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\fancyhead[RE,LO]{Brian Tomasik}
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\fancyfoot[RE,LO]{Essays on Reducing Suffering}
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\LARGE
Killing Animals May Increase Deaths Per Unit Time\\
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by Brian Tomasik\\
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First written: 22 Apr. 2006; last update: 3 Aug. 2014\\
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\begin{abstract}
It's sometimes claimed that killing animals doesn't matter because they would have died naturally, perhaps by a means at least as painful. This analysis ignores the point that killing animals before they would have died on their own may increase ``turnover rates", i.e., the number of deaths per unit time. This may increase total suffering.
\end{abstract}
\section{Introduction}
Is the suffering caused by killing an animal only the difference between how painful it is to be killed vs. how painful it would have been to die naturally?
\section{Factors}
The total harm caused by human activity is the difference between the sum-total of animal utilities when the activity occurs and when it doesn't.% This is the basis of equation \eqref{difference} below.
\subsection{Difference in Pain of Death}
The first factor to consider is the difference in the painfulness of death in the two situations. In some cases, human-caused animal deaths are less painful than deaths in the wild. Animals killed by buildings or cars may suffer less than those eaten alive by predators.
\subsection{Increase in Total Deaths}
The second consideration is, How much does human activity increase the total number of animal deaths that occur? To take an example, suppose that Species X has an average lifespan in the wild of 2 years. A combine comes along and kills 100 members of Species X; the average age of the animals killed was probably around the halfway point: 1 year. Suppose those 100 animals are immediately replaced. Then, the next year, when the combine harvests the field again, another 100 animals are killed. If instead the combine had not been present, the animals would have lived for about two years before dying (i.e., after 2 years, 100 animals would have died). So when people kill animals before they naturally would have died, they thereby increase the total number of deaths. In this case, the combine caused twice as many animals to endure the pain of death as would otherwise have happened. If death by a combine is anything more than half as bad as death by natural causes, then the combine caused net harm.
\subsection{Precluded Utility}
What I said above is true only in a special case: when one maintains the assumption of instantaneous replacement. Perhaps this is unrealistic. Perhaps, to continue the example above, when a member of Species X is killed by the combine, it is not replaced until a year later. In this case, the combine doesn't cause a net increase in deaths after all (since only 100 animals die within two years). But this time, there's a different cost: preclusion of a total of 100 life-years for the hundred members of Species X that died early.
\begin{comment}
I'm omitting the below because I don't understand what my original thinking was or if it's correct.
\section{Equations}
Below is sort of an intuitive model that captures each of these three components. I'm not completely sure these equations are right because I developed them sort of \textit{ad hoc}. I'd have to think about this model more carefully to make sure that these equations are right.
\begin{itemize}
\item \begin{math} \Delta U \equiv \end{math} overall change in aggregated utility.
\item \begin{math} A \equiv \end{math} a given human activity that kills animals (e.g., mechanically harvesting crops).
\item \begin{math} \neg A \equiv \end{math} the absence of the given human activity.
\item \begin{math} u \equiv \end{math} average utility of living per organism per unit time.
\item \begin{math} d \equiv \end{math} pain of death.
\item \begin{math} L \equiv \end{math} average lifespan in the wild.
\item \begin{math} L_k \equiv \end{math} average age of the animals at the time of being killed.
\item \begin{math} n \equiv \end{math} number of deaths.
\item \begin{math} k \equiv \end{math} number of animals killed.
\item \begin{math} t \equiv \end{math} average replacement time.
\end{itemize}
By definition
\begin{equation}\label{difference}
\Delta U \equiv \Delta U_{A} - \Delta U_{\neg A}.
\end{equation}
With the anthropogenic source of mortality, the change in utility involved is the pain of death minus the foregone utility of those killed early:
\begin{equation} \label{ua}
\Delta U_{A} = n_Ad_A - k_Aut.
\end{equation}
Without the anthropogenic source of mortality, the utility impact is just due to deaths:
\begin{equation}
\Delta U_{\neg A} = n_{\neg A}d_{\neg A}.
\end{equation}
\begin{equation} \label{na}
n_A = n_{\neg A} + k_A \frac{L - (L_k + t)}{L}.
\end{equation}
\\
Substituting (4) into (2),
\[\Delta U_{A} = (n_{\neg A} + k_A \frac{L - (L_k + t)}{L})d_A - k_Aut.\]
\[\Delta U_{A} = n_{\neg A}d_A + k_Ad_A \frac{L - L_k - t}{L} - k_Aut.\]
\\
Then, subtracting off \begin{math} U_{\neg A} \end{math},
\[\Delta U = n_{\neg A}d_A + k_Ad_A \frac{L - L_k - t}{L} - k_Aut - n_{\neg A}d_{\neg A}.\]
\begin{equation}
\Delta U = n_{\neg A}(d_A - d_{\neg A}) + k_Ad_A \frac{L - L_k - t}{L} - k_Aut.
\end{equation}
(5) includes, in order, all three of the factors described above: the difference between the painfulness of death in the two situations, the pain caused by increased deaths, and utility denied by noninstantaneous replacement.
\section{Numbers from Example}
Try inserting the values for the Species-X example into (5):\\
\\
\begin{math}
n_{\neg A} = 100,\\
k_A = 200,\\
L = 2,\\
L_k = 1.\\
\end{math}
\[\Delta U = 100(d_A - d_{\neg A}) + 200d_A \frac{1 - t}{2} - 200ut.\]
To simplify further, suppose that\\
\\
\begin{math}
d_{\neg A} = 2d_A,\\
1u = -d_A.
\end{math}
\[\Delta U = -100d_A + 200d_A \frac{1 - t}{2} + 200d_At.\]
\[\Delta U = d_A(-100 + 200\frac{1 - t}{2} + 200t).\]
\[\Delta U = d_A(-100 + 100 - 100t + 200t).\]
\[\Delta U = 100d_At.\]
\\
Since \begin{math} d_A < 0 \end{math} and \begin{math} t \geq 0 \end{math}, \begin{math} \Delta U \leq 0 \end{math} in this case. \begin{math} \blacksquare \end{math}
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