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chnage general notions to standard metaphysical terms, eg tautology -…

…> necessary truth
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rzach committed Dec 15, 2016
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@@ -4,13 +4,15 @@ \part{Key notions of logic}
\addtocontents{toc}{\protect\mbox{}\protect\hrulefill\par}
\chapter{Arguments}\label{argRaining}\label{s:Arguments}
\chapter{Arguments}
\label{s:Arguments}
Logic is the business of evaluating arguments; sorting the good from the bad.
In everyday language, we sometimes use the word `argument' to talk about belligerent shouting matches. If you and a friend have an argument in this sense, things are not going well between the two of you. Logic is not concerned with such teeth-gnashing and hair-pulling. They are not arguments, in our sense; they are disagreements.
An argument, as we will understand it, is something more like this:
\begin{earg}
\begin{earg}\label{argRaining}
\item[] It is raining heavily.
\item[] If you do not take an umbrella, you will get soaked.
\item[\therefore] You should take an umbrella.
@@ -99,9 +101,9 @@ \section{Sentences}
\practiceproblems
At the end of some sections, there are problems that review and explore the material covered in the chapter. There is no substitute for actually working through some problems, because logic is more about a way of thinking than it is about memorising facts.
\medskip
\
\\Highlight the phrase which expresses the conclusion of each of these arguments:
Highlight the phrase which expresses the conclusion of each of these arguments:
\begin{earg}
\item It is sunny. So I should take my sunglasses.
\item It must have been sunny. I did wear my sunglasses, after all.
@@ -110,7 +112,9 @@ \section{Sentences}
\end{earg}
\chapter{Valid arguments}\label{s:Valid}
\chapter{Valid arguments}
\label{s:Valid}
In \S\ref{s:Arguments}, we gave a very permissive account of what an argument is. To see just how permissive it is, consider the following:
\begin{earg}
\item[] There is a bassoon-playing dragon in the \emph{Cathedra Romana}.
@@ -132,6 +136,7 @@ \section{Two ways that arguments can go wrong}
% \end{earg}
%This is not a terrible argument. Both of the premises are true. And most people who read this book are logic students. Yet, it is possible for someone besides a logic student to read this book. If your roommate picked up the book and thumbed through it, they would not immediately become a logic student. So the premises of this argument, even though they are true, do not guarantee the truth of the conclusion.
%
The general point is as follows. For any argument, there are two ways that it might go wrong:
\begin{ebullet}
\item One or more of the premises might be false.
@@ -141,13 +146,13 @@ \section{Two ways that arguments can go wrong}
So: we are interested in whether or not a conclusion \emph{follows from} some premises. Don't, though, say that the premises \emph{infer} the conclusion. Entailment is a relation between premises and conclusions; inference is something we do. (So if you want to mention inference when the conclusion follows from the premises, you could say that \emph{one may infer} the conclusion from the premises.)
\section{Validity}
As logicians, we want to be able to determine when the conclusion of an argument follows from the premises. One way to put this is as follows. We want to know whether, if all the premises were true, the conclusion would also have to be true. This motivates a definition:
\factoidbox{
An argument is \define{valid} if and only if it is impossible for all of the premises to be true and the conclusion false.
}
\newglossaryentry{valid}
{
name=valid,
@@ -159,8 +164,6 @@ \section{Validity}
name=invalid,
description={A property of arguments that holds when it is possible for the premises to be true without the conclusion being true; the opposite of \gls{valid}.}
}
The crucial thing about a valid argument is that it is impossible for the premises to be true whilst the conclusion is false. Consider this example:
\begin{earg}
@@ -179,15 +182,13 @@ \section{Validity}
The premises and conclusion of this argument are, as a matter of fact, all true, but the argument is invalid. If Paris were to declare independence from the rest of France, then the conclusion would be false, even though both of the premises would remain true. Thus, it is \emph{possible} for the premises of this argument to be true and the conclusion false. The argument is therefore invalid.
The important thing to remember is that validity is not about the actual truth or falsity of the sentences in the argument. It is about whether it is \emph{possible} for all the premises to be true and the conclusion false. Nonetheless, we will say that an argument is \define{sound} if and only if it is both valid and all of its premises are true.
\newglossaryentry{sound}
{
name=sound,
description={A property of arguments that holds if the argument is valid and has all true premises.}
}
\section{Inductive arguments}
Many good arguments are invalid. Consider this one:
\begin{earg}
@@ -201,7 +202,6 @@ \section{Inductive arguments}
The point of all this is that inductive arguments---even good inductive arguments---are not (deductively) valid. They are not \emph{watertight}. Unlikely though it might be, it is \emph{possible} for their conclusion to be false, even when all of their premises are true. In this book, we will set aside (entirely) the question of what makes for a good inductive argument. Our interest is simply in sorting the (deductively) valid arguments from the invalid ones.
\practiceproblems
\problempart
Which of the following arguments are valid? Which are invalid?
@@ -271,9 +271,8 @@ \chapter{Other logical notions}\label{s:BasicNotions}
%The general point is that, the premises and conclusion of an argument must be capable of having a \define{truth value}. The two truth values that concern us are just True and False.
\section{Consistency}
Consider these two sentences:
\begin{ebullet}
\item[B1.] Jane's only brother is shorter than her.
@@ -301,77 +300,77 @@ \section{Consistency}
\end{ebullet}
G1 and G4 together entail that there are at least four martian giraffes at the park. This conflicts with G3, which implies that there are no more than two martian giraffes there. So the sentences G1--G4 are jointly inconsistent. They cannot all be true together. (Note that the sentences G1, G3 and G4 are jointly inconsistent. But if sentences are already jointly inconsistent, adding an extra sentence to the mix will not make them consistent!)
\section{Necessary truths, necessary falsehoods, and contingency}
\section{Contradictions, tautologies and contingency}
In assessing arguments for validity, we care about what would be true \emph{if} the premises were true, but some sentences just \emph{must} be true. Consider these sentences:
\begin{earg}
\item[\ex{Acontingent}] It is raining.
\item[\ex{Atautology}] Either it is raining here, or it is not.
\item[\ex{Acontradiction}] It is both raining here and not raining here.
\end{earg}
In order to know if sentence \ref{Acontingent} is true, you would need to look outside or check the weather channel. It might be true; it might be false. A sentence which is capable of being true or false, but which is neither tautological nor contradictory, is \define{contingent}.
In order to know if sentence \ref{Acontingent} is true, you would need to look outside or check the weather channel. It might be true; it might be false. A sentence which is capable of being true and capable of being false (in different circumstances, of course) is called \define{contingent}.
\newglossaryentry{contingent sentence}
{
name=contingent sentence,
description={A sentence that is neither a \gls{necessary truth} nor a \gls{necessary falsehood}; a sentence that in some situations is true and in others false.}
}
Sentence \ref{Atautology} is different. You do not need to look outside to know that it is true. Regardless of what the weather is like, it is either raining or it is not. That is a \define{tautology}.
Sentence \ref{Atautology} is different. You do not need to look outside to know that it is true. Regardless of what the weather is like, it is either raining or it is not. That is a \define{necessary truth}.
\newglossaryentry{tautology}
\newglossaryentry{necessary truth}
{
name=tautology,
name={necessary truth},
description={A sentence that must be true}
}
Equally, you do not need to check the weather to determine whether or not sentence \ref{Acontradiction} is true. It must be false, simply as a matter of logic. It might be raining here and not raining across town; it might be raining now but stop raining even as you finish this sentence; but it is impossible for it to be both raining and not raining in the same place and at the same time. So, whatever the world is like, it is not both raining here and not raining here. It is a \define{contradiction}.
Equally, you do not need to check the weather to determine whether or not sentence \ref{Acontradiction} is true. It must be false, simply as a matter of logic. It might be raining here and not raining across town; it might be raining now but stop raining even as you finish this sentence; but it is impossible for it to be both raining and not raining in the same place and at the same time. So, whatever the world is like, it is not both raining here and not raining here. It is a \define{necessary falsehood}.
\newglossaryentry{contradiction}
\newglossaryentry{necessary falsehood}
{
name=contradiction,
description={A sentence that must be false, as a matter of logic.}
name={necessary falsehood},
description={A sentence that must be false}
}
%Something might \emph{always} be true and still be contingent. For instance, if there never were a time when the universe contained fewer than seven things, then the sentence `At least seven things exist' would always be true. Yet the sentence is contingent: the world could have been much, much smaller than it is, and then the sentence would have been false.
\subsection{Logical equivalence}
\subsection{Necessary equivalence}
We can also ask about the logical relations \emph{between} two sentences. For example:
\begin{earg}
\item[] John went to the store after he washed the dishes.
\item[] John washed the dishes before he went to the store.
\end{earg}
These two sentences are both contingent, since John might not have gone to the store or washed dishes at all. Yet they must have the same truth-value. If either of the sentences is true, then they both are; if either of the sentences is false, then they both are. When two sentences necessarily have the same truth value, we say that they are \define{logically equivalent}.
These two sentences are both contingent, since John might not have gone to the store or washed dishes at all. Yet they must have the same truth-value. If either of the sentences is true, then they both are; if either of the sentences is false, then they both are. When two sentences necessarily have the same truth value, we say that they are \define{necessarily equivalent}.
\newglossaryentry{logical equivalence}
\newglossaryentry{necessary equivalence}
{
name={logical equivalence},
text={logically equivalent},
name={necessary equivalence},
text={necessarily equivalent},
description={A property held by a pair of sentences that must always have the same truth value.}
}
\section*{Summary of logical notions}
\begin{itemize}
\item An argument is (deductively) \define{valid} if it is impossible for the premises to be true and the conclusion false; it is \define{invalid} otherwise.
\item A \define{tautology} is a sentence that must be true, as a matter of logic.
\item A \define{necessary truth} is a sentence that must be true, that could not possibly be false.
\item A \define{contradiction} is a sentence that must be false, as a matter of logic.
\item A \define{necessary falsehood} is a sentence that must be false, that could not possibly be true.
\item A \define{contingent sentence} is neither a tautology nor a contradiction.
\item A \define{contingent sentence} is neither a necessary truth nor a necessary falsehood. It may be true but could have been false, or vice versa.
\item Two sentences are \define{logically equivalent} if they necessarily have the same truth value.
\item Two sentences are \define{necessarily equivalent} if they must have the same truth value.
\item A collection of sentences is \define{consistent} if it is logically possible for all the members of the set to be true at the same time; it is \define{inconsistent} otherwise.
\item A collection of sentences is \define{jointly possible} if it is possible for all these sentences to be true together; it is \define{jointly impossible} otherwise.
\end{itemize}
\practiceproblems
\problempart
\label{pr.EnglishTautology}
For each of the following: Is it tautological, contradictory, or contingent?
For each of the following: Is it a necessary truth, a necessary falsehood, or contingent?
\begin{earg}
\item Caesar crossed the Rubicon.
\item Someone once crossed the Rubicon.
@@ -381,7 +380,8 @@ \section*{Summary of logical notions}
\item If anyone has ever crossed the Rubicon, it was Caesar.
\end{earg}
\problempart Label the following tautology, contradiction, or contingent.
\problempart
For each of the following: Is it a necessary truth, a necessary falsehood, or contingent?
\begin{earg}
\item Elephants dissolve in water.
\item Wood is a light, durable substance useful for building things.
@@ -390,7 +390,7 @@ \section*{Summary of logical notions}
\item If gerbils were mammals they would nurse their young.
\end{earg}
\problempart Which of the following pairs of sentences are logically equivalent?
\problempart Which of the following pairs of sentences are necessarily equivalent?
\begin{earg}
\item Elephants dissolve in water. \\
@@ -404,7 +404,7 @@ \section*{Summary of logical notions}
\item Elephants dissolve in water. \\
All mammals dissolve in water.
\end{earg}
\problempart Which of the following pairs of sentences are logically equivalent?
\problempart Which of the following pairs of sentences are necessarily equivalent?
\begin{earg}
\item Thelonious Monk played piano. \\
@@ -427,7 +427,7 @@ \section*{Summary of logical notions}
\item[G4] \label{itm:martians} Every giraffe at the wild animal park is a Martian.
\end{enumerate}
Now consider each of the following collections of sentences. Which are consistent? Which are inconsistent?
Now consider each of the following collections of sentences. Which are jointly possible? Which are jointly impossible?
\begin{earg}
\item Sentences G2, G3, and G4
\item Sentences G1, G3, and G4
@@ -442,7 +442,7 @@ \section*{Summary of logical notions}
\item[M3] \label{itm:socnotdie} Socrates will never die.
\item[M4] \label{itm:socmortal} Socrates is mortal.
\end{enumerate}
Which combinations of sentences are jointly consistent? Mark each ``consistent'' or ``inconsistent.''
Which combinations of sentences are jointly possible? Mark each ``possible'' or ``impossible.''
\begin{earg}
\item Sentences M1, M2, and M3
\item Sentences M2, M3, and M4
@@ -459,34 +459,34 @@ \section*{Summary of logical notions}
\item A valid argument that has a false conclusion
\item A valid argument, the conclusion of which is a contradiction
\item A valid argument, the conclusion of which is a necessary falsehood
\item An invalid argument, the conclusion of which is a tautology
\item An invalid argument, the conclusion of which is a necessary truth
\item A tautology that is contingent
\item A necessary truth that is contingent
\item Two logically equivalent sentences, both of which are tautologies
\item Two necessarily equivalent sentences, both of which are necessary truths
\item Two logically equivalent sentences, one of which is a tautology and one of which is contingent
\item Two necessarily equivalent sentences, one of which is a necessary truth and one of which is contingent
\item Two logically equivalent sentences that together are an inconsistent set
\item Two necessarily equivalent sentences that together are jointly impossible
\item A consistent set of sentences that contains a contradiction
\item A jointly possible collection of sentences that contains a necessary falsehood
\item An inconsistent set of sentences that contains a tautology
\item A jointly impossible set of sentences that contains a necessary truth
\end{earg}
\problempart
Which of the following is possible? If it is possible, give an example. If it is not possible, explain why.
\begin{earg}
\item A valid argument, whose premises are all tautologies, and whose conclusion is contingent
\item A valid argument, whose premises are all necessary truths, and whose conclusion is contingent
\item A valid argument with true premises and a false conclusion
\item A consistent set of sentences that contains two sentences that are not logically equivalent
\item A consistent set of sentences, all of which are contingent
\item A false tautology
\item A jointly possible collection of sentences that contains two sentences that are not necessarily equivalent
\item A jointly possible collection of sentences, all of which are contingent
\item A false necessary truth
\item A valid argument with false premises
\item A logically equivalent pair of sentences that are not consistent
\item A tautological contradiction
\item A consistent set of sentences that are all contradictions
\item A necessarily equivalent pair of sentences that are not jointly possible
\item A necessary truth that is also a necessary falsehood
\item A jointly possible collection of sentences that are all necessary falsehoods
\end{earg}

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