forallx-yyc-truthtables.tex

%!TEX root = forallxyyc.tex
\part{Truth tables}
\label{ch.TruthTables}
\addtocontents{toc}{\protect\mbox{}\protect\hrulefill\par}

\chapter{Characteristic truth tables}
\label{s:CharacteristicTruthTables}

Any sentence of TFL is composed of sentence letters, possibly combined using sentential connectives. The truth value of the compound sentence depends only on the truth value of the sentence letters that comprise it. In order to know the truth value of `$(D \eand E)$', for instance, you only need to know the truth value of `$D$' and the truth value of `$E$'.

We introduced five connectives in \cref{s:TFLConnectives}. So we just need to explain how they map between truth values. For convenience, we abbreviate `True' with `T' and `False' with `F'. (But, to be clear, the two truth values are True and False; the truth values are not \emph{letters}!)

\newglossaryentry{truth value}
                 {
                   name = truth value,
                   description = {One of the two logical values sentences can have: True and False}
                   }

\paragraph{Negation} For any sentence \metav{A}: If \metav{A} is true, then \enot\metav{A} is false; and if \enot\metav{A} is true, then \metav{A} is false. We can summarize this in the \emph{characteristic truth table} for negation:
\begin{center}
\begin{tabular}{c|c}
\metav{A} & \enot\metav{A}\\
\hline
T & F\\
F & T 
\end{tabular}
\end{center}

\paragraph{Conjunction} For any sentences \metav{A} and \metav{B}, \metav{A}\eand\metav{B} is true if and only if both \metav{A} and \metav{B} are true. We can summarize this in the {characteristic truth table} for conjunction:
\begin{center}
\begin{tabular}{c c |c}
\metav{A} & \metav{B} & $\metav{A}\eand\metav{B}$\\
\hline
T & T & T\\
T & F & F\\
F & T & F\\
F & F & F
\end{tabular}
\end{center}
Note that the truth value for $\metav{A} \eand \metav{B}$ is always
the same as the truth value for $\metav{B} \eand \metav{A}$.
Connectives that have this property are called \emph{commutative}.

\paragraph{Disjunction} Recall that `$\eor$' always represents inclusive or. So, for any sentences \metav{A} and \metav{B}, $\metav{A}\eor \metav{B}$ is true if and only if either \metav{A} or \metav{B} is true. We can summarize this in the {characteristic truth table} for disjunction:
\begin{center}
\begin{tabular}{c c|c}
\metav{A} & \metav{B} & $\metav{A}\eor\metav{B}$ \\
\hline
T & T & T\\
T & F & T\\
F & T & T\\
F & F & F
\end{tabular}
\end{center}
Like conjunction, disjunction is commutative.

\paragraph{Conditional} We're just going to come clean and admit it: Conditionals are a mess in TFL. Exactly how much of a mess they are is \emph{philosophically} contentious. We'll discuss a few of the subtleties in \cref{s:IndicativeSubjunctive,s:ParadoxesOfMaterialConditional}. For now, we are going to stipulate the following: $\metav{A}\eif\metav{B}$ is false if and only if \metav{A} is true and \metav{B} is false. We can summarize this with a characteristic truth table for the conditional.
\begin{center}
\begin{tabular}{c c|c}
\metav{A} & \metav{B} & $\metav{A}\eif\metav{B}$\\
\hline
T & T & T\\
T & F & F\\
F & T & T\\
F & F & T
\end{tabular}
\end{center}
The conditional is \emph{not commutative}. You cannot swap the antecedent and consequent without changing the meaning of the sentence; $\metav{A}\eif\metav{B}$ and $\metav{B} \eif \metav{A}$ have different truth tables.

\paragraph{Biconditional} Since a biconditional is to be the same as the conjunction of the conditionals running in both directions, we will want the truth table for the biconditional to be:
\begin{center}
\begin{tabular}{c c|c}
\metav{A} & \metav{B} & $\metav{A}\eiff\metav{B}$\\
\hline
T & T & T\\
T & F & F\\
F & T & F\\
F & F & T
\end{tabular}
\end{center}
Unsurprisingly, the biconditional is commutative.

\chapter{Truth-functional connectives}
\label{s:TruthFunctionality}

\section{The idea of truth-functionality}
Let's introduce an important idea.
	\factoidbox{
		A connective is \define{truth-functional} \ifeff{} the truth value of a sentence with that connective as its main logical operator is uniquely determined by the truth value(s) of the constituent sentence(s).
	}
\newglossaryentry{truth-functional connective}
{
name=truth-functional connective,
description={An operator that builds larger sentences out of smaller ones and fixes the \gls{truth value} of the resulting sentence based only on the truth value of the component sentences}
}
        
Every connective in TFL is truth-functional. The truth value of a negation is uniquely determined by the truth value of the unnegated sentence. The truth value of a conjunction is uniquely determined by the truth value of both conjuncts. The truth value of a disjunction is uniquely determined by the truth value of both disjuncts, and so on. To determine the truth value of some TFL sentence, we only need to know the truth value of its components.

This is what gives TFL its name: it is \emph{truth-functional logic}.

Many languages use connectives that are not truth-functional. In English, for example, we can form a new sentence from any simpler sentence by prefixing it with `It is necessarily the case that\ldots'. The truth value of this new sentence is not fixed solely by the truth value of the original sentence. For consider two true sentences:
	\begin{compactlist}
		\item $2 + 2 = 4$
		\item Shostakovich wrote fifteen string quartets.
	\end{compactlist}
Whereas it is necessarily the case that $2 + 2 = 4$, it is not \emph{necessarily} the case that Shostakovich wrote fifteen string quartets. If Shostakovich had died earlier, he would have failed to finish Quartet no.~15; if he had lived longer, he might have written a few more. So `It is necessarily the case that\ldots' is not \emph{truth-functional}.


\section{Symbolizing versus translating}
All of the connectives of TFL are truth-functional, but more than that: they really do nothing \emph{but} map us between truth values.  

When we symbolize a sentence or an argument in TFL, we ignore everything \emph{besides} the contribution that the truth values of a component might make to the truth value of the whole. There are subtleties to our ordinary claims that far outstrip their mere truth values. Sarcasm; poetry; snide implicature; emphasis; these are important parts of everyday discourse, but none of this is retained in TFL. As remarked in \cref{s:TFLConnectives}, TFL cannot capture the subtle differences between the following English sentences:
	\begin{compactlist}
		\item Dana is a logician and Dana is a nice person
		\item Although Dana is a logician, Dana is a nice person
		\item Dana is a logician despite being a nice person
		\item Dana is a nice person, but also a logician
		\item Dana's being a logician notwithstanding, he is a nice person
	\end{compactlist}
All of the above sentences will be symbolized with the same TFL sentence, perhaps `$L \eand N$'.

Now, we keep saying that we use TFL sentences to \emph{symbolize} English sentences. Many other textbooks talk about \emph{translating} English sentences into TFL. However, a good translation should preserve certain facets of meaning, and---as we just saw---TFL just cannot do that. This is why we will speak of \emph{symbolizing} English sentences, rather than of \emph{translating} them.

This affects how we should understand our symbolization keys. Consider a key like:
	\begin{ekey}
		\item[L] Dana is a logician.
		\item[N] Dana is a nice person.
	\end{ekey}
Other textbooks will understand this as a stipulation that the TFL sentence `$L$' should \emph{mean} that Dana is a logician, and that the TFL sentence `$N$' should \emph{mean} that Dana is a nice person. But TFL just is totally unequipped to deal with \emph{meaning}. The preceding symbolization key is doing no more and no less than stipulating that the TFL sentence `$L$' should take the same truth value as the English sentence `Dana is a logician' (whatever that might be), and that the TFL sentence `$N$' should take the same truth value as the English sentence `Dana is a nice person' (whatever that might be).
	\factoidbox{
		When we treat a TFL sentence as \emph{symbolizing} an English sentence, we are stipulating that the TFL sentence is to take the same truth value as that English sentence.
	}


\section{Indicative versus subjunctive conditionals}\label{s:IndicativeSubjunctive}
We want to bring home the point that TFL can \emph{only} deal with
truth functions by considering the case of the conditional. When we
introduced the characteristic truth table for the material conditional
in \cref{s:CharacteristicTruthTables}, we did not say anything to
justify it. Let's now offer a justification, which follows Dorothy
Edgington.\footnote{Dorothy Edgington, ``Conditionals'',
\textit{Stanford Encyclopedia of Philosophy} (Fall 2020)
(\url{https://plato.stanford.edu/archives/fall2020/entries/conditionals/}).} 

Suppose that Lara has drawn some shapes on a piece of paper, and coloured some of them in. We have not seen them, but nevertheless claim:
	\begin{quote}
		If any shape is grey, then that shape is also circular.
	\end{quote}
As it happens, Lara has drawn the following:

\begin{center}
\begin{arialabel}{There are three shapes: the first shape is a grey circle labelled A, some empty space, the second shape is a white circle labelled C, and the fourth a white square labelled D.}
\begin{tikzpicture}
	\node[circle, grey_shape] (cat1) {A};
	\node[right=10pt of cat1, diamond, phantom_shape] (cat2)  { } ;
	\node[right=10pt of cat2, circle, white_shape] (cat3)  {C} ;
	\node[right=10pt of cat3, white_shape] (cat4)  {D};
\end{tikzpicture}
\end{arialabel}
\end{center}

\noindent In this case, our claim is surely true.  Shapes C and D are not grey, and so can hardly present \emph{counterexamples} to our claim. Shape A \emph{is} grey, but fortunately it is also circular. So our claim has no counterexamples. It must be true. That means that each of the following \emph{instances} of our claim must be true too:
	\begin{itemize}
		\item If A is grey, then it is circular

		\qquad (true antecedent, true consequent)
		\item If C is grey, then it is circular

		\qquad (false antecedent, true consequent)
		\item If D is grey, then it is circular

		\qquad (false antecedent, false consequent)
	\end{itemize}
However, if Lara had drawn a fourth shape, thus:

\begin{center}
\begin{arialabel}{There are four shapes: the first shape is
	a grey circle labelled A, the second a grey square labelled B, the
	third a white circle labelled C, and the fourth a white square
	labelled D.}
\begin{tikzpicture}
	\node[circle, grey_shape] (cat1) {A};
	\node[right=10pt of cat1, grey_shape] (cat2)  {B};
	\node[right=10pt of cat2, circle, white_shape] (cat3)  {C};
	\node[right=10pt of cat3, white_shape] (cat4)  {D};
\end{tikzpicture}
\end{arialabel}
\end{center}

\noindent then our claim would have been false. So this claim must also be false:
	\begin{itemize}
		\item If B is grey, then it is circular

		\qquad (true antecedent, false consequent)
	\end{itemize}
Now, recall that every connective of TFL has to be truth-functional. This means that the truth values of the antecedent and consequent alone must uniquely determine the truth value of the conditional as a whole. Thus, from the truth values of our four claims---which provide us with all possible combinations of truth and falsity in antecedent and consequent---we can read off the truth table for the material conditional.

What this argument shows is that `$\eif$' is the \emph{best} candidate for a truth-functional conditional. Otherwise put, \emph{it is the best conditional that TFL can provide}. But is it any good, as a surrogate for the conditionals we use in everyday language? Consider two sentences:
	\begin{enumerate}
		\item\label{brownwins1} If Hillary Clinton had won the 2016 election, then she would have been the first woman president of the USA.
		\item\label{brownwins2} If Hillary Clinton had won the 2016 election, then she would have turned into a helium-filled balloon and floated away into the night sky.
	\end{enumerate}
\Cref*{brownwins1} is true; \cref*{brownwins2} is false, but both have false antecedents and false consequents. (Hillary did not win; she did not become the first woman president of the US; and she did not fill with helium and float away.) So the truth value of the whole sentence is not uniquely determined by the truth value of the parts.

The crucial point is that \cref*{brownwins1,brownwins2} employ \emph{subjunctive} conditionals, rather than \emph{indicative} conditionals. They ask us to imagine something contrary to fact---after all, Hillary Clinton lost the 2016 election---and then ask us to evaluate what \emph{would} have happened in that case. Such considerations simply cannot be tackled using `$\eif$'.

We will say more about the difficulties with conditionals in \cref{s:ParadoxesOfMaterialConditional}. For now, we will content ourselves with the observation that `$\eif$' is the only candidate for a truth-functional conditional for TFL, but that many English conditionals cannot be represented adequately using `$\eif$'. TFL is an intrinsically limited language.


\chapter{Complete truth tables}
\label{s:CompleteTruthTables}

So far, we have used symbolization keys to assign truth values to TFL sentences \emph{indirectly}. For example, we might say that the TFL sentence `$B$' is to be true \ifeff{} Big Ben is in London. Since Big Ben \emph{is} in London, this symbolisation would make `$B$' true. But we can also assign truth values \emph{directly}. We can simply stipulate that `$B$' is to be true, or stipulate that it is to be false. Such stipulations are called \emph{valuations}: 
	\factoidbox{
		A \define{valuation} is any assignment of truth values to particular sentence letters of TFL.
	}

\newglossaryentry{valuation}
{
name=valuation,
description={An assignment of \glspl{truth value} to particular \glspl{sentence letter}}
}

The power of truth tables lies in the following. Each row of a truth table represents a possible valuation. The complete truth table represents all possible valuations. And the truth table provides us with a means to calculate the truth value of complex sentences, on each possible valuation. But all of this is easiest to explain by example.

\section{A worked example}
Consider the sentence `$(H\eand I)\eif H$'. There are four possible ways to assign True and False to the sentence letter `$H$' and `$I$'---four valuations---which we can represent as follows:
\begin{center}
\begin{tabular}{c c|d e e e f}
$H$&$I$&$(H$&\eand&$I)$&\eif&$H$\\
\hline
 T & T\\
 T & F\\
 F & T\\
 F & F
\end{tabular}
\end{center}
To calculate the truth value of the entire sentence `$(H \eand I) \eif H$', we first copy the truth values for the sentence letters and write them underneath the letters in the sentence:
\begin{center}
\begin{tabular}{c c|d e e e f}
$H$&$I$&$(H$&\eand&$I)$&\eif&$H$\\
\hline
 T & T & {T} & & {T} & & {T}\\
 T & F & {T} & & {F} & & {T}\\
 F & T & {F} & & {T} & & {F}\\
 F & F & {F} & & {F} & & {F}
\end{tabular}
\end{center}
Now consider the subsentence `$(H\eand I)$'. This is a conjunction, $(\metav{A}\eand\metav{B})$, with `$H$' as \metav{A} and with `$I$' as \metav{B}. The characteristic truth table for conjunction gives the truth conditions for \emph{any} sentence of the form $(\metav{A}\eand\metav{B})$, whatever $\metav{A}$ and $\metav{B}$ might be. It represents the point that a conjunction is true \ifeff{} both conjuncts are true. In this case, our conjuncts are just `$H$' and `$I$'. They are both true on (and only on) the first line of the truth table. Accordingly, we can calculate the truth value of the conjunction on all four rows.
\begin{center}
\begin{tabular}{c c|d e e e f}
 & & \metav{A} & \eand & \metav{B} & & \\
$H$&$I$&$(H$&\eand&$I)$&\eif&$H$\\
\hline
 T & T & T & {T} & T & & T\\
 T & F & T & {F} & F & & T\\
 F & T & F & {F} & T & & F\\
 F & F & F & {F} & F & & F
\end{tabular}
\end{center}
Now, the entire sentence that we are dealing with is a conditional, $\metav{A}\eif\metav{B}$, with `$(H \eand I)$' as \metav{A} and with `$H$' as \metav{B}. On the second row, for example, `$(H\eand I)$' is false and `$H$' is true. Since a conditional is true when the antecedent is false, we write a `T' in the second row underneath the conditional symbol. We continue for the other three rows and get this:
\begin{center}
\begin{tabular}{c c| d e e e f}
 & &  & \metav{A} &  &\eif &\metav{B} \\
$H$&$I$&$(H$&\eand&$I)$&\eif&$H$\\
\hline
 T & T &  & {T} &  &{T} & T\\
 T & F &  & {F} &  &{T} & T\\
 F & T &  & {F} &  &{T} & F\\
 F & F &  & {F} &  &{T} & F
\end{tabular}
\end{center}
The conditional is the main logical operator of the sentence, so the column of `T's underneath the conditional tells us that the sentence `$(H \eand I)\eif H$' is true regardless of the truth values of `$H$' and `$I$'. They can be true or false in any combination, and the compound sentence still comes out true. Since we have considered all four possible assignments of truth and falsity to `$H$' and `$I$'---since, that is, we have considered all the different \emph{valuations}---we can say that `$(H \eand I)\eif H$' is true on every valuation.

In this example, we have not repeated all of the entries in every column in every successive table. When actually writing truth tables on paper, however, it is impractical to erase whole columns or rewrite the whole table for every step. Although it is more crowded, the truth table can be written in this way:
\begin{center}
\begin{tabular}{c c| d e e e f}
$H$&$I$&$(H$&\eand&$I)$&\eif&$H$\\
\hline
 T & T & T & {T} & T & \TTbf{T} & T\\
 T & F & T & {F} & F & \TTbf{T} & T\\
 F & T & F & {F} & T & \TTbf{T} & F\\
 F & F & F & {F} & F & \TTbf{T} & F
\end{tabular}
\end{center}
Most of the columns underneath the sentence are only there for bookkeeping purposes. The column that matters most is the column underneath the \emph{main logical operator} for the sentence, since this tells you the truth value of the entire sentence. We have emphasized this, by putting this column in bold. When you work through truth tables yourself, you should similarly emphasize it (perhaps by highlighting).

\section{Building complete truth tables}
A \define{complete truth table} has a line for every possible assignment of True and False to the relevant sentence letters. Each line represents a \emph{valuation}, and a complete truth table has a line for all the different valuations.

\newglossaryentry{complete truth table}
{
name=complete truth table,
description={A table that gives all the possible \glspl{truth value} for a \gls{sentence of TFL} or sentences in TFL, with a line for every possible \gls{valuation} of all sentence letters}
}

The size of the complete truth table depends on the number of different sentence letters in the table. A sentence that contains only one sentence letter requires only two rows, as in the characteristic truth table for negation. This is true even if the same letter is repeated many times, as in the sentence
`$[(C\eiff C) \eif C] \eand \enot(C \eif C)$'.
The complete truth table requires only two lines because there are only two possibilities: `$C$' can be true or it can be false. The truth table for this sentence looks like this:
\begin{center}
\begin{tabular}{c| d e e e e e e e e e e e e e e f}
$C$&$[($&$C$&\eiff&$C$&$)$&\eif&$C$&$]$&\eand&\enot&$($&$C$&\eif&$C$&$)$\\
\hline
 T &    & T &  T  & T &   & T  & T & &\TTbf{F}&  F& &   T &  T  & T &   \\
 F &    & F &  T  & F &   & F  & F & &\TTbf{F}&  F& &   F &  T  & F &   \\
\end{tabular}
\end{center}
Looking at the column underneath the main logical operator, we see that the sentence is false on both rows of the table; i.e., the sentence is false regardless of whether `$C$' is true or false. It is false on every valuation.

There will be four lines in the complete truth table for a sentence containing two sentence letters, as in the characteristic truth tables, or the truth table for `$(H \eand I)\eif H$'.

There will be eight lines in the complete truth table for a sentence containing three sentence letters, e.g.: 
\begin{center}
\begin{tabular}{c c c|d e e e f}
$M$&$N$&$P$&$M$&\eand&$(N$&\eor&$P)$\\
\hline
%           M        &     N   v   P
T & T & T & T & \TTbf{T} & T & T & T\\
T & T & F & T & \TTbf{T} & T & T & F\\
T & F & T & T & \TTbf{T} & F & T & T\\
T & F & F & T & \TTbf{F} & F & F & F\\
F & T & T & F & \TTbf{F} & T & T & T\\
F & T & F & F & \TTbf{F} & T & T & F\\
F & F & T & F & \TTbf{F} & F & T & T\\
F & F & F & F & \TTbf{F} & F & F & F
\end{tabular}
\end{center}
From this table, we know that the sentence `$M\eand(N\eor P)$' can be true or false, depending on the truth values of `$M$', `$N$', and `$P$'.

A complete truth table for a sentence that contains four different sentence letters requires 16 lines. Five letters, 32 lines. Six letters, 64 lines. And so on. To be perfectly general: If a complete truth table has $n$ different sentence letters, then it must have $2^n$ lines.

In order to fill in the columns of a complete truth table, begin with the right-most sentence letter and alternate between `T' and `F'. In the next column to the left, write two `T's, write two `F's, and repeat. For the third sentence letter, write four `T's followed by four `F's. This yields an eight line truth table like the one above. For a 16 line truth table, the next column of sentence letters should have eight `T's followed by eight `F's. For a 32 line table, the next column would have 16 `T's followed by 16 `F's, and so on.


\section{More about brackets}\label{s:MoreBracketingConventions}
Consider these two sentences:
	\begin{align*}
		((A \eand B) \eand C)\\
		(A \eand (B \eand C))
	\end{align*}
These are truth functionally equivalent. Consequently, it will never make any difference from the perspective of truth value---which is all that TFL cares about (see \cref{s:TruthFunctionality})---which of the two sentences we assert (or deny). Even though the order of the brackets does not matter as to their truth, we should not just drop them. The expression
	\begin{align*}
		A \eand B \eand C
	\end{align*}
is ambiguous between the two sentences above.  The same observation holds for disjunctions. The following sentences are logically equivalent:
	\begin{align*}
		((A \eor B) \eor C)\\
		(A \eor (B \eor C))
	\end{align*}
But we should not simply write:
	\begin{align*}
		A \eor B \eor C
	\end{align*}
In fact, it is a specific fact about the characteristic truth table of $\eor$ and $\eand$ that guarantees that any two conjunctions (or disjunctions) of the same sentences are truth functionally equivalent, however you place the brackets. \emph{This is only true of conjunctions and disjunctions}, however. The following two sentences have \emph{different} truth tables:
	\begin{align*}
		((A \eif B) \eif C)\\
		(A \eif (B \eif C))
	\end{align*}
So if we were to write:
	\begin{align*}
		A \eif B \eif C
	\end{align*}
it would be dangerously ambiguous. Leaving out brackets in this case would be disastrous. Equally, these sentences have different truth tables:
	\begin{align*}
		((A \eor B) \eand C)\\
		(A \eor (B \eand C))
	\end{align*}
So if we were to write:
	\begin{align*}
		A \eor B \eand C
	\end{align*}
it would be dangerously ambiguous. \emph{Never write this.} The moral is: never drop brackets (except the outermost ones).

\practiceproblems\label{pr.TT.TTorC}
\problempart
Offer complete truth tables for each of the following:
\begin{compactlist}
\item $A \eif A$ %taut
\item $C \eif\enot C$ %contingent
\item $(A \eiff B) \eiff \enot(A\eiff \enot B)$ %tautology
\item $(A \eif B) \eor (B \eif A)$ % taut
\item $(A \eand B) \eif (B \eor A)$  %taut
\item $\enot(A \eor B) \eiff (\enot A \eand \enot B)$ %taut
\item $\bigl[(A\eand B) \eand\enot(A\eand B)\bigr] \eand C$ %contradiction
\item $[(A \eand B) \eand C] \eif B$ %taut
\item $\enot\bigl[(C\eor A) \eor B\bigr]$ %contingent
\end{compactlist}
\problempart
Check all the claims made in \cref{s:MoreBracketingConventions}, i.e., show that:
\begin{compactlist}
	\item `$((A \eand B) \eand C)$' and `$(A \eand (B \eand C))$' have the same truth table
	\item `$((A \eor B) \eor C)$' and `$(A \eor (B \eor C))$' have the same truth table
	\item `$((A \eor B) \eand C)$' and `$(A \eor (B \eand C))$' do not have the same truth table
	\item `$((A \eif B) \eif C)$' and `$(A \eif (B \eif C))$' do not have the same truth table
\end{compactlist}
Also, check whether:
\begin{compactlist}
	\item[5.] `$((A \eiff B) \eiff C)$' and `$(A \eiff (B \eiff C))$' have the same truth table
\end{compactlist}

\problempart
Write complete truth tables for the following sentences and mark the column that represents the possible truth values for the whole sentence.

\begin{compactlist}

\item $\enot (S \eiff (P \eif S))$

%\begin{tabular}{c|c|ccccc}
%\cline{2-2}
%1.	&	\enot 	&	(S 	&	\eiff	&	(P 	&	\eif	&	S))	\\ 
%\cline{2-7}
%	& 	F 		&	T	&	T	&	T	&	T	&	T	\\
%	& 	F 		&	T	&	T	&	F	&	T	&	T	\\
%	& 	F 		&	F	&	T	&	T	&	F	&	F	\\
%	& 	T 		&	F	&	F	&	F	&	T	&	F	\\
%\cline{2-2}
%\end{tabular}


 \item $\enot [(X \eand Y) \eor (X \eor Y)]$

%\begin{tabular}{c|c|ccccccc}
%\cline{2-2}
%2.	&	\enot	&	 [(X 	&	\eand& 	Y) 	&	\eor 	&	(X 	&	\eor 	&	Y)] \\
%\cline{2-9}
%	&	F	&	T	&	T	&	T	&	T	&	T	&	T	&	T	\\
%	&	F	&	T	&	F	&	F	&	T	&	T	&	T	&	F	\\
%	&	F	&	F	&	F	&	T	&	T	&	F	&	T	&	T	\\
%	&	T	&	F	&	F	&	F	&	F	&	F	&	F	&	F	\\
%\cline{2-2}
%\end{tabular}


\item $(A \eif B) \eiff (\enot B\eiff \enot A)$
%\begin{tabular}{cccc|c|ccccc}
%\cline{5-5}
%3.	&	(A 	&	\eif	&	B)	&	 \eiff 	&	(\enot&	B 	&	\eiff 	&	 \enot 	& 	 A) \\
%\cline{2-10}
%	&	T	&	T	&	T	&	T		&	F	 &	T	&	T	&	F		&	T	\\	
%	&	T	&	F	&	F	&	T		&	T	 &	F	&	F	&	F		&	T	\\
%	&	F	&	T	&	T	&	F		&	F	 &	T	&	F	&	T		&	F	\\
%	&	F	&	T	&	F	&	T		&	T	 &	F	&	T	&	T		&	F	\\
%\cline{5-5}
%\end{tabular}

\item $[C \eiff (D \eor E)] \eand \enot C$

%\begin{tabular}{cccccc|c|cc}
%\cline{7-7}
%4.	&	[C 	&	\eiff 	&	(D 	&	\eor 	&	E)] 	&	\eand 	&	 \enot 	&	 C \\
%\cline{2-9}
%	&	T	&	T	&	T	&	T	&	T	&	F		&	F		&	T	\\
%	&	T	&	T	&	T	&	T	&	F	&	F		&	F		&	T	\\
%	&	T	&	T	&	F	&	T	&	T	&	F		&	F		&	T	\\
%	&	T	&	F	&	F	&	F	&	F	&	F		&	F		&	T	\\
%	&	F	&	F	&	T	&	T	&	T	&	F		&	T		&	F	\\
%	&	F	&	F	&	T	&	T	&	F	&	F		&	T		&	F	\\
%	&	F	&	F	&	F	&	T	&	T	&	F		&	T		&	F	\\
%	&	F	&	T	&	F	&	F	&	F	&	T		&	T		&	F	\\
%\cline{7-7}
%\end{tabular}

\item $\enot(G \eand (B \eand H)) \eiff (G \eor (B \eor H))$
%
%\begin{tabular}{ccccccc|c|ccccc}
%\cline{8-8}
%5.	&\enot&	(G 	&\eand &	(B 	&	 \eand 	&	 H))	&	\eiff 	&	(G 	& \eor 	& (B 	& \eor	& H))	\\
%\cline{2-13}
%	&F	   &	T	&	  T &	T	&	T		&	T	&	F	&	T	&	T	&	T	&	T	&	T	\\
%	&T	   &	T	&	  F &	T	&	F		&	F	&	T	&	T	&	T	&	T	&	T	&	F	\\	
%	&T	   &	T	&	 F  &	F	&	F		&	T	&	T	&	T	&	T	&	F	&	T	&	T	\\
%	&T	   &	T	&	 F  &	F	&	F		&	F	&	T	&	T	&	T	&	F	&	F	&	F	\\
%	&T	   &	F	&	F   &	T	&	T		&	T	&	T	&	F	&	T	&	T	&	T	&	T	\\
%	&T	   &	F	&	F   &	T	&	F		&	F	&	T	&	F	&	T	&	T	&	T	&	F	\\
%	&T	   &	F	&	F   &	F	&	F		&	T	&	T	&	F	&	T	&	F	&	T	&	T	\\
%	&T	   &	F	&	F   &	F	&	F		&	F	&	F	&	F	&	F	&	F	&	F	&	F	\\
%\cline{8-8}
%\end{tabular}

%\vspace{1em}

\end{compactlist}

\problempart
Write complete truth tables for the following sentences and mark the column that represents the possible truth values for the whole sentence.

\begin{compactlist}

\item	$(D \eand \enot D) \eif G $

%\vspace{1em}

%\begin{tabular}{ccccc|c|c}
%\cline{6-6}
%1.	&	(D 	&	 \eand 	& 	 \enot	&	 D) 	&	 \eif 	&	 G \\
%	&	T	&	F		&	F		&	T	&	T	&	T	\\
%	&	T	&	F		&	F		&	T	&	T	&	F	\\
%	&	F	&	F		&	T		&	F	&	T	&	T	\\
%	&	F	&	F		&	T		&	F	&	T	&	F	\\
%\cline{6-6}
%\end{tabular}
%\vspace{1em}


\item	$(\enot P \eor \enot M) \eiff M $

%\begin{tabular}{cccccc|c|c}
%\cline{7-7}
%2.	&	(\enot 	&	P 	&	\eor 	&	\enot 	& 	 M) 	& 	\eiff 	&	 M \\
%	&	F		&	T	&	F	&	F		&	T	&	T	&	T	\\
%	&	F		&	T	&	T	&	T		&	F	&	F	&	F	\\
%	&	T		&	F	&	T	&	F		&	T	&	T	&	T	\\
%	&	T		&	F	&	T	&	T		&	F	&	T	&	F	\\
%\cline{7-7}
%\end{tabular}
%\vspace{1em}



\item	$\enot \enot (\enot A \eand \enot B)  $

%\begin{tabular}{c|c|cccccc}
%\cline{2-2}
%3.	&	\enot		&	 \enot 	&	(\enot 	& 	 A 	& \eand 	& 	\enot 	&	 B)  \\
%	&	F		&	T		&	F		&	T	&	F	&	F		&	T	\\
%	&	F		&	T		&	F		&	T	&	F	&	T		&	F	\\
%	&	F		&	T		&	T		&	F	&	F	&	F		&	T	\\
%	&	T		&	F		&	T		&	F	&	T	&	T		&	F	\\
%\cline{2-2}
%\end{tabular}
%\vspace{1em}



\item 	$[(D \eand R) \eif I] \eif \enot(D \eor R) $

%\begin{tabular}{cccccc|c|cccc}
%\cline{7-7}
%4.	&	[(D 	& 	 \eand 	& 	 R)	& 	\eif 	&	I] 	&	\eif 	&	 \enot 	&	(D 	&	 \eor 	& R) \\
%	&	T	&	T		&	T	&	T	&	T	&	F	&	F		&	T	&	T		&T	\\
%	&	T	&	T		&	T	&	F	&	F	&	T	&	F		&	T	&	T		&T	\\
%	&	T	&	F		&	F	&	T	&	T	&	F	&	F		&	T	&	T		&F	\\
%	&	T	&	F		&	F	&	T	&	F	&	F	&	F		&	T	&	T		&F	\\
%	&	F	&	F		&	T	&	T	&	T	&	F	&	F		&	F	&	T		&T	\\
%	&	F	&	F		&	T	&	T	&	F	&	F	&	F		&	F	&	T		&T	\\
%	&	F	&	F		&	F	&	T	&	T	&	T	&	T		&	F	&	F		&F	\\
%	&	F	&	F		&	F	&	T	&	F	&	T	&	T		&	F	&	F		&F	\\
%\cline{7-7}
%\end{tabular}
%	
%\vspace{1em}


\item	$\enot [(D \eiff O) \eiff A] \eif (\enot D \eand O) $

%\begin{tabular}{ccccccc|c|cccc}
%\cline{8-8}
%5.	&	\enot 	&	[(D 	&	\eiff 	&	O) 	&	\eiff 	&	 A]	& 	\eif 	 &	(\enot 	& 	D 	 & 	 \eand &O) \\ 
%	&	F		&	T	&	T	&	T	&	T	&	T	&	T	&	F		&	T	&	F	&T	\\
%	&	T		&	T	&	T	&	T	&	F	&	F	&	F	&	F		&	T	&	F	&T	\\
%	&	T		&	T	&	F	&	F	&	F	&	T	&	F	&	F		&	T	&	F	&F	\\
%	&	F		&	T	&	F	&	F	&	T	&	F	&	T	&	F		&	T	&	F	&F	\\
%	&	T		&	F	&	F	&	T	&	F	&	T	&	T	&	T		&	F	&	T	&T	\\
%	&	F		&	F	&	F	&	T	&	T	&	F	&	T	&	T		&	F	&	T	&T	\\
%	&	F		&	F	&	T	&	F	&	T	&	T	&	T	&	T		&	F	&	F	&F	\\
%	&	T		&	F	&	T	&	F	&	F	&	F	&	T	&	T		&	F	&	F	&F	\\
%\cline{8-8}
%\end{tabular}
%\vspace{1em}
\end{compactlist}


If you want additional practice, you can construct truth tables for any of the sentences and arguments in the exercises for the previous chapter.


\chapter{Semantic concepts}
\label{s:SemanticConcepts}

In the previous chapter, we introduced the idea of a valuation and showed how to determine the truth value of any TFL sentence, on any valuation, using a truth table. In this chapter, we will introduce some related ideas, and show how to use truth tables to test whether or not they apply.


\section{Tautologies and contradictions}
In \cref{s:BasicNotions}, we explained \emph{necessary truth} and \emph{necessary falsity}. Both notions have surrogates in TFL. We will start with a surrogate for necessary truth.
	\factoidbox{
		$\metav{A}$ is a \define{tautology} \ifeff{} it is true on every valuation.
	}

\newglossaryentry{tautology}
{
name=tautology,
description={A sentence that has only Ts in the column under the main logical operator of its \gls{complete truth table}; a sentence that is true on every \gls{valuation}}
}

We can use truth tables to decide whether a sentence is a tautology. If the sentence is true on every line of its complete truth table, then it is true on every valuation, so it is a tautology. In the example of \cref{s:CompleteTruthTables}, `$(H \eand I) \eif H$' is a tautology.

This is only, though, a \emph{surrogate} for necessary truth. There are some necessary truths that we cannot adequately symbolize in TFL. One example is `$2 + 2 = 4$'. This \emph{must} be true, but if we try to symbolize it in TFL, the best we can offer is a sentence letter, and no sentence letter is a tautology. Still, if we can adequately symbolize some English sentence using a TFL sentence which is a tautology, then that English sentence expresses a necessary truth.

We have a similar surrogate for necessary falsity:
	\factoidbox{
		$\metav{A}$ is a \define{contradiction} (in TFL) \ifeff{} it is false on every valuation.
	}
\newglossaryentry{contradiction of TFL}
{
  name=contradiction (of TFL),
  text = contradiction,
description={A sentence that has only Fs in the column under the main logical operator of its \gls{complete truth table}; a sentence that is false on every \gls{valuation}}
}

We can use truth tables to decide whether a sentence is a contradiction. If the sentence is false on every line of its complete truth table, then it is false on every valuation, so it is a contradiction. In the example of \cref{s:CompleteTruthTables}, `$[(C\eiff C) \eif C] \eand \enot(C \eif C)$' is a contradiction.

\section{Equivalence}\label{sec:equivalent}
Here is a similar useful notion:
	\factoidbox{
		$\metav{A}$ and $\metav{B}$ are \define{equivalent} (in TFL) iff, for every valuation, their truth values agree, i.e., if there is no valuation in which they have opposite truth values.
	}
\newglossaryentry{equivalent}
{
  name=equivalence (in TFL),
  text = equivalent,
description={A property held by pairs of sentences if and only if the \gls{complete truth table} for those sentences has identical columns under the two main logical operators, i.e., if the sentences have the same truth value on every valuation}
}

We have already made use of this notion, in effect, in \cref{s:MoreBracketingConventions}; the point was that `$(A \eand B) \eand C$' and  `$A \eand (B \eand C)$' are equivalent. Again, it is easy to test for equivalence using truth tables. Consider the sentences `$\enot(P \eor Q)$' and `$\enot P \eand \enot Q$'. Are they equivalent? To find out, we construct a truth table.
\begin{center}
\begin{tabular}{c c|d e e f |d e e e f}
$P$&$Q$&\enot&$(P$&\eor&$Q)$&\enot&$P$&\eand&\enot&$Q$\\
\hline
 T & T & \TTbf{F} & T & T & T & F & T & \TTbf{F} & F & T\\
 T & F & \TTbf{F} & T & T & F & F & T & \TTbf{F} & T & F\\
 F & T & \TTbf{F} & F & T & T & T & F & \TTbf{F} & F & T\\
 F & F & \TTbf{T} & F & F & F & T & F & \TTbf{T} & T & F
\end{tabular}
\end{center}
Look at the columns for the main logical operators; negation for the first sentence, conjunction for the second. Go through the rows in the table one by one and compare the truth values in the columns for the main logical operators. On the first three rows, both are false. On the final row, both are true. Since they match on every row, the two sentences are equivalent.


\section{Satisfiability}
In \cref{s:BasicNotions}, we said that sentences are jointly possible \ifeff{} it is possible for all of them to be true at once. We can offer a surrogate for this notion too:
	\factoidbox{
		$\metav{A}_1, \metav{A}_2, \ldots, \metav{A}_n$ are \define{jointly satisfiable} (in TFL) \ifeff{} there is some valuation which makes them all true.
	}

        \newglossaryentry{satisfiability in TFL}
{
  name=satisfiability (in TFL),
  text=jointly satisfiable,
description={A property held by sentences if and only if the \gls{complete truth table} for those sentences contains one line on which all the sentences are true, i.e., if some \gls{valuation} makes all the sentences true}
}

Derivatively, sentences are \define{jointly unsatisfiable} \ifeff{} no valuation makes them all true. Again, it is easy to test for joint satisfiability using truth tables.

\section{Entailment and validity}
The following idea is closely related to that of joint satisfiability:
	\factoidbox{
		The sentences $\metav{A}_1, \metav{A}_2, \ldots, \metav{A}_n$ \define{entail} (in TFL) the sentence $\metav{C}$ \ifeff{} no valuation of the relevant sentence letters makes all of $\metav{A}_1, \metav{A}_2, \ldots, \metav{A}_n$ true and $\metav{C}$ false.
	}
       \newglossaryentry{valid in TFL}
{
  name= validity of arguments (in TFL),
  text = valid,
description={A property held by arguments if and only if the \gls{complete truth table} for the argument contains no rows where the \glspl{premise} are all true and the \gls{conclusion} false, i.e., if no \gls{valuation} makes all premises true and the conclusion false}
}
 
Again, it is easy to test this with a truth table. To check whether `$\enot L \eif (J \eor L)$' and `$\enot L$' entail `$J$', we simply need to check whether there is any valuation which makes both `$\enot L \eif (J \eor L)$' and `$\enot L$' true whilst making `$J$' false. So we use a truth table: 
\begin{center}
\begin{tabular}{c c|d e e e e f|d f| c}
$J$&$L$&\enot&$L$&\eif&$(J$&\eor&$L)$&\enot&$L$&$J$\\
\hline
%J   L   -   L      ->     (J   v   L)
 T & T & F & T & \TTbf{T} & T & T & T & \TTbf{F} & T & \TTbf{T}\\
 T & F & T & F & \TTbf{T} & T & T & F & \TTbf{T} & F & \TTbf{T}\\
 F & T & F & T & \TTbf{T} & F & T & T & \TTbf{F} & T & \TTbf{F}\\
 F & F & T & F & \TTbf{F} & F & F & F & \TTbf{T} & F & \TTbf{F}
\end{tabular}
\end{center}
The only row on which both`$\enot L \eif (J \eor L)$' and `$\enot L$' are true is the second row, and that is a row on which `$J$' is also true. So `$\enot L \eif (J \eor L)$' and `$\enot L$' entail `$J$'.

We now make an important observation:
	\factoidbox{
	If $\metav{A}_1, \metav{A}_2, \ldots, \metav{A}_n$ entail
	$\metav{C}$ in TFL, then $\metav{A}_1, \metav{A}_2, \ldots,
	\metav{A}_n \therefore \metav{C}$ is valid.
	}
Here's why. Suppose $\metav{A}_1, \metav{A}_2, \ldots, \metav{A}_n$
entail $\metav{C}$ in TFL. Can it happen that the argument
$\metav{A}_1, \metav{A}_2, \ldots, \metav{A}_n \therefore \metav{C}$
is invalid? Then there would be a \emph{case} which makes all of
$\metav{A}_1, \metav{A}_2, \ldots, \metav{A}_n$ true and $\metav{C}$
false, relative to some symbolization key. This case would generate a
valuation of the sentence letters occurring in $\metav{A}_1,
\metav{A}_2, \ldots, \metav{A}_n$, and $\metav{C}$: take the truth
value of any sentence letter to be just the truth value of the
corresponding sentence in the case in question. This valuation would
also make $\metav{A}_1, \metav{A}_2, \ldots, \metav{A}_n$ true and
$\metav{C}$ false, since truth values are determined from the truth
values of sentence letters by the truth tables of the connectives, and
the truth values of the sentence letters are the same in the valuation
as they are in the (imagined) case. But this is impossible, since
we've assumed that $\metav{A}_1, \metav{A}_2, \ldots, \metav{A}_n$
entail $\metav{C}$ in TFL, and so there is no valuation which makes
all of $\metav{A}_1, \metav{A}_2, \ldots, \metav{A}_n$ true and also
makes $\metav{C}$ false. So it can't happen that the argument
$\metav{A}_1, \metav{A}_2, \ldots, \metav{A}_n \therefore \metav{C}$
is invalid; consequently, it must be valid.

In short, we have a way to test for the validity of English arguments. First, we symbolize them in TFL; then we test for entailment in TFL using truth tables.



\section{The double turnstile}
In what follow, we will use the notion of entailment rather a lot in this book. It will help us, then, to introduce a symbol that abbreviates it. Rather than saying that the TFL sentences $\metav{A}_1$, $\metav{A}_2$, \dots{} and $\metav{A}_n$ together entail $\metav{C}$, we will abbreviate this by:
	$$\metav{A}_1, \metav{A}_2, \dots, \metav{A}_n \entails \metav{C}$$
The symbol `$\entails$' is known as \emph{the double turnstile}, since it looks like a turnstile with two horizontal beams.

Let's be clear. `$\entails$' is not a symbol of TFL. Rather, it is a symbol of our metalanguage, augmented English (recall the difference between object language and metalanguage from \cref{s:UseMention}). So the metalanguage sentence:
	\[\metav{A}, \metav{A} \eif \metav{B} \entails \metav{B}\]
is \emph{just} an abbreviation for this metalanguage sentence: 
	\begin{center}
		The TFL sentences $\metav{A}$ and $\metav{A} \eif \metav{B}$ entail $\metav{B}$
	\end{center}
Note that there is no limit on the number of TFL sentences that can be mentioned before the symbol~`$\entails$'. Indeed, we can even consider the limiting case:
	$$\entails \metav{C}$$
This says that there is no valuation which makes all the sentences
mentioned on the left side of `$\entails$' true whilst making all
sentences on the right side (in this case, $\metav{C}$) false. Since \emph{no} sentences are mentioned on the left
side of `$\entails$' in this case, this just means that there is no
valuation which makes $\metav{C}$ false. Otherwise put, it says that
every valuation makes $\metav{C}$ true. Otherwise put, it says that
$\metav{C}$ is a tautology. Equally, to say that $\metav{A}$ is a
contradiction, we can write:
	$$\metav{A} \entails\phantom{\metav{C}}$$
For this says that no valuation makes $\metav{A}$ true.

Sometimes, we will want to deny that there is a tautological entailment, and say something of this shape: 
\begin{center}
	it is \emph{not} the case that $\metav{A}_1, \ldots, \metav{A}_n \entails \metav{C}$
\end{center}
In that case, we can just slash the turnstile through, and write: 
$$\metav{A}_1, \metav{A}_2, \ldots, \metav{A}_n \nentails\metav{C}$$
This means that \emph{some} valuation makes all of $\metav{A}_1,
\ldots, \metav{A}_n$ true whilst making $\metav{C}$ false. Note that
it does \emph{not} follow that $\metav{A}_1,\ldots, \metav{A}_n
\entails \enot \metav{C}$, for it is possible that some \emph{other}
valuation makes all of $\metav{A}_1, \ldots, \metav{A}_n$ true and
makes $\metav{C}$ \emph{true}. For instance, $P \nentails Q$ but also
$P \nentails \enot Q$.

\section{`$\entails$' versus `$\eif$'}\label{entails-v-iff}
We now want to compare and contrast `$\entails$' and `$\eif$'.

Observe: $\metav{A} \entails \metav{C}$ \ifeff{} no valuation of the sentence letters makes $\metav{A}$ true and $\metav{C}$ false.

Observe: $\metav{A} \eif \metav{C}$ is a tautology \ifeff{} no valuation of the sentence letters makes $\metav{A} \eif \metav{C}$ false. Since a conditional is true except when its antecedent is true and its consequent false, $\metav{A} \eif \metav{C}$ is a tautology \ifeff{} no valuation makes $\metav{A}$ true and $\metav{C}$ false.

Combining these two observations, we see that $\metav{A} \eif \metav{C}$  is a tautology \ifeff{}  $\metav{A} \entails \metav{C}$. But there is a really, really important difference between `$\entails$' and `$\eif$':
	\factoidbox{
		\begin{itemize}
			\item `$\eif$' is a sentential connective of TFL.
			\item `$\entails$' is a symbol of augmented English.
		\end{itemize}
	}
Indeed, when `$\eif$' is flanked with two TFL sentences, the result is a longer TFL sentence. By contrast, when we use `$\entails$', we form a metalinguistic sentence that \emph{mentions} the surrounding TFL sentences.

The English word `implies' is often used as a synonym for `entails',
i.e.,~`$\entails$'. Some logicians \emph{also} use it for the
conditional, i.e.,~`$\eif$'. Since this is liable to commit the
confusion warned about above, we avoid the use of `implies' as much as
possible. (Whenever it is used, it means `entails'.)


\practiceproblems
\problempart
Revisit your answers to
\hyperref[pr.TT.TTorC]{exercise~\ref*{s:CompleteTruthTables}A}. Determine which
sentences were tautologies, which were contradictions, and which were
neither tautologies nor contradictions.
\solutions

\

\problempart
\label{pr.TT.satisfiable}
Use truth tables to determine whether these sentences are jointly satisfiable, or jointly unsatisfiable:
\begin{compactlist}
\item $A\eif A$, $\enot A \eif \enot A$, $A\eand A$, $A\eor A$ %satisfiable
\item $A\eor B$, $A\eif C$, $B\eif C$ %satisfiable
\item $B\eand(C\eor A)$, $A\eif B$, $\enot(B\eor C)$  %unsatisfiable
\item $A\eiff(B\eor C)$, $C\eif \enot A$, $A\eif \enot B$ %satisfiable
\end{compactlist}


\solutions
\problempart
\label{pr.TT.valid}
Use truth tables to determine whether each argument is valid or invalid.
\begin{compactlist}
\item $A\eif A \therefore A$ %invalid
\item $A\eif(A\eand\enot A) \therefore \enot A$ %valid
\item $A\eor(B\eif A) \therefore \enot A \eif \enot B$ %valid
\item $A\eor B, B\eor C, \enot A \therefore B \eand C$ %invalid
\item $(B\eand A)\eif C, (C\eand A)\eif B \therefore (C\eand B)\eif A$ %invalid
\end{compactlist}

\problempart Determine whether each sentence is a tautology, a contradiction, or a contingent sentence, using a complete truth table.
\begin{compactlist}
\item $\enot B \eand B$ \vspace{.5ex}%contra


\item $\enot D \eor D$ \vspace{.5ex}%taut


\item $(A\eand B) \eor (B\eand A)$\vspace{.5ex} %contingent


\item $\enot[A \eif (B \eif A)]$\vspace{.5ex} %contra


\item $A \eiff [A \eif (B \eand \enot B)]$ \vspace{.5ex}%contra


\item $[(A \eand B) \eiff B] \eif (A \eif B)$ \vspace{.5ex}% contingent.

\end{compactlist}



\noindent\problempart
\label{pr.TT.equiv}
Determine whether each the following sentences are logically equivalent using complete truth tables. If the two sentences really are logically equivalent, write ``equivalent.'' Otherwise write, ``Not equivalent.'' 
\begin{compactlist}
\item $A$ and $\enot A$
\item $A \eand \enot A$ and $\enot B \eiff B$
\item $[(A \eor B) \eor C]$ and $[A \eor (B \eor C)]$
\item $A \eor (B \eand C)$ and $(A \eor B) \eand (A \eor C)$
\item $[A \eand (A \eor B)] \eif B$ and $A \eif B$\end{compactlist}


\problempart
\label{pr.TT.equiv2}
Determine whether each the following sentences are logically equivalent using complete truth tables. If the two sentences really are equivalent, write ``equivalent.'' Otherwise write, ``not equivalent.''
\begin{compactlist}
\item $A\eif A$ and $A \eiff A$
\item $\enot(A \eif B)$ and $\enot A \eif \enot B$
\item $A \eor B$ and $\enot A \eif B$
\item$(A \eif B) \eif C$ and $A \eif (B \eif C)$
\item $A \eiff (B \eiff C)$ and $A \eand (B \eand C)$
\end{compactlist}


\problempart
\label{pr.TT.satisfiable2}
Determine whether each collection of sentences is jointly satisfiable or jointly unsatisfiable using a complete truth table.
\begin{compactlist}
\item $A \eand \enot B$, $\enot(A \eif B)$, $B \eif A$\vspace{.5ex} %Consistent

%\begin{tabular}{ccccccccccccccc} 
%1. 	&	A 					 & \eand 		&  \enot & B & & \enot  		& 	 (A	  & 	 \eif	 	 & 	 B)		 & 	 & 	 B	 	 & 	\eif 	 	 & 	A 	 	 & 	 Consistent \\ 
%\cline{2-5} \cline{7-10}\cline{12-14} 
%	& 	T 					 & 	 F	 		&  F	 & T & & F	 		& 	 T	  & 	 T	 	 & 	T 	 	 & 	 & 	 T	 	 & 	 T	 	 & T	 	 	&	  \\ 
%\cline{2-14}
%	& \multicolumn{1}{|r}{T}& 	\textbf{T}	 & T	 & F & & \textbf{T}	 & 	 T	 & 	 F	 	 & 	 F	 	 & 	 & 	 F	 	 & 	 \textbf{T}	 	 & 	 \multicolumn{1}{r|}{T}	 	 & 	  \\ 
%\cline{2-14}
%	& 	 F	 				 & 	 F	 & 	 F	 & T & 	& 	 F	 & 	 F	 & 	 T	 	 & 	 T	 	 & 	  & 	 T	 	 & 	 F	 	 & 	 F	 	 & 	  \\ 
%	& 	 F	  				& 	 F	 & 	 T	 & 	F&  & 	 F	 & 	 F	 & 	 T	 	 & 	 F	 	 & 	  & 	 F	 	 & 	 T	 	 & 	 F	 	 & 	  \\ 
%\end{tabular}

\item $A \eor B$, $A \eif \enot A$, $B \eif \enot B$ \vspace{.5ex}%unsatisfiable.

%\begin{tabular}{ccccccccccccccc} 
%2. &A	 & \eor 	 & B 	 & 	 	 & A 	 & \eif 	 & 	\enot & A 	 & 	 	 & B 	 & \eif 	 & \enot	 & 	B 	 & 	Insatisfiable \\ 
%\cline{2-4}\cline{6- 9} \cline{11-14}
%   &	T	 & 	 T	 &T  	 & 	 	 & T	 & 	 F	 & 	F 	 & T 	 & 	 	 & 	T 	 & 	F 	 & 	 F	 & 	T 	 & 	 \\ 
%   &	 T	& 	 T	 & F 	 & 	 	 & 	T 	 & 	 F	 & 	 F	 & 	 T	 & 	 	 & 	F 	 & 	 T	 & 	 T	 & 	 F	 & 	 \\ 
%   &	 F	& 	 T	 & 	 T	 & 	 	 & 	F 	 & 	 T	 & 	 T	 & 	F 	 & 	 	 & 	 T	 & 	 F	 & 	 F	 & 	 T	 & 	 \\ 
%   &	 F	& 	 F	 & 	 F	 & 	 	 & 	 F	 & 	 T	 & 	 T	 & 	 F	 & 	 	 & 	 F	 & 	 T	 & 	 T	 & 	 F	 & 	 \\ 
%\end{tabular}

\item $\enot(\enot A \eor B) $, $A \eif \enot C$, $A \eif (B \eif C)$\vspace{.5ex} %Insatisfiable

%3. &\enot & (\enot & A & \eor &B) &  &A  & \eif 	 &\enot 	 &C & 	 & A &\eif 	& (B 	 &\eif 	& C)	 &Consistent \\ 
%\cline{2-6}\cline{8-11} \cline{13-17} 
%   &	F 	& 	F	 & 	T & T	 & T & 	  & T & F	 & 	 F&T 	 & 	 &T & T	 & T	 &T 	 &T 	 & \\ 
%   &	 F	& 	F	 & 	T & T	 & T & 	  & T & T	 & 	 T& F	 & 	 &T & F	 & T	 & F	 &F 	 & \\ 
% 
%  &	 T & 	F 	& 	T & F	 & F & 	  & T & F	 & 	 F& T	 & 	 &T & T	 & F	 & T	 &T 	 & \\ 
%\cline{2-17}
%   &	 \multicolumn{1}{|r}{{\color{red}T}}		&  F	 & 	T & F	 & 	F &  & 	T & {\color{red}T}	 & 	 T&F 	& 	 &T & {\color{red}T}	 & F	 & T	 &\multicolumn{1}{r|}{F} 	 & \\ 
%\cline{2-17}
%   &	 F	& 	T	 & 	F & T	 & 	T &  & 	F & T	 & 	 F& T	 & 	 &F	 & F	 & T	 & T	 &T 	 & \\ 
%   &	 F	& 	 T	& 	F & T	 & 	T &  & 	F & T	 & 	T & F 	& 	 &F	 & T	 & T	 &F 	 &F 	 & \\ 
%   &	 F	& 	 T	& 	F & T	 & 	F &  & 	F & T	 & 	F & T	 & 	 &F	 & T	 & F	 & T	 &T 	 & \\ 
%   &	 F	& 	 T	& 	F & T	 & 	F &  & 	F & T	 & 	T & F	 & 	 &F	 & T	 & F	 & T	 &F 	 & \\ 
%\end{tabular}
%


\item $A \eif B$, $A \eand \enot B$\vspace{.5ex} %Insatisfiable

\item $A \eif (B \eif C)$, $(A \eif B) \eif C$, $A \eif C$\vspace{.5ex} % satisfiable.

\end{compactlist}

\noindent\problempart
\label{pr.TT.satisfiable3}
Determine whether each collection of sentences is jointly satisfiable or jointly unsatisfiable, using a complete truth table.
\begin{compactlist}
\item $\enot B$, $A \eif B$, $A$ \vspace{.5ex}%unsatisfiable.
\item $\enot(A \eor B)$, $A \eiff B$, $B \eif A$\vspace{.5ex} %Consistent
\item $A \eor B$, $\enot B$, $\enot B \eif \enot A$\vspace{.5ex} %Insatisfiable
\item $A \eiff B$, $\enot B \eor \enot A$, $A \eif B$\vspace{.5ex} %satisfiable.
\item $(A \eor B) \eor C$, $\enot A \eor \enot B$, $\enot C \eor \enot B$\vspace{.5ex} %satisfiable
\end{compactlist}




\noindent\problempart
\label{pr.TT.valid2}
Determine whether each argument is valid or invalid, using a complete truth table.
\begin{compactlist}
\item $A\eif B$, $B \therefore  A$ %invalid

\item $A\eiff B$, $B\eiff C \therefore A\eiff C$ %valid

\item $A \eif B$, $A \eif C\therefore B \eif C$ %invalid.

\item $A \eif B$, $B \eif A\therefore A \eiff B$ %valid.

\end{compactlist}

\noindent\problempart
\label{pr.TT.valid3}
Determine whether each argument is valid or invalid, using a complete truth table.
\begin{compactlist}
\item $A\eor\bigl[A\eif(A\eiff A)\bigr] \therefore  A $\vspace{.5ex}%invalid
\item $A\eor B$, $B\eor C$, $\enot B \therefore A \eand C$\vspace{.5ex} %valid
\item $A \eif B$, $\enot A\therefore \enot B$ \vspace{.5ex}%invalid
\item $A$, $B\therefore \enot(A\eif \enot B)$ \vspace{.5ex}%valid
\item $\enot(A \eand B)$, $A \eor B$, $A \eiff B\therefore C$ \vspace{.5ex}%valid 
\end{compactlist}

\solutions
\problempart
\label{pr.TT.concepts}
Answer each of the questions below and justify your answer.
\begin{compactlist}
\item Suppose that \metav{A} and \metav{B} are logically equivalent. What can you say about $\metav{A}\eiff\metav{B}$?
%\metav{A} and \metav{B} have the same truth value on every line of a complete truth table, so $\metav{A}\eiff\metav{B}$ is true on every line. It is a tautology.
\item Suppose that $(\metav{A}\eand\metav{B})\eif\metav{C}$ is neither a tautology nor a contradiction. What can you say about whether $\metav{A}, \metav{B} \therefore\metav{C}$ is valid?
%The sentence is false on some line of a complete truth table. On that line, \metav{A} and \metav{B} are true and \metav{C} is false. So the argument is invalid.
\item Suppose that $\metav{A}$, $\metav{B}$ and $\metav{C}$  are jointly unsatisfiable. What can you say about $(\metav{A}\eand\metav{B}\eand\metav{C})$?
\item Suppose that \metav{A} is a contradiction. What can you say about whether $\metav{A}, \metav{B} \entails \metav{C}$?
%Since \metav{A} is false on every line of a complete truth table, there is no line on which \metav{A} and \metav{B} are true and \metav{C} is false. So the argument is valid.
\item Suppose that \metav{C} is a tautology. What can you say about whether $\metav{A}, \metav{B}\entails \metav{C}$?
%Since \metav{C} is true on every line of a complete truth table, there is no line on which \metav{A} and \metav{B} are true and \metav{C} is false. So the argument is valid.
\item Suppose that \metav{A} and \metav{B} are logically equivalent. What can you say about $(\metav{A}\eor\metav{B})$?
%Not much. $(\metav{A}\eor\metav{B})$ is a tautology if \metav{A} and \metav{B} are tautologies; it is a contradiction if they are contradictions; it is contingent if they are contingent.
\item Suppose that \metav{A} and \metav{B} are \emph{not} logically equivalent. What can you say about $(\metav{A}\eor\metav{B})$?
%\metav{A} and \metav{B} have different truth values on at least one line of a complete truth table, and $(\metav{A}\eor\metav{B})$ will be true on that line. On other lines, it might be true or false. So $(\metav{A}\eor\metav{B})$ is either a tautology or it is contingent; it is \emph{not} a contradiction.
\end{compactlist}
\problempart 
Consider the following principle:
	\begin{quote}
		Suppose $\metav{A}$ and $\metav{B}$ are logically equivalent. Suppose an argument contains $\metav{A}$ (either as a premise, or as the conclusion). The validity of the argument would be unaffected, if we replaced $\metav{A}$ with $\metav{B}$.
	\end{quote}
Is this principle correct? Explain your answer.

\chapter{Limitations of TFL}\label{s:ParadoxesOfMaterialConditional}

We have reached an important milestone: a test for the validity of
arguments! However, we should not get carried away just yet. It is
important to understand the \emph{limits} of our achievement. We will
illustrate these limits with four examples.

First, consider the argument: 
	\begin{earg}
		\item Daisy has four legs.
		\item[\texttherefore] Daisy has more than two legs.
	\end{earg}
To symbolize this argument in TFL, we would have to use two different sentence letters---perhaps `$F$'  and `$T$'---for the premise and the conclusion respectively. Now, it is obvious that `$F$' does not entail~`$T$'. But the English argument is surely valid!

Second, consider the sentence:
	\begin{enumerate}
		\item\label{n:JanBald} Jan is neither bald nor not bald.
	\end{enumerate}
To symbolize this sentence in TFL, we would offer something like
`$\enot J \eand \enot \enot J$'. This a contradiction (check this with
a truth-table), but \cref*{n:JanBald} does not itself seem like a
contradiction: we might have happily added `Jan is on the borderline
of baldness'!

Third, consider the following sentence:
	\begin{enumerate}
	\item\label{n:GodParadox}	It's not the case that, if God exists, She answers malevolent prayers.
	\end{enumerate}
Symbolizing this in TFL, we would offer something like `$\enot (G \eif M)$'. Now, `$\enot (G \eif M)$' entails `$G$' (again, check this with a truth table). So if we symbolize \cref*{n:GodParadox} in TFL, it seems to entail that God exists. But that's strange: surely even an atheist can accept \cref*{n:GodParadox}, without contradicting herself!

One lesson of this is that the symbolization of \cref*{n:GodParadox} as `$\enot(G \eif M)$' shows that \cref*{n:GodParadox} does not express what we intend. Perhaps we should rephrase it as
	\begin{enumerate}
	\item\label{n:GodParadox2} If God exists, She does not answer malevolent prayers.
	\end{enumerate}
and symbolize \cref*{n:GodParadox2} as `$G \eif \enot M$'.  Now, if atheists are right, and there is no God, then `$G$' is false and so `$G \eif \enot M$' is true, and the puzzle disappears. However, if `$G$' is false, `$G \eif M$', i.e., `If God exists, She answers malevolent prayers', is \emph{also} true!

In different ways, these four examples highlight some of the limits of
TFL, symbolization of English in TFL, and the tests based on truth
tables we've devised. Our first example showed that TFL is not
expressive enough to symbolize everything we might want to in a way
that allows us to apply our logical toolkit to even the best possible
symbolization. We will see later (\cref{sec.identity}) that we
\emph{can} properly symbolize `Daisy has four legs' in the more
expressive language FOL, and then the test we will devise for FOL will
apply (and give the correct answer).

In \cref{n:JanBald} we similarly had an example where the
straightforward symbolization in TFL does not quite work. The best we
can do in TFL is to symbolize `Jan is not bald' using its own sentence
letter, say,~`$N$' (contrary to our standing practice). But this also
does not work. For instance, we would then symbolize `Jan is both bald
and isn't' as `$J \eand N$'. But `Jan is both bald and isn't' arguably
is a self-contradiction, yet the alternate symbolization `$J \eand N$'
is not. Some logicians have proposed that in cases of sentences like
`Jan is bald' which allow borderline cases, we have to adjust our
semantics and allow that the sentence letter~`$J$' that symbolizes it
should be allowed to take truth values other than T and~F. But this is
by no means universally accepted.

\Cref{n:GodParadox} also showed that symbolization is often tricky,
and that the straightforward symbolization of and English sentence
sometimes does not capture its intended truth conditions. But even our
improvement, \cref{n:GodParadox2}, and its symbolization `$G \eif
\enot M$' turned out to not be good enough. The phenomenon we
encountered here is one of the so-called paradoxes of the material
conditional. The paradox is that material conditionals like `$G \eif
\enot M$' and `$G \eif \enot M$' are true whenever their
antecedent~`$G$' is false, even though intuitively the two sentences
`If God exists, She answers malevolent prayers' and `If God exists,
She does \emph{not} answer malevolent prayers' should not both be true. This
indicates that the `if \dots then' in these two examples is not
properly captured by the TFL connective~`\eif'. The
truth-functionality of TFL here is a real limitation. The solution
would be to treat `if \dots then' in this case as a subjunctive
conditional (see \cref{s:IndicativeSubjunctive}), but TFL cannot
capture those.\footnote{Logicians have devised logics that deal with
subjunctive conditionals better. They are called `conditional logics'
or `logics of counterfactuals'.}

The case of Jan's baldness (or otherwise) raises the general question
of what logic we should use when dealing with \emph{vague} discourse.
The case of the atheist raises the question of how to deal with the
(so-called) \emph{paradoxes of the material conditional}. Part of the
purpose of this book is to equip you with the tools to explore these
questions of \emph{philosophical logic}. But we have to walk before we
can run; we have to become proficient in using TFL, before we can
adequately discuss its limits, and consider alternatives.

\chapter{Truth table shortcuts}
With practice, you will quickly become adept at filling out truth tables. In this chapter, we consider (and justify) some shortcuts which will help you along the way.

\section{Working through truth tables}
You will quickly find that you do not need to copy the truth value of each sentence letter, but can simply refer back to them. So you can speed things up by writing:
\begin{center}
\begin{tabular}{c c|d e e e e f}
$P$&$Q$&$(P$&\eor&$Q)$&\eiff&\enot&$P$\\
\hline
 T & T &  & T &  & \TTbf{F} & F\\
 T & F &  & T &  & \TTbf{F} & F\\
 F & T &  & T & & \TTbf{T} & T\\
 F & F &  & F &  & \TTbf{F} & T
\end{tabular}
\end{center}
You also know for sure that a disjunction is true whenever one of the disjuncts is true. So if you find a true disjunct, there is no need to work out the truth value of the other disjunct. Thus you might offer:
\begin{center}
\begin{tabular}{c c|d e e e e e e f}
$P$&$Q$& $(\enot$ & $P$&\eor&\enot&$Q)$&\eor&\enot&$P$\\
\hline
 T & T & F & & F & F& & \TTbf{F} & F\\
 T & F &  F & & T& T& &  \TTbf{T} & F\\
 F & T & & &  & & & \TTbf{T} & T\\
 F & F & & & & & &\TTbf{T} & T
\end{tabular}
\end{center}
Equally, you know for sure that a conjunction is false whenever one of the conjuncts is false. So if you find a false conjunct, there is no need to work out the truth value of the other conjunct. Thus you might offer:
\begin{center}
\begin{tabular}{c c|d e e e e e e f}
$P$&$Q$&\enot &$(P$&\eand&\enot&$Q)$&\eand&\enot&$P$\\
\hline
 T & T &  &  & &  & & \TTbf{F} & F\\
 T & F &   &  &&  & & \TTbf{F} & F\\
 F & T & T &  & F &  & & \TTbf{T} & T\\
 F & F & T &  & F & & & \TTbf{T} & T
\end{tabular}
\end{center}
A similar short cut is available for conditionals. You immediately know that a conditional is true if either its consequent is true, or its antecedent is false. Thus you might present:
\begin{center}
\begin{tabular}{c c|d e e e e e f}
$P$&$Q$& $((P$&\eif&$Q$)&\eif&$P)$&\eif&$P$\\
\hline
 T & T & &  & & & & \TTbf{T} & \\
 T & F &  &  & && & \TTbf{T} & \\
 F & T & & T & & F & & \TTbf{T} & \\
 F & F & & T & & F & &\TTbf{T} & 
\end{tabular}
\end{center}
So `$((P \eif Q) \eif P) \eif P$' is a tautology. In fact, it is an instance of \emph{Peirce's Law}, named after Charles Sanders Peirce.

\section{Testing for validity and entailment}
In \cref{s:SemanticConcepts}, we saw how to use truth tables to test for validity. In that test, we look for \emph{bad} lines: lines where the premises are all true and the conclusion is false. Now:
\begin{compactlist}
	\item[\textbullet] If the conclusion is true on a line, then that line is not bad. (And we don't need to evaluate anything \emph{else} on that line to confirm this.)
	\item[\textbullet] If any premise is false on a line, then that line is not bad. (And we don't need to evaluate anything \emph{else} on that line to confirm this.)
\end{compactlist}
With this in mind, we can speed up our tests for validity quite considerably.

Let's consider how we might test the following:
$$\enot L \eif (J \eor L), \enot L \therefore J$$
The \emph{first} thing we should do is evaluate the conclusion. If we find that the conclusion is \emph{true} on some line, then that is not a bad line. So we can simply ignore the rest of the line. So, after our first stage, we are left with something like this:
\begin{center}
	\begin{tabular}{c c|d e e e e f |d f|c}
		$J$&$L$&\enot&$L$&\eif&$(J$&\eor&$L)$&\enot&$L$&$J$\\
		\hline
		%J   L   -   L      ->     (J   v   L)
		T & T & &&&&&&&& {T}\\
		T & F & &&&&&&&& {T}\\
		F & T & &&?&&&&?&& {F}\\
		F & F & &&?&&&&?&& {F}
	\end{tabular}
\end{center}
where the blanks indicate that we won't bother with any more investigation (since the line is not bad), and the question marks indicate that we need to keep digging.

The easiest premise to evaluate is the second, so we do that next, and get:
\begin{center}
	\begin{tabular}{c c|d e e e e f |d f|c}
		$J$&$L$&\enot&$L$&\eif&$(J$&\eor&$L)$&\enot&$L$&$J$\\
		\hline
		%J   L   -   L      ->     (J   v   L)
		T & T & &&&&&&&& {T}\\
		T & F & &&&&&&&& {T}\\
		F & T & &&&&&&{F}&& {F}\\
		F & F & &&?&&&&{T}&& {F}
	\end{tabular}
\end{center}
Note that we no longer need to consider the third line on the table: it is certainly not bad, because some premise is false on that line. And finally, we complete the truth table:
\begin{center}
	\begin{tabular}{c c|d e e e e f |d f|c}
		$J$&$L$&\enot&$L$&\eif&$(J$&\eor&$L)$&\enot&$L$&$J$\\
		\hline
		%J   L   -   L      ->     (J   v   L)
		T & T & &&&&&&&& {T}\\
		T & F & &&&&&&&& {T}\\
		F & T & &&&&&&{F}& & {F}\\
		F & F & T &  & \TTbf{F} &  & F & & {T} & & {F}
	\end{tabular}
\end{center}
The truth table has no bad lines, so the argument is valid. Any valuation which makes every premise true makes the conclusion true.

It's probably worth illustrating the tactic again. Consider this argument:
$$A\eor B, \enot (B\eand C) \therefore (A \eor \enot C)$$
Again, we start by evaluating the conclusion. Since this is a disjunction, it is true whenever either disjunct is true, so we can speed things along a bit.
\begin{center}
\begin{tabular}[t]{c c c| c|c|d e e f }
$A$ & $B$ & $C$ & $A\eor B$ & $\enot (B \eand C)$ & $(A$ &$\eor $& $\enot $ & $C)$\\
\hline
T & T & T &  &  & & \TTbf{T} & & \\
T & T & F &  &  & & \TTbf{T} & & \\
T & F & T &  &  & & \TTbf{T} & & \\
T & F & F &  &  & & \TTbf{T} & & \\
F & T & T & ? & ? & & \TTbf{F} &F & \\
F & T & F &  &  && \TTbf{T} & T& \\
F & F & T & ? & ? && \TTbf{F} & F& \\
F & F & F &  &  & & \TTbf{T} & T& \\
\end{tabular}
\end{center}
We can now ignore all but the two lines where the sentence after the turnstile is false. Evaluating the two sentences on the left of the turnstile, we get:
\begin{center}
	\begin{tabular}[t]{c c c| c|d e e f |d e e f }
		$A$ & $B$ & $C$ & $A\eor B$ & $\enot ($&$B$&$ \eand$&$ C)$ & $(A$ &$\eor $& $\enot $ & $C)$\\
		\hline
		T & T & T &  & &&& & & \TTbf{T} & & \\
		T & T & F &  & &&& & & \TTbf{T} & & \\
		T & F & T &  & &&& & & \TTbf{T} & & \\
		T & F & F &  & &&& & & \TTbf{T} & & \\
		F & T & T & \textbf{T} & \textbf{F}&&T& & & \TTbf{F} &F & \\
		F & T & F & &&& & && \TTbf{T} & T& \\
		F & F & T & \textbf{F} & &&& & & \TTbf{F} & F& \\
		F & F & F & &&&& && \TTbf{T} & T& \\
	\end{tabular}
\end{center}
So the entailment holds! And our shortcuts saved us a \emph{lot} of work.
 
We have been discussing shortcuts in testing for validity. But exactly the same shortcuts can be used in testing for entailment. By employing a similar notion of bad lines, you can save yourself a huge amount of work.
 
\practiceproblems
\problempart
Using shortcuts, check whether each sentence is a tautology, a contradiction, or neither.
\begin{compactlist}
	\item $\enot B \eand B$ %contra
	\item $\enot D \eor D$ %taut
	\item $(A\eand B) \eor (B\eand A)$ %contingent
	\item $\enot[A \eif (B \eif A)]$ %contra
	\item $A \eiff [A \eif (B \eand \enot B)]$ %contra
	\item $\enot(A\eand B) \eiff A$ %contingent
	\item $A\eif(B\eor C)$ %contingent
	\item $(A \eand\enot A) \eif (B \eor C)$ %tautology
	\item $(B\eand D) \eiff [A \eiff(A \eor C)]$%contingent
\end{compactlist}

\chapter{Partial truth tables}\label{s:PartialTruthTable}

Sometimes, we do not need to know what happens on every line of a truth table. Sometimes, just a line or two will do.

\paragraph{Tautology} 
In order to show that a sentence is a tautology, we need to show that it is true on every valuation. That is to say, we need to know that it comes out true on every line of the truth table. So we need a complete truth table.

To show that a sentence is \emph{not} a tautology, however, we only need one line: a line on which the sentence is false. Therefore, in order to show that some sentence is not a tautology, it is enough to provide a single valuation---a single line of the truth table---which makes the sentence false.

Suppose that we want to show that the sentence `$(U \eand T) \eif (S \eand W)$' is \emph{not} a tautology. We set up a \define{partial truth table}:
\begin{center}
\begin{tabular}{c c c c |d e e e e e f}
$S$&$T$&$U$&$W$&$(U$&\eand&$T)$&\eif    &$(S$&\eand&$W)$\\
\hline
   &   &   &   &    &   &    &\TTbf{F}&    &   &   
\end{tabular}
\end{center}
We have only left space for one line, rather than 16, since we are only looking for one line, on which the sentence is false (hence, also, the `F').

The main logical operator of the sentence is a conditional. In order for the conditional to be false, the antecedent must be true and the consequent must be false. So we fill these in on the table:
\begin{center}
\begin{tabular}{c c c c |d e e e e e f}
$S$&$T$&$U$&$W$&$(U$&\eand&$T)$&\eif    &$(S$&\eand&$W)$\\
\hline
   &   &   &   &    &  T  &    &\TTbf{F}&    &   F &   
\end{tabular}
\end{center}
In order for the `$(U\eand T)$' to be true, both `$U$' and `$T$' must be true.
\begin{center}
\begin{tabular}{c c c c|d e e e e e f}
$S$&$T$&$U$&$W$&$(U$&\eand&$T)$&\eif    &$(S$&\eand&$W)$\\
\hline
   & T & T &   &  T &  T  & T  &\TTbf{F}&    &   F &   
\end{tabular}
\end{center}
Now we just need to make `$(S\eand W)$' false. To do this, we need to make at least one of `$S$' and `$W$' false. We can make both `$S$' and `$W$' false if we want. All that matters is that the whole sentence turns out false on this line. Making an arbitrary decision, we finish the table in this way:
\begin{center}
\begin{tabular}{c c c c|d e e e e e f}
$S$&$T$&$U$&$W$&$(U$&\eand&$T)$&\eif    &$(S$&\eand&$W)$\\
\hline
 F & T & T & F &  T &  T  & T  &\TTbf{F}&  F &   F & F  
\end{tabular}
\end{center}
We now have a partial truth table, which shows that `$(U \eand T) \eif (S \eand W)$' is not a tautology. Put otherwise, we have shown that there is a valuation which makes `$(U \eand T) \eif (S \eand W)$' false, namely, the valuation which makes `$S$' false, `$T$' true, `$U$' true and `$W$' false.

\paragraph{Contradictions}
Showing that something is a contradiction in TFL requires a complete truth table: we need to show that there is no valuation which makes the sentence true; that is, we need to show that the sentence is false on every line of the truth table.

However, to show that something is \emph{not} a contradiction, all we need to do is find a valuation which makes the sentence true, and a single line of a truth table will suffice. We can illustrate this with the same example.
\begin{center}
\begin{tabular}{c c c c|d e e e e e f}
$S$&$T$&$U$&$W$&$(U$&\eand&$T)$&\eif    &$(S$&\eand&$W)$\\
\hline
  &  &  &  &   &   &   &\TTbf{T}&  &  &
\end{tabular}
\end{center}
To make the sentence true, it will suffice to ensure that the antecedent is false. Since the antecedent is a conjunction, we can just make one of them false. Making an arbitrary choice, let's make `$U$' false; we can then assign any truth value we like to the other sentence letters.
\begin{center}
\begin{tabular}{c c c c|d e e e e e f}
$S$&$T$&$U$&$W$&$(U$&\eand&$T)$&\eif    &$(S$&\eand&$W)$\\
\hline
 F & T & F & F &  F &  F  & T  &\TTbf{T}&  F &   F & F
\end{tabular}
\end{center}

\paragraph{Equivalence}
To show that two sentences are equivalent, we must show that the sentences have the same truth value on every valuation. So this requires a  complete truth table.

To show that two sentences are \emph{not} equivalent, we only need to show that there is a valuation on which they have different truth values. So this requires only a one-line partial truth table: make the table so that one sentence is true and the other false.

\paragraph{Joint satisfiability}
To show that some sentences are jointly satisfiable, we must show that
there is a valuation which makes all of the sentences true, so this
requires only a partial truth table with a single line.

To show that some sentences are jointly unsatisfiable, we must show that there is no valuation which makes all of the sentence true. So this requires a complete truth table: You must show that on every row of the table at least one of the sentences is false.

\paragraph{Validity and entailment}
To show that an argument is valid, we must show that there is no valuation which makes all of the premises true and the conclusion false. So this  requires a complete truth table.  (Likewise for entailment.)

To show that argument is \emph{invalid}, we must show that there is a valuation which makes all of the premises true and the conclusion false. So this requires only a one-line partial truth table on which all of the premises are true and the conclusion is false. (Likewise for a failure of entailment.)

This table summarizes what is required:

\begin{center}
\begin{tabular}{l l l}
%\cline{2-3}
 & \textbf{Yes} & \textbf{No}\\
 \hline
%\cline{2-3}
tautology? & complete & one-line partial \\
contradiction? &  complete & one-line partial \\
%contingent? & two-line partial truth table & complete truth table\\
equivalent? & complete  & one-line partial \\
satisfiable? & one-line partial & complete \\
valid? & complete & one-line partial \\
entailment? & complete & one-line partial\\
\end{tabular}
\end{center}
\label{table.CompleteVsPartial}


\practiceproblems

\problempart
\label{pr.TT.equiv3}
Use complete or partial truth tables (as appropriate) to determine whether these pairs of sentences are logically equivalent:
\begin{compactlist}
\item $A$, $\enot A$ %No
\item $A$, $A \eor A$ %Yes
\item $A\eif A$, $A \eiff A$ %Yes
\item $A \eor \enot B$, $A\eif B$ %No
\item $A \eand \enot A$, $\enot B \eiff B$ %Yes
\item $\enot(A \eand B)$, $\enot A \eor \enot B$ %Yes
\item $\enot(A \eif B)$, $\enot A \eif \enot B$ %No
\item $(A \eif B)$, $(\enot B \eif \enot A)$ %Yes
\end{compactlist}

\problempart
\label{pr.TT.satisfiable4}
Use complete or partial truth tables (as appropriate) to determine whether these sentences are jointly satisfiable, or jointly unsatisfiable:
\begin{compactlist}
\item $A \eand B$, $C\eif \enot B$, $C$ %unsatisfiable
\item $A\eif B$, $B\eif C$, $A$, $\enot C$ %unsatisfiable
\item $A \eor B$, $B\eor C$, $C\eif \enot A$ %satisfiable
\item $A$, $B$, $C$, $\enot D$, $\enot E$, $F$ %satisfiable
\item $A \eand (B \eor C)$, $\enot(A \eand C)$, $\enot(B \eand C)$ %satisfiable
\item $A \eif B$, $B \eif C$, $\enot(A \eif C)$ %unsatisfiable
\end{compactlist}

\problempart
\label{pr.TT.valid4}
Use complete or partial truth tables (as appropriate) to determine whether each argument is valid or invalid:
\begin{compactlist}
\item $A\eor\bigl[A\eif(A\eiff A)\bigr] \therefore A$ %invalid
\item $A\eiff\enot(B\eiff A) \therefore A$ %invalid
\item $A\eif B, B \therefore A$ %invalid
\item $A\eor B, B\eor C, \enot B \therefore A \eand C$ %valid
\item $A\eiff B, B\eiff C \therefore A\eiff C$ %valid
\end{compactlist}

\problempart
\label{pr.TT.TTorC3}
Determine whether each sentence is a tautology, a contradiction, or a contingent sentence. Justify your answer with a complete or partial truth table as appropriate.

% truth tables in LaTeX generated by http://www.curtisbright.com/logic/. Be sure to give him a shout out.

\begin{compactlist}
\item  $A \eif \enot A$ \vspace{.5ex}							

%{\color{red}
%$
%\begin{array}{c|cccc}
%A&A&\eif&\enot&A\\\hline
%T&T&\mathbf{F}&F&T\\
%F&F&\mathbf{T}&T&F
%\end{array}
%$ 
%
%Contingent	 \vspace{6pt}
%}
%	T letter, 2 connectives
\item $A \eif (A \eand (A \eor B))$ \vspace{.5ex}	

%{\color{red}
%$
%\begin{array}{cc|ccc@{}ccc@{}ccc@{}c@{}c}
%A&B&A&\eif&(&A&\eand&(&A&\eor&B&)&)\\\hline
%T&T&T&\mathbf{T}&&T&T&&T&T&T&&\\
%T&F&T&\mathbf{T}&&T&T&&T&T&F&&\\
%F&T&F&\mathbf{T}&&F&F&&F&T&T&&\\
%F&F&F&\mathbf{T}&&F&F&&F&F&F&&
%\end{array}
%$
%
%Tautology \vspace{6pt}
%}
%			2 letters, 3 connectives

\item $(A \eif B) \eiff (B \eif A)$ 	\vspace{.5ex}				%
%
%{\color{red}
%$
%\begin{array}{cc|c@{}ccc@{}ccc@{}ccc@{}c}
%a&b&(&a&\rightarrow&b&)&\leftrightarrow&(&b&\rightarrow&a&)\\\hline
%T&T&&T&T&T&&\mathbf{T}&&T&T&T&\\
%T&F&&T&F&F&&\mathbf{F}&&F&T&T&\\
%F&T&&F&T&T&&\mathbf{F}&&T&F&F&\\
%F&F&&F&T&F&&\mathbf{T}&&F&T&F&
%\end{array}
%$
%
%Contingent \vspace{6pt}
%
%}
%		2 letters, 3 connectives

\item $A \eif \enot(A \eand (A \eor B)) $	\vspace{.5ex}	

%{\color{red}
%$
%\begin{array}{cc|cccc@{}ccc@{}ccc@{}c@{}c}
%a&b&a&\rightarrow&\enot&(&a&\eand&(&a&\eor&b&)&)\\\hline
%T&T&T&\mathbf{F}&F&&T&T&&T&T&T&&\\
%T&F&T&\mathbf{F}&F&&T&T&&T&T&F&&\\
%F&T&F&\mathbf{T}&T&&F&F&&F&T&T&&\\
%F&F&F&\mathbf{T}&T&&F&F&&F&F&F&&
%\end{array}
%$
%
%Contingent	\vspace{6pt}
%
%}
%
% 2 letters, 4 connectives

\item $\enot B \eif [(\enot A \eand A) \eor B]$\vspace{.5ex} 

%{\color{red}
%$
%\begin{array}{cc|cccc@{}c@{}cccc@{}ccc@{}c}
%a&b&\enot&b&\rightarrow&(&(&\enot&a&\eand&a&)&\eor&b&)\\\hline
%T&T&F&T&\mathbf{T}&&&F&T&F&T&&T&T&\\
%T&F&T&F&\mathbf{F}&&&F&T&F&T&&F&F&\\
%F&T&F&T&\mathbf{T}&&&T&F&F&F&&T&T&\\
%F&F&T&F&\mathbf{F}&&&T&F&F&F&&F&F&
%\end{array}
%$
%Contingent	 \vspace{6pt}
%
%}
%	2 letters, 5 connectives

\item $\enot(A \eor B) \eiff (\enot A \eand \enot B)$ \vspace{.5ex}

%{\color{red}
%$
%\begin{array}{cc|cc@{}ccc@{}ccc@{}ccccc@{}c}
%a&b&\enot&(&a&\eor&b&)&\leftrightarrow&(&\enot&a&\eand&\enot&b&)\\\hline
%T&T&F&&T&T&T&&\mathbf{T}&&F&T&F&F&T&\\
%T&F&F&&T&T&F&&\mathbf{T}&&F&T&F&T&F&\\
%F&T&F&&F&T&T&&\mathbf{T}&&T&F&F&F&T&\\
%F&F&T&&F&F&F&&\mathbf{T}&&T&F&T&T&F&
%\end{array}
%$
%
%Tautology \vspace{6pt}
%}
%2 letters, 6 connectives

\item $[(A \eand B) \eand C] \eif B$\vspace{.5ex}							
%
%{\color{red}
%$
%\begin{array}{ccc|c@{}c@{}ccc@{}ccc@{}ccc}
%a&b&c&(&(&a&\eand&b&)&\eand&c&)&\rightarrow&b\\\hline
%T&T&T&&&T&T&T&&T&T&&\mathbf{T}&T\\
%T&T&F&&&T&T&T&&F&F&&\mathbf{T}&T\\
%T&F&T&&&T&F&F&&F&T&&\mathbf{T}&F\\
%T&F&F&&&T&F&F&&F&F&&\mathbf{T}&F\\
%F&T&T&&&F&F&T&&F&T&&\mathbf{T}&T\\
%F&T&F&&&F&F&T&&F&F&&\mathbf{T}&T\\
%F&F&T&&&F&F&F&&F&T&&\mathbf{T}&F\\
%F&F&F&&&F&F&F&&F&F&&\mathbf{T}&F
%\end{array}
%$
%
%Tautology \vspace{6pt}
%}
%
%3 letters, 3 connectives

\item $\enot\bigl[(C\eor A) \eor B\bigr]$\vspace{.5ex} 						
%
%{\color{red}
%$
%\begin{array}{ccc|cc@{}c@{}ccc@{}ccc@{}c}
%a&b&c&\enot&(&(&c&\eor&a&)&\eor&b&)\\\hline
%T&T&T&\mathbf{F}&&&T&T&T&&T&T&\\
%T&T&F&\mathbf{F}&&&F&T&T&&T&T&\\
%T&F&T&\mathbf{F}&&&T&T&T&&T&F&\\
%T&F&F&\mathbf{F}&&&F&T&T&&T&F&\\
%F&T&T&\mathbf{F}&&&T&T&F&&T&T&\\
%F&T&F&\mathbf{F}&&&F&F&F&&T&T&\\
%F&F&T&\mathbf{F}&&&T&T&F&&T&F&\\
%F&F&F&\mathbf{T}&&&F&F&F&&F&F&
%\end{array}
%$
%
%Contingent \vspace{6pt}
%
%}
%	 	3 letters, 3 connectives

\item $\bigl[(A\eand B) \eand\enot(A\eand B)\bigr] \eand C$ \vspace{.5ex}	
%
%{\color{red}
%$
%\begin{array}{ccc|c@{}c@{}ccc@{}cccc@{}ccc@{}c@{}ccc}
%a&b&c&(&(&a&\eand&b&)&\eand&\enot&(&a&\eand&b&)&)&\eand&c\\\hline
%T&T&T&&&T&T&T&&F&F&&T&T&T&&&\mathbf{F}&T\\
%T&T&F&&&T&T&T&&F&F&&T&T&T&&&\mathbf{F}&F\\
%T&F&T&&&T&F&F&&F&T&&T&F&F&&&\mathbf{F}&T\\
%T&F&F&&&T&F&F&&F&T&&T&F&F&&&\mathbf{F}&F\\
%F&T&T&&&F&F&T&&F&T&&F&F&T&&&\mathbf{F}&T\\
%F&T&F&&&F&F&T&&F&T&&F&F&T&&&\mathbf{F}&F\\
%F&F&T&&&F&F&F&&F&T&&F&F&F&&&\mathbf{F}&T\\
%F&F&F&&&F&F&F&&F&T&&F&F&F&&&\mathbf{F}&F
%\end{array}
%$
%
%Contradiction \vspace{6pt}
%
%}
%
%% 	3 letters, 5 connectives
%
\item $(A \eand B) ]\eif[(A \eand C) \eor (B \eand D)]$ \vspace{.5ex}		
%
%{\color{red}
%$
%\begin{array}{cccc|c@{}c@{}ccc@{}c@{}ccc@{}c@{}ccc@{}ccc@{}ccc@{}c@{}c}
%a&b&c&d&(&(&a&\eand&b&)&)&\eif&(&(&a&\eand&c&)&\eor&(&b&\eand&d&)&)\\\hline
%T&T&T&T&&&T&T&T&&&\mathbf{T}&&&T&T&T&&T&&T&T&T&&\\
%T&T&F&F&&&T&T&T&&&\mathbf{F}&&&T&F&F&&F&&T&F&F&&\\
%\end{array}
%$
%
%Contingent \vspace{6pt}
%}
%
%	4 letters, 5 connectives
\end{compactlist}

\noindent\problempart
\label{pr.TT.TTorC4}
Determine whether each sentence is a tautology, a contradiction, or a contingent sentence. Justify your answer with a complete or partial truth table as appropriate.
\begin{compactlist}
\item  $\enot (A \eor A)$\vspace{.5ex}							%	Contradiction		1 letter, 2 connectives
\item $(A \eif B) \eor (B \eif A)$\vspace{.5ex}					%	Tautology			2 letters, 2 connectives
\item $[(A \eif B) \eif A] \eif A$\vspace{.5ex}					%	Tautology			2 letters, 3 connectives
\item $\enot[( A \eif B) \eor (B \eif A)]$\vspace{.5ex}			%	Contradiction		2 letters, 4 connectives
\item $(A \eand B) \eor (A \eor B)$\vspace{.5ex} 				%	Contingent		2 letters, 5 connectives
\item $\enot(A\eand B) \eiff A$\vspace{.5ex} 					%contingent			2 letters, 3 connectives
\item $A\eif(B\eor C)$\vspace{.5ex} 							%contingent			3 letters, 2 connectives
\item $(A \eand\enot A) \eif (B \eor C)$\vspace{.5ex} 			%tautology			3 letters, 4 connectives 
\item $(B\eand D) \eiff [A \eiff(A \eor C)]$\vspace{.5ex}			%contingent			4 letters, 4 connectives
\item $\enot[(A \eif B) \eor (C \eif D)]$\vspace{.5ex} 			% Contingent. 		4 letters, 4 connectives
\end{compactlist}



\noindent\problempart
Determine whether each the following pairs of sentences are logically equivalent using complete truth tables. Justify your answer with a complete or partial truth table as appropriate.
\begin{compactlist}
\item $A$ and $A \eor A$
\item $A$ and $A \eand A$
\item $A \eor \enot B$ and $A\eif B$
\item $(A \eif B)$ and $(\enot B \eif \enot A)$
\item $\enot(A \eand B)$ and $\enot A \eor \enot B$
\item $ ((U \eif (X \eor X)) \eor U)$ and $\enot (X \eand (X \eand U))$
\item $ ((C \eand (N \eiff C)) \eiff C)$ and $(\enot \enot \enot N \eif C)$
\item $[(A \eor B) \eand C]$ and $[A \eor (B \eand C)]$
\item $((L \eand C) \eand I)$ and $L \eor C$
\end{compactlist}


\noindent\problempart
\label{pr.TT.satisfiable5}
Determine whether each collection of sentences is jointly satisfiable or jointly unsatisfiable. Justify your answer with a complete or partial truth table as appropriate.
\begin{compactlist}
\item $A\eif A$, $\enot A \eif \enot A$, $A\eand A$, $A\eor A$ %satisfiable
\item $A \eif \enot A$, $\enot A \eif A$%unsatisfiable.
\item $A\eor B$, $A\eif C$, $B\eif C$ %satisfiable
\item $A \eor B$, $A \eif C$, $B \eif C$, $\enot C$ %	Insatisfiable
\item $B\eand(C\eor A)$, $A\eif B$, $\enot(B\eor C)$  %unsatisfiable
\item $(A \eiff B) \eif B$,  $B \eif \enot (A \eiff B)$, $A \eor B$  %	Consistent
\item $A\eiff(B\eor C)$, $C\eif \enot A$, $A\eif \enot B$ %satisfiable
\item  $A \eiff B$,  $\enot B \eor \enot A$,  $A \eif  B$ % Consistent
\item $A \eiff B$, $A \eif C$, $B \eif D$, $\enot(C \eor D)$ %consitent
\item $\enot (A \eand \enot B)$,  $B \eif \enot A$, $\enot B$   %Consistent
\end{compactlist}

\noindent\problempart Determine whether each argument is valid or invalid. Justify your answer with a complete or partial truth table as appropriate.
\label{pr.TT.valid5} 
\begin{compactlist}
\item $A\eif(A\eand\enot A)\therefore \enot A$% valid
\item $A \eor B$, $A \eif B$, $B \eif A \therefore  A \eiff B$  % Valid
\item $A\eor(B\eif A)\therefore \enot A \eif \enot B$ %valid
\item $A \eor B$, $A \eif B$, $ B \eif A \therefore  A \eand B$ %valid
\item $(B\eand A)\eif C$, $(C\eand A)\eif B\therefore (C\eand B)\eif A$ % invalid
\item $\enot (\enot A \eor \enot B)$, $A \eif \enot C \therefore  A \eif (B \eif C)$ % invalid.
\item $A \eand (B \eif C)$, $\enot C \eand (\enot B \eif \enot A)\therefore C \eand \enot C$ % valid
\item $A \eand B$, $\enot A \eif \enot C$, $B \eif \enot D \therefore  A \eor B$ % Invalid
\item $A \eif B\therefore (A \eand B) \eor (\enot A \eand \enot B)$ % invalid
\item $\enot A \eif B$,$ \enot B \eif C $,$ \enot C \eif A \therefore  \enot A \eif (\enot B \eor \enot C) $% Invalid

\end{compactlist}

\noindent\problempart Determine whether each argument is valid or invalid. Justify your answer with a complete or partial truth table as appropriate.
\label{pr.TT.valid6} 
\begin{compactlist}
\item $A\eiff\enot(B\eiff A)\therefore A$ % invalid
\item $A\eor B$, $B\eor C$, $\enot A\therefore B \eand C$ % invalid
\item $A \eif C$, $E \eif (D \eor B)$, $B \eif \enot D\therefore (A \eor C) \eor (B \eif (E \eand D))$ % invalid
\item $A \eor B$, $C \eif A$, $C \eif B\therefore A \eif (B \eif C)$ % invalid
\item $A \eif B$, $\enot B \eor A\therefore A \eiff B$ % valid
\end{compactlist}