi)$ | $(\varphi \leftrightarrow \psi)$ | | ---- | ---- | ---- | ---- | ---- | ---- | ---- | | T | T | F | T | T | T | T | | T | F | F | F | T | F | F | | F | T | T | F | T | T | F | | F | F | T | F | F | T | T | # Properties of $L_{1}$ Sentences > [!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.) ^f198dd > [!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 (i.e., possible assignments of truth values to the sentence letters) in which it is true. > [!NOTE] Definition (logical equivalence) > 1. Two sentences are ***[[Logical Equivalence|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] Theorem (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 $\lnot(R\lor(\lnot P\lor \lnot Q)) \iff \lnot(R\lor \lnot(P\land Q))$. > [!NOTE] Definition (Translation between English and $L_{1}$) > An English sentence $S$ and an $L_{1}$ sentence $\varphi$ under a given interpretation (/dictionary) are correct ***translations*** of each other only if they have the same truth conditions. > [!NOTE] Definition (Lexical Ambiguity) > The lexical of a word or phrase applies to it having more than one meaning in the language to which the word belongs (Wikipedia). > [!NOTE] Definition (Structural Ambiguity) >  arises when a sentence can have two (or more) different meanings because of the structure of the sentence > [!NOTE] Definition (Scope & Subsentence) > The **scope** of (an occurrence of) a connective is the shortest sub-sentence in which it occurs. > >A **subsentence** is any part of a sentence which is itself a sentence. > >We say that the scope of (an occurrence of) a connective is wider than another iff the latter occurs within the former and not vice versa. > [!NOTE] Definition (Scope Ambiguity) > A type of structural ambiguity. > > There are is no scope ambiguity in $L_{1}$. > [!info] Why do we need Proof systems in $L_{1}$? > See [[Fitch System for L1p]]. # Applications - [[L2 (FOL)]].