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update some examples

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rzach committed Dec 17, 2016
1 parent afd969d commit 2a95058d577ac7242c503d24d0e60b87add045ba
Showing with 31 additions and 38 deletions.
  1. +12 −19 forallx-yyc-fol.tex
  2. +8 −8 forallx-yyc-interpretations.tex
  3. +2 −2 forallx-yyc-preface.tex
  4. +9 −9 forallx-yyc-tfl.tex
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@@ -1186,24 +1186,17 @@ \chapter{Sentences of FOL}\label{s:FOLSentences}
\section{Expressions}
There are six kinds of symbols in FOL:
-\begin{center}
-\begin{tabular}{l l}
-Predicates & $A,B,C,\ldots,Z$\\
-with subscripts, as needed & $A_1, B_1,Z_1,A_2,A_{25},J_{375},\ldots$\\
-\\
-Constants & $a,b,c,\ldots, r$\\
-with subscripts, as needed & $a_1, b_{224}, h_7, m_{32},\ldots$\\
-\\
-Variables & $s, t, u, v, w, x,y,z$\\
-with subscripts, as needed & $x_1, y_1, z_1, x_2,\ldots$\\
-\\
-Connectives & $\enot,\eand,\eor,\eif,\eiff$\\
-\\
-Brackets &( , )\\
-\\
-Quantifiers & $\forall, \exists$\\
-\end{tabular}
-\end{center}
+\begin{description}
+\item[Predicates] $A,B,C,\ldots,Z$, or
+with subscripts, as needed: $A_1, B_1,Z_1,A_2,A_{25},J_{375},\ldots$
+\item[Constants] $a,b,c,\ldots, r$, or
+with subscripts, as needed $a_1, b_{224}, h_7, m_{32},\ldots$
+\item[Variables] $s, t, u, v, w, x,y,z$, or
+with subscripts, as needed $x_1, y_1, z_1, x_2,\ldots$
+\item[Connectives] $\enot,\eand,\eor,\eif,\eiff$
+\item[Brackets] ( , )
+\item[Quantifiers] $\forall, \exists$
+\end{description}
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}
@@ -1236,7 +1229,7 @@ \section{Terms and formulas}
}
-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:
+The use of script letterss 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}
$x = a$\\
$a = b$\\
@@ -41,17 +41,17 @@ \section{Symbolising versus translating}
\section{A word on extensions}
We can stipulate directly what predicates are to be true of, so it is worth noting that our stipulations can be as arbitrary as we like. For example, we could stipulate that `$Hx$' should be true of, and only of, the following objects:
\begin{center}
- Theresa May\\
+ Justin Trudeau\\
the number $\pi$\\
every top-F key on every piano ever made
\end{center}
Now, the objects that we have listed have nothing particularly in common. But this doesn't matter. Logic doesn't care about what strikes us mere humans as `natural' or `similar'. Armed with this interpretation of `$Hx$', suppose we now add to our symbolisation key:
\begin{ekey}
- \item[d] Theresa May
- \item[n] Jeremy Corbyn
+ \item[j] Justin Trudeau
+ \item[r] Rachel Notley
\item[p] the number $\pi$
\end{ekey}
-Then `$Hd$' and `$Hp$' will both be true, on this interpretation, but `$Hn$' will be false, since Jeremy Corbyn was not among the stipulated objects.
+Then `$Hj$' and `$Hp$' will both be true, on this interpretation, but `$Hr$' will be false, since Rachel Notley was not among the stipulated objects.
(This process of explicit stipulation is sometimes described as stipulating the \emph{extension} of a predicate.)
@@ -190,7 +190,7 @@ \chapter{Truth in FOL}\label{s:TruthFOL}
\begin{ekey}
\item[\text{domain}] all people born before 2000\textsc{ce}
\item[a] Aristotle
- \item[b] Bush
+ \item[b] Beyonc\'e
\item[Px] \gap{x} is a philosopher
\item[Rxy] \gap{x} was born before \gap{y}
\end{ekey}
@@ -199,7 +199,7 @@ \chapter{Truth in FOL}\label{s:TruthFOL}
\section{Atomic sentences}
The truth of atomic sentences should be fairly straightforward. The sentence `$Pa$' should be true just in case `$Px$' is true of `$a$'. Given our go-to interpretation, this is true iff Aristotle is a philosopher. Aristotle is a philosopher. So the sentence is true. Equally, `$Pb$' is false on our go-to interpretation.
-Likewise, on this interpretation, `$Rab$' is true iff the object named by `$a$' was born before the object named by `$b$'. Well, Aristotle was born before Bush. So `$Rab$' is true. Equally, `$Raa$' is false: Aristotle was not born before Aristotle.
+Likewise, on this interpretation, `$Rab$' is true iff the object named by `$a$' was born before the object named by `$b$'. Well, Aristotle was born before Beyonc\'e. So `$Rab$' is true. Equally, `$Raa$' is false: Aristotle was not born before Aristotle.
Dealing with atomic sentences, then, is very intuitive. When \meta{R} is an $n$-place predicate and $\meta{a}_1, \meta{a}_{2}, \ldots, \meta{a}_{n}$ are names,
@@ -212,7 +212,7 @@ \section{Atomic sentences}
$\meta{a} = \meta{b}$ is true in an interpretation \textbf{iff}\\
\meta{a} and \meta{b} name the very same object in that interpretation
}
-So in our go-to interpretation, `$a = b$' is false, since Aristotle is distinct from Bush.
+So in our go-to interpretation, `$a = b$' is false, since Aristotle is distinct from Beyonc\'e.
\section{Sentential connectives}
@@ -256,7 +256,7 @@ \section{When the main logical operator is a quantifier}
So here is a third thought. (And this thought is not na\"{i}ve, but correct.) Although it is not the case that we have named \emph{everyone}, each person \emph{could} have been given a name. So we should focus on this possibility of extending an interpretation, by adding a new name. We will offer a few examples of how this might work, centring on our go-to interpretation, and we will then present the formal definition.
-In our go-to interpretation, `$\exists x Rbx$' should be true. After all, in the domain, there is certainly someone who was born after Bush. Lady Gaga is one of those people. Indeed, if we were to extend our go-to interpretation---temporarily, mind---by adding the name `$c$' to refer to Lady Gaga, then `$Rbc$' would be true on this extended interpretation. This, surely, should suffice to make `$\exists x Rbx$' true on the original go-to interpretation.
+In our go-to interpretation, `$\exists x Rbx$' should be true. After all, in the domain, there is certainly someone who was born after Beyonc\'e. Lady Gaga is one of those people. Indeed, if we were to extend our go-to interpretation---temporarily, mind---by adding the name `$c$' to refer to Lady Gaga, then `$Rbc$' would be true on this extended interpretation. This, surely, should suffice to make `$\exists x Rbx$' true on the original go-to interpretation.
In our go-to interpretation, `$\exists x (Px \eand Rxa)$' should also be true. After all, in the domain, there is certainly someone who was both a philosopher and born before Aristotle. Socrates is one such person. Indeed, if we were to extend our go-to interpretation by letting a new name, `$c$', denote Socrates, then `$Wc \eand Rca$' would be true on this extended interpretation. Again, this should surely suffice to make `$\exists x (Px \eand Rxa)$' true on the original go-to interpretation.
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@@ -1,11 +1,11 @@
\chapter{Preface}
-As the title indicates, this is a textbook on formal logic. Formal logic concerns the study of a certain kind of language which, like any language, can serve to express states of affairs. It is a formal language, i.e., its expressions (such as sentences) are defined formally. This makes it a very useful language for being very precise about the states of affairs its sentences describe. In particular, in formal logic is is impossible to be ambiguous. The study of these languages centres on the relationship of entailment between sentences, i.e., which sentences follow from which other sentences. Entailment is central because by understanding it better we can tell when some states of affairs must obtain provided some other states of affairs obtain. But entailment is not the only important notion. We will also consider the relationship of being consistent, i.e., of not being mutually contradictory. These notions can be defined semantically---using precise definitions of entailment based on interpretation of the language---or proof-theoretically, using formal systems of deduction.
+As the title indicates, this is a textbook on formal logic. Formal logic concerns the study of a certain kind of language which, like any language, can serve to express states of affairs. It is a formal language, i.e., its expressions (such as sentences) are defined formally. This makes it a very useful language for being very precise about the states of affairs its sentences describe. In particular, in formal logic is is impossible to be ambiguous. The study of these languages centres on the relationship of entailment between sentences, i.e., which sentences follow from which other sentences. Entailment is central because by understanding it better we can tell when some states of affairs must obtain provided some other states of affairs obtain. But entailment is not the only important notion. We will also consider the relationship of being consistent, i.e., of not being mutually contradictory. These notions can be defined semantically, using precise definitions of entailment based on interpretations of the language---or proof-theoretically, using formal systems of deduction.
Formal logic is of course a central sub-discipline of philosophy, where the logical relationship of assumptions to conclusions reached from them is important. Philosophers investigate the consequences of definitions and assumptions and evaluate these definitions and assumptions on the basis of their consequences. It is also important in mathematics and computer science. In mathematics, formal languages are used to describe not ``everyday'' states of affairs, but mathematical states of affairs. Mathematicians are also interested in the consequences of definitions and assumptions, and for them it is equally important to establish these consequences (which they call ``theorems'') using completely precise and rigorous methods. Formal logic provides such methods. In computer science, formal logic is applied to describe the state and behaviours of computational systems, e.g., circuits, programs, databases, etc. Methods of formal logic can likewise be used to establish consequences of such descriptions, such as whether a circuit is error-free, whether a program does what it's intended to do, whether a database is consistent or if something is true of the data in it.
The book is divided into eight parts. The first part deals introduces the topic and notions of logic in an informal way, without introducing a formal language yet. Parts II--IV concern truth-functional languages. In it, sentences are formed from basic sentences using a number of connectives (`or', `and', `not', `if \dots then') which just combine sentences into more complicated ones. We discuss logical notions such as entailment in two ways: semantically, using the method of truth tables (in Part~III) and proof-theoretically, using a system of formal derivations (in Part~IV). Parts V--VII deal with a more complicated language, that of first-order logic. It includes, in addition to the connectives of truth-functional logic, also names, predicates, identity, and the so-called quantifiers. These additional elements of the language make it much more expressive that the truth-functional language, and we'll spend a fair amount of time investigating just how much one can express in it. Again, logical notions for the language of first-order logic are defined semantically, using interpretations, and prof-theoretically, using a more complex version of the formal derivation system introduced in Part~IV. Part~VIII covers an advanced topic: that of expressive adequacy of the truth-functional connectives.
In the appendices you'll find a discussion of alternative notations for the languages we discuss in this text, of alternative derivation systems, and a quick reference listing most of the important rules and definitions. The central terms are listed in a glossary at the very end.
-This book is based on a text originally written by P.~D. Magnus and revised and expanded by Tim Button and independently by J.~Robert Loftis. Aaron Thomas-Bolduc and Richard Zach have combined elements of these texts into the present version, changed some of the terminology and examples, and added material of their own. This resulting text is provided free of charge under a Creative Commons Attribution-ShareAlike 4.0 license.
+This book is based on a text originally written by P.~D. Magnus and revised and expanded by Tim Button and independently by J.~Robert Loftis. Aaron Thomas-Bolduc and Richard Zach have combined elements of these texts into the present version, changed some of the terminology and examples, and added material of their own. The resulting text is licensed under a Creative Commons Attribution-ShareAlike 4.0 license.
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@@ -704,22 +704,22 @@ \chapter{Use and mention}\label{s:UseMention}
\section{Quotation conventions}
Consider these two sentences:
\begin{ebullet}
- \item Theresa May is the Prime Minister.
- \item The expression `Theresa May' is composed of two uppercase letters and eight lowercase letters
+ \item Justin Trudeau is the Prime Minister.
+ \item The expression `Justin Trudeau' is composed of two uppercase letters and eleven lowercase letters
\end{ebullet}
-When we want to talk about the Prime Minister, we \emph{use} her name. When we want to talk about the Prime Minister's name, we \emph{mention} that name, which we do by putting it in quotation marks.
+When we want to talk about the Prime Minister, we \emph{use} his name. When we want to talk about the Prime Minister's name, we \emph{mention} that name, which we do by putting it in quotation marks.
There is a general point here. When we want to talk about things in the world, we just \emph{use} words. When we want to talk about words, we typically have to \emph{mention} those words. We need to indicate that we are mentioning them, rather than using them. To do this, some convention is needed. We can put them in quotation marks, or display them centrally in the page (say). So this sentence:
\begin{ebullet}
- \item `Theresa May' is the Prime Minister.
+ \item `Justin Trudeau' is the Prime Minister.
\end{ebullet}
-says that some \emph{expression} is the Prime Minister. That's false. The \emph{woman} is the Prime Minister; her \emph{name} isn't. Conversely, this sentence:
+says that some \emph{expression} is the Prime Minister. That's false. The \emph{man} is the Prime Minister; his \emph{name} isn't. Conversely, this sentence:
\begin{ebullet}
- \item Theresa May is composed of two uppercase letters and eight lowercase letters.
+ \item Justin Trudeau is composed of two uppercase letters and eleven lowercase letters.
\end{ebullet}
-also says something false: Theresa May is a woman, made of flesh rather than letters. One final example:
+also says something false: Justin Trudeau is a man, made of flesh rather than letters. One final example:
\begin{ebullet}
- \item ``\,`Theresa May'\,'' is the name of `Theresa May'.
+ \item ``\,`Justin Trudeau'\,'' is the name of `Justin Trudeau'.
\end{ebullet}
On the left-hand-side, here, we have the name of a name. On the right hand side, we have a name. Perhaps this kind of sentence only occurs in logic textbooks, but it is true nonetheless.
@@ -768,7 +768,7 @@ \section{Object language and metalanguage}
\section{Metavariable}
-However, we do not just want to talk about \emph{specific} expressions of TFL. We also want to be able to talk about \emph{any arbitrary} sentence of TFL. Indeed, we had to do this in \S\ref{s:TFLSentences}, when we presented the recursive definition of a sentence of TFL. We used uppercase swash-font letters to do this, namely:
+However, we do not just want to talk about \emph{specific} expressions of TFL. We also want to be able to talk about \emph{any arbitrary} sentence of TFL. Indeed, we had to do this in \S\ref{s:TFLSentences}, when we presented the recursive definition of a sentence of TFL. We used uppercase script letters to do this, namely:
$$\script{A}, \script{B}, \script{C}, \script{D}, \ldots$$
These symbols do not belong to TFL. Rather, they are part of our (augmented) metalanguage that we use to talk about \emph{any} expression of TFL. To repeat the second clause of the recursive definition of a sentence of TFL, we said:
\begin{earg}

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