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complete glossary

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rzach committed Dec 16, 2016
1 parent 0721158 commit 367d614e78638ff243195a9090d6b9651ac05edc
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@@ -44,6 +44,11 @@ \section{Names}
$$a,b,c,\ldots, r, a_1, f_{32}, j_{390}, m_{12}$$
These should be thought of along the lines of proper names in English, but with one difference. `Tim Button' is a proper name, but there are several people with this name. (Equally, there are at least two people with the name `P.D.\ Magnus'.) We live with this kind of ambiguity in English, allowing context to individuate the fact that `Tim Button' refers to an author of this book, and not some other Tim. In FOL, we do not tolerate any such ambiguity. Each name must pick out \emph{exactly} one thing. (However, two different names may pick out the same thing.)
\newglossaryentry{name}{
name = name,
description = {a symbol of FOL used to pick out an object of the \gls{domain}}
}
As with TFL, we can provide symbolisation keys. These indicate, temporarily, what a name will pick out. So we might offer:
\begin{ekey}
\item[e] Elsa
@@ -65,15 +70,20 @@ \section{Predicates}
Vinnie borrowed \blank\ from Nunzio\\
Vinnie borrowed the family car from \blank
\end{quote}
FOL predicates are capital letters $A$ through $Z$, with or without subscripts. We might write a symbolisation key for predicates thus:
In FOL, \define{predicates} are capital letters $A$ through $Z$, with or without subscripts. We might write a symbolisation key for predicates thus:
\begin{ekey}
\item[Ax] \gap{x} is angry
\item[Hx] \gap{x} is happy
% \item[T_1xy] \gap{x} is as tall or taller than \gap{y}
% \item[T_2xy] \gap{x} is as tough or tougher than \gap{y}
% \item[Bxyz] \gap{y} is between \gap{x} and \gap{z}
\end{ekey}
(Why the subscripts on the gaps? We will return to this in \S\ref{s:MultipleGenerality}.)
(Why the subscripts on the gaps? We will return to this in \S\ref{s:MultipleGenerality}.)
\newglossaryentry{predicate}{
name = predicate,
description = {a symbol of FOL used to symbolize a property or relation}
}
If we combine our two symbolisation keys, we can start to symbolise some English sentences that use these names and predicates in combination. For example, consider the English sentences:
\begin{earg}
@@ -96,12 +106,28 @@ \section{Quantifiers}
\end{earg}
It might be tempting to symbolise sentence \ref{q.a} as `$He \eand Hg \eand Hm$'. Yet this would only say that Elsa, Gregor, and Marybeth are happy. We want to say that \emph{everyone} is happy, even those with no names. In order to do this, we introduce the `$\forall$' symbol. This is called the \define{universal quantifier}.
A quantifier must always be followed by a variable. In FOL, variables are italic lowercase letters `$s$' through `$z$', with or without subscripts. So we might symbolise sentence \ref{q.a} as `$\forall x Hx$'. The variable `$x$' is serving as a kind of placeholder. The expression `$\forall x$' intuitively means that you can pick anyone and put them in as `$x$'. The subsequent `$Hx$' indicates, of that thing you picked out, that it is happy.
\newglossaryentry{universal quantifier}{
name = universal quantifier,
description = {the symbol $\forall$ of FOL used to symbolize generality; $\forall x\, Fx$ is true iff every member of the domain is~$F$}
}
A quantifier must always be followed by a \define{variable}. In FOL, variables are italic lowercase letters `$s$' through `$z$', with or without subscripts. So we might symbolise sentence \ref{q.a} as `$\forall x Hx$'. The variable `$x$' is serving as a kind of placeholder. The expression `$\forall x$' intuitively means that you can pick anyone and put them in as `$x$'. The subsequent `$Hx$' indicates, of that thing you picked out, that it is happy.
\newglossaryentry{variable}{
name = variable,
description = {a symbol of FOL used following quantifiers and as placeholders in atomic formulas; lowercase letters between $s$ and $z$}
}
It should be pointed out that there is no special reason to use `$x$' rather than some other variable. The sentences `$\forall x Hx$', `$\forall y Hy$', `$\forall z Hz$', and `$\forall x_5 Hx_5$' use different variables, but they will all be logically equivalent.
To symbolise sentence \ref{q.e}, we introduce another new symbol: the \define{existential quantifier}, `$\exists$'. Like the universal quantifier, the existential quantifier requires a variable. Sentence \ref{q.e} can be symbolised by `$\exists x Ax$'. Whereas `$\forall x Ax$' is read naturally as `for all x, x is angry', `$\exists x Ax$' is read naturally as `there is something, x, such that x is angry'. Once again, the variable is a kind of placeholder; we could just as easily have symbolised sentence \ref{q.e} with `$\exists z Az$', `$\exists w_{256} Aw_{256}$', or whatever.
\newglossaryentry{existential quantifier}{
name = existential quantifier,
description = {the symbol $\exists$ of FOL used to symbolize existence; $\exists x\, Fx$ is true iff at least one member of the domain is~$F$}
}
Some more examples will help. Consider these further sentences:
\begin{earg}
\item[\ex{q.ne}] No one is angry.
@@ -122,6 +148,11 @@ \section{Domains}
\end{ekey}
The quantifiers \emph{range over} the domain. Given this domain, `$\forall x$' is to be read roughly as `Every person in Chicago is such that\ldots' and `$\exists x$' is to be read roughly as `Some person in Chicago is such that\ldots'.
\newglossaryentry{domain}{
name = domain,
description = {the collection of objects assumed for a symbolization in FOL, or that gives the range of the quantifiers in an \gls{interpretation}}
}
In FOL, the domain must always include at least one thing. Moreover, in English we can infer `something is angry' from `Gregor is angry'. In FOL, then, we will want to be able to infer `$\exists x Ax$' from `$Ag$'. So we will insist that each name must pick out exactly one thing in the domain. If we want to name people in places beside Chicago, then we need to include those people in the domain.
\factoidbox{
A domain must have \emph{at least} one member. A name must pick out \emph{exactly} one member of the domain, but a member of the domain may be picked out by one name, many names, or none at all.
@@ -227,7 +258,13 @@ \section{Common quantifier phrases}
\section{Empty predicates}
In \S\ref{s:FOLBuildingBlocks}, we emphasised that a name must pick out exactly one object in the domain. However, a predicate need not apply to anything in the domain. A predicate that applies to nothing in the domain is called an \define{empty} predicate. This is worth exploring.
In \S\ref{s:FOLBuildingBlocks}, we emphasised that a name must pick out exactly one object in the domain. However, a predicate need not apply to anything in the domain. A predicate that applies to nothing in the domain is called an \define{empty predicate}. This is worth exploring.
\newglossaryentry{empty predicate}{
name = {empty predicate},
description = {a \gls{predicate} that applies to no object in the \gls{domain}}
}
Suppose we want to symbolise these two sentences:
\begin{earg}
@@ -1170,7 +1207,9 @@ \section{Expressions}
We define an \define{expression of FOL} as any string of symbols of FOL. Take any of the symbols of FOL and write them down, in any order, and you have an expression.
\section{Terms and formulas}
In \S\ref{s:TFLSentences}, we went straight from the statement of the vocabulary of TFL to the definition of a sentence of TFL. In FOL, we will have to go via an intermediary stage: via the notion of a \emph{formula}. The intuitive idea is that a formula is any sentence, or anything which can be turned into a sentence by adding quantifiers out front. But this will take some unpacking.
\label{s:TermsFormulas}
In \S\ref{s:TFLSentences}, we went straight from the statement of the vocabulary of TFL to the definition of a sentence of TFL. In FOL, we will have to go via an intermediary stage: via the notion of a \define{formula}. The intuitive idea is that a formula is any sentence, or anything which can be turned into a sentence by adding quantifiers out front. But this will take some unpacking.
We start by defining the notion of a term.
\factoidbox{
@@ -1185,6 +1224,18 @@ \section{Terms and formulas}
\item Nothing else is an atomic formula.
\end{enumerate}
}
\newglossaryentry{term}{
name = term,
description = {either a \gls{name} or a \gls{variable}}
}
\newglossaryentry{formula}{
name = formula,
description = {an expression of FOL built according to the recursive rules in \S\ref{s:TermsFormulas}}
}
The use of swashfonts here follows the conventions laid down in \S\ref{s:UseMention}. So, `$\meta{R}$' is not itself a predicate of FOL. Rather, it is a symbol of our metalanguage (augmented English) that we use to talk about any predicate of FOL. Similarly, `$\meta{t}_1$' is not a term of FOL, but a symbol of the metalanguage that we can use to talk about any term of FOL. So, where `$F$' is a one-place predicate, `$G$' is a three-place predicate, and `$S$' is a six-place predicate, here are some atomic formulas:
\begin{center}
$x = a$\\
@@ -1230,12 +1281,23 @@ \section{Terms and formulas}
\factoidbox{
The \define{main logical operator} in a formula is the operator that was introduced last, when that formula was constructed using the recursion rules.
\
\bigskip
The \define{scope} of a logical operator in a formula is the subformula for which that operator is the main logical operator.
}
So we can graphically illustrate the scope of the quantifiers in the preceding example thus:
$$\overbrace{\forall x \overbrace{\exists y (Fx \eiff \overbrace{\forall z (Gayz \eif Syzyayx)}^{\text{scope of `}\forall z\text{'}}}^{\text{scope of `}\exists y\text{'}})}^{\text{scope of `}\forall x\text{'}}$$
$$\overbrace{\forall x \overbrace{\exists y (Fx \eiff \overbrace{\forall z (Gayz \eif Syzyayx)}^{\text{scope of `}\forall z\text{'}}}^{\text{scope of `}\exists y\text{'}})}^{\text{scope of `$\forall x$'}}$$
\newglossaryentry{main logical operator}{
name = main logical operator,
description = {the operator used last in the construction of a \gls{formula}}
}
\newglossaryentry{scope}{
name = scope,
description = {the subformula of a \gls{formula} of FOL for which the \gls{main logical operator} is the operator}
}
\section{Sentences}
Recall that we are largely concerned in logic with assertoric sentences: sentences that can be either true or false. Many formulas are not sentences. Consider the following symbolisation key:
@@ -1252,10 +1314,22 @@ \section{Sentences}
\factoidbox{
A \define{bound variable} is an occurrence of a variable \meta{x} that is within the scope of either $\forall\meta{x}$ or $\exists\meta{x}$.
\
\bigskip
A \define{free variable} is any variable that is not bound.
}
\newglossaryentry{bound variable}{
name = bound variable,
description = {an occurrence of a variable in a \gls{formula} which is in the scope of a quantifier followed by the same variable}
}
\newglossaryentry{free variable}{
name = free variable,
description = {an occurrence of a variable in a \gls{formula} which is not a \gls{bound variable}}
}
For example, consider the formula
$$\forall x(Ex \eor Dy) \eif \exists z(Ex \eif Lzx)$$
The scope of the universal quantifier `$\forall x$' is `$\forall x (Ex \eor Dy)$', so the first `$x$' is bound by the universal quantifier. However, the second and third occurrence of `$x$' are free. Equally, the `$y$' is free. The scope of the existential quantifier `$\exists z$' is `$(Ex \eif Lzx)$', so `$z$' is bound.
@@ -1265,6 +1339,11 @@ \section{Sentences}
A \define{sentence} of FOL is any formula of FOL that contains no free variables.
}
\newglossaryentry{sentence of FOL}{
name = sentence of FOL,
description = {a \gls{formula} of FOL which has no \glspl{bound variable}}
}
\section{Bracketing conventions}
@@ -115,14 +115,19 @@ \section{Semantics for identity}
Suppose `$A$' is a one-place predicate; then `$Aa$' is false and `$Ab$' is false, so `$Aa \eiff Ab$' is true. Similarly, if `$R$' is a two-place predicate, then `$Raa$' is false and `$Rab$' is false, so that `$Raa \eiff Rab$' is true. And so it goes: every atomic sentence not involving `$=$' is false, so every biconditional linking such sentences is true. For all that, Tim Button and P.D.\ Magnus are two distinct people, not one and the same!
\section{Interpretation}
We defined a \emph{valuation} in TFL as any assignment of truth and falsity to atomic sentences. In FOL, we are going to define an \define{interpretation} as consisting of three things:
We defined a \define{valuation} in TFL as any assignment of truth and falsity to atomic sentences. In FOL, we are going to define an \define{interpretation} as consisting of three things:
\begin{ebullet}
\item the specification of a domain
\item for each name that we care to consider, an assignment of exactly one object within the domain
\item for each predicate that we care to consider---other than `$=$'---a specification of what things (in what order) the predicate is to be true of
\end{ebullet}
The symbolisation keys that we considered in chapter \ref{ch.FOL} consequently give us one very convenient way to present an interpretation. We will continue to use them throughout this chapter. However, it is sometimes also convenient to present an interpretation \emph{diagrammatically}.
\newglossaryentry{interpretation}{
name = {interpretation},
description = {a specification of a \gls{domain} together with the objects the \glspl{name} pick out and which objects the \glspl{predicate} are true of}
}
Suppose we want to consider just a single two-place predicate, `$Rxy$'. Then we can represent it just by drawing an arrow between two objects, and stipulate that `$Rxy$' is to hold of x and y just in case there is an arrow running from x to y in our diagram. As an example, we might offer:
\begin{center}
\begin{tikzpicture}
@@ -269,6 +274,11 @@ \section{When the main logical operator is a quantifier}
$$\forall y \exists x (Ryx \eiff Fx)$$
with the instantiating name `$e$'.
\newglossaryentry{substitution instance}{
name = substitution instance,
description = {the result of replacing every occurrence of a \gls{free variable} in a \gls{formula} with a \gls{name}}
}
Armed with this notation, the rough idea is as follows. The sentence $\forall \meta{x}\meta{A}(\ldots \meta{x} \ldots \meta{x} \ldots)$ will be true iff $\meta{A}(\ldots \meta{c} \ldots \meta{c}\ldots)$ is true no matter what object (in the domain) we name with $\meta{c}$. Similarly, the sentence $\exists \meta{x}\meta{A}$ will be true iff there is \emph{some} way to assign the name $\meta{c}$ to an object that makes $\meta{A}(\ldots \meta{c} \ldots \meta{c} \ldots)$ true. More precisely, we stipulate:
\factoidbox{
$\forall \meta{x}\meta{A}(\ldots \meta{x}\ldots\meta{x}\ldots)$ is true in an interpretation \textbf{iff}\\
@@ -374,30 +384,58 @@ \section{When the main logical operator is a quantifier}
\chapter{Semantic concepts}
Offering a precise definition of truth in FOL was more than a little fiddly, but now that we are done, we can define various central logical notions. These will look very similar to the definitions we offered for TFL. However, remember that they concern \emph{interpretations}, rather than valuations.
\
\\We will use the symbol `$\entails$' for FOL much as we did for TFL. So:
We will use the symbol `$\entails$' for FOL much as we did for TFL. So:
$$\meta{A}_1, \meta{A}_2, \ldots, \meta{A}_n \entails\meta{C}$$
means that there is no interpretation in which all of $\meta{A}_1, \meta{A}_2, \ldots, \meta{A}_n$ are true and in which \meta{C} is false. Derivatively,
$$\entails\meta{A}$$
means that \meta{A} is true in every interpretation.
\
\\An FOL sentence $\meta{A}$ is a \define{logical truth} iff $\meta{A}$ is true in every interpretation; i.e., $\entails\meta{A}$.
\
\\$\meta{A}$ is a \define{contradiction} iff $\meta{A}$ is false in every interpretation; i.e., $\entails\enot\meta{A}$.
\
\\$\meta{A}_1, \meta{A}_2, \ldots \meta{A}_n \therefore \meta{C}$ is \define{valid in FOL} iff there is no interpretation in which all of the premises are true and the conclusion is false; i.e., $\meta{A}_1,\meta{A}_2,\ldots \meta{A}_n \entails\meta{C}$. It is \define{invalid in FOL} otherwise.
\
\\Two FOL sentences \meta{A} and \meta{B} are \define{logically equivalent} iff they are true in exactly the same interpretations as each other; i.e., both $\meta{A}\entails\meta{B}$ and $\meta{B}\entails\meta{A}$.
\
\\The FOL sentences $\meta{A}_1,\meta{A}_2,\ldots, \meta{A}_n$ are \define{jointly consistent} iff there is some interpretation in which all of the sentences are true. They are \define{jointly inconsistent} iff there is no such interpretation.
The other logical notions also have corresponding definitions in FOL:
\begin{itemize}
\item An FOL sentence $\meta{A}$ is a \define{logical truth} iff $\meta{A}$ is true in every interpretation; i.e., $\entails\meta{A}$.
\newglossaryentry{logical truth}
{
name=logical truth,
description={A \gls{sentence of FOL} that is true in every \gls{interpretation}}
}
\item $\meta{A}$ is a \define{contradiction} iff $\meta{A}$ is false in every interpretation; i.e., $\entails\enot\meta{A}$.
\newglossaryentry{contradiction of FOL}
{
name=contradiction (of FOL),
text=contradiction,
description={A \gls{sentence of FOL} that is false in every \gls{interpretation}}
}
\item $\meta{A}_1, \meta{A}_2, \ldots \meta{A}_n \therefore \meta{C}$ is \define{valid in FOL} iff there is no interpretation in which all of the premises are true and the conclusion is false; i.e., $\meta{A}_1,\meta{A}_2,\ldots \meta{A}_n \entails\meta{C}$. It is \define{invalid in FOL} otherwise.
\newglossaryentry{logically valid in FOL}
{
name=logical validity (in FOL),
text = logically valid,
description={A property held by arguments if and only if no \gls{interpretation} makes all premises true and the conclusion false}
}
\item Two FOL sentences \meta{A} and \meta{B} are \define{logically equivalent} iff they are true in exactly the same interpretations as each other; i.e., both $\meta{A}\entails\meta{B}$ and $\meta{B}\entails\meta{A}$.
\newglossaryentry{logically equivalent in FOL}
{
name=logical equivalence (in FOL),
text = logically equivalent,
description={A property held by pairs of \glspl{sentence of FOL} if and only if the sentences have the same truth value in every \gls{interpretation}.}
}
\item The FOL sentences $\meta{A}_1,\meta{A}_2,\ldots, \meta{A}_n$ are \define{jointly consistent} iff there is some interpretation in which all of the sentences are true. They are \define{jointly inconsistent} iff there is no such interpretation.
\newglossaryentry{logically consistent in FOL}
{
name=logical consistency (in FOL),
text=jointly consistent,
description={A property held by \glspl{sentence of FOL} if and only if some \gls{interpretation} makes all the sentences true}
}
\end{itemize}
\chapter{Using interpretations}
\label{sec.UsingModels}
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