<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+24.118x+2T2020" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+24.118x+2T2020+type@vertical+block@eacde0b7970244ba964fc0674e424811" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-request-token="583cde46edb311eeb45c1299a322540b" data-graded="False">
<h2 class="hd hd-2 unit-title">The Bomber's Paradox</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+24.118x+2T2020+type@html+block@0f6550aacfec4b8da945b714cca68b7a">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-course-id="course-v1:MITx+24.118x+2T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+24.118x+2T2020+type@html+block@0f6550aacfec4b8da945b714cca68b7a" data-has-score="False" data-runtime-version="1" data-block-type="html" data-request-token="583cde46edb311eeb45c1299a322540b" data-graded="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><span style="font-family: 'book antiqua', palatino;">In this lecture we’ll talk about some paradoxes based on reverse <span class="math inline">\(\omega\)</span>-sequences.</span></p>
<p><span style="font-family: 'book antiqua', palatino;"><a href="http://dailynous.com/2017/04/12/josh-parsons-1973-2017/" target="_blank">Josh Parsons</a>, who was a fellow at Oxford until shortly before his untimely death, once told me about the following puzzle. (It is a version of <a href="http://en.wikipedia.org/wiki/Supertask#Benardete.E2.80.99s_paradox" target="[object Object]">Benardete’s Paradox</a>.)</span></p>
<p><span style="font-family: 'book antiqua', palatino;">There are infinitely many electronic bombs, <span class="math inline">\(B_0,B_1,B_2,\ldots\)</span>, one for each natural number. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">They are set to go off on the following schedule: <span class="math display">\[\begin{array}{cc} \text{Bomb} & \text{When bomb is set to go off} \\ \text{$B_0$} & \text{12:00pm} \\ \text{$B_1$} & \text{11:30am} \\ \text{$B_2$} & \text{11:15am} \\ \text{$\vdots$} & \text{$\vdots$}\\ \text{$B_k$} & \text{\(\dfrac{1}{2^k}\) hours after 11:00am} \\ \text{$\vdots$} & \text{$\vdots$}\\ \end{array}\]</span> Our bombs are of a special kind: they target electronics. Should one of the bombs go off, it will instantaneously disable all nearby electronic devices, <em>including other bombs</em>. This means that a bomb goes off if and only if no bombs have gone off before it. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">More specifically:</span></p>
<p><span class="math display" style="font-family: 'book antiqua', palatino;">\[\begin{array}{cc} (0) & \text{\(B_0\) goes off if and only if, for each $n > 0$, \(B_n\) fails to go off}. \\ (1) & \text{\(B_1\) goes off if and only if, for each $n > 1$, \(B_n\) fails to go off}. \\ (2) & \text{\(B_2\) goes off if and only if, for each $n > 2$, \(B_n\) fails to go off}. \\ \vdots & \vdots \\ \mathbf{(k)} & \text{\(B_k\) goes off if and only if, for each $n > k$, \(B_n\) fails to go off}. \\ \mathbf{(k+1)} & \text{\(B_{k+1}\) goes off if and only if, for each $n > k+1$, \(B_n\) fails to go off}. \\ \end{array}\]</span></p>
<p></p>
<p><span style="font-family: 'book antiqua', palatino;">Will any bombs go off? If so, which ones? Here’s a proof that bomb <span class="math inline">\(B_k\)</span> can’t go off:</span></p>
<blockquote>
<p><span style="font-family: 'book antiqua', palatino;">Suppose that bomb <span class="math inline">\(B_k\)</span> goes off. It follows from statement <span class="math inline">\((k)\)</span> above that <span class="math inline">\(B_n\)</span> must fail to go off for each <span class="math inline">\(n > k\)</span>. This means, in particular, that <span class="math inline">\(B_{k+1}\)</span> must have failed to go off. But it follows from statement (<span class="math inline">\(k+1\)</span>) that the only way for that to happen is for <span class="math inline">\(B_m\)</span> to go off for some <span class="math inline">\(m>k+1\)</span>. And that’s impossible: we concluded earlier that <span class="math inline">\(B_n\)</span> must fail to go off for each <span class="math inline">\(n > k\)</span>.</span></p>
</blockquote>
<p><span style="font-family: 'book antiqua', palatino;">But wait! Here’s a proof that bomb <span class="math inline">\(B_k\)</span> must go off:</span></p>
<blockquote>
<p><span style="font-family: 'book antiqua', palatino;">Suppose that <span class="math inline">\(B_k\)</span> fails to go off. It follows from statement <span class="math inline">\((k)\)</span> above that <span class="math inline">\(B_n\)</span> must go off for some <span class="math inline">\(n > k\)</span>, and the previous argument shows that that is impossible.</span></p>
</blockquote>
<p><span style="font-family: 'book antiqua', palatino;">What’s going on?</span></p>
<p></p>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+24.118x+2T2020+type@video+block@1f96a8dae17f400b851636e09a249c95">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-course-id="course-v1:MITx+24.118x+2T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+24.118x+2T2020+type@video+block@1f96a8dae17f400b851636e09a249c95" data-has-score="False" data-runtime-version="1" data-block-type="video" data-request-token="583cde46edb311eeb45c1299a322540b" data-graded="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Video Review: The Bombers</h3>
<div
id="video_1f96a8dae17f400b851636e09a249c95"
class="video closed"
data-metadata='{"ytTestTimeout": 1500, "autohideHtml5": false, "generalSpeed": 1.0, "sources": [], "end": 0.0, "saveStateUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@1f96a8dae17f400b851636e09a249c95/handler/xmodule_handler/save_user_state", "speed": null, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@1f96a8dae17f400b851636e09a249c95/handler/transcript/available_translations", "completionEnabled": false, "transcriptLanguage": "en", "streams": "1.00:vBmKqmkQaqU", "captionDataDir": null, "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "showCaptions": "true", "ytMetadataEndpoint": "", "autoplay": false, "recordedYoutubeIsAvailable": true, "savedVideoPosition": 0.0, "transcriptLanguages": {"en": "English"}, "poster": null, "autoAdvance": false, "prioritizeHls": false, "start": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@1f96a8dae17f400b851636e09a249c95/handler/transcript/translation/__lang__", "completionPercentage": 0.95, "duration": 0.0, "saveStateEnabled": false, "publishCompletionUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@1f96a8dae17f400b851636e09a249c95/handler/publish_completion"}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="1f96a8dae17f400b851636e09a249c95"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_1f96a8dae17f400b851636e09a249c95">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_1f96a8dae17f400b851636e09a249c95">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@1f96a8dae17f400b851636e09a249c95/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@1f96a8dae17f400b851636e09a249c95/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+24.118x+2T2020" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+24.118x+2T2020+type@vertical+block@cd184ccc0fa54791bcf19ca59a27595e" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-request-token="583cde46edb311eeb45c1299a322540b" data-graded="False">
<h2 class="hd hd-2 unit-title">Assessment: the Bomber's Paradox</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+24.118x+2T2020+type@html+block@3ca4a7a2e66641dd89ff8c531a6477dd">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-course-id="course-v1:MITx+24.118x+2T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+24.118x+2T2020+type@html+block@3ca4a7a2e66641dd89ff8c531a6477dd" data-has-score="False" data-runtime-version="1" data-block-type="html" data-request-token="583cde46edb311eeb45c1299a322540b" data-graded="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<h3><span style="font-family: 'book antiqua', palatino;">The Bomber's Paradox </span><br /><span style="font-family: 'book antiqua', palatino;">Paradox Grade: 6</span></h3>
<p><span style="font-family: 'book antiqua', palatino;">I have a preferred response to the Bomber’s Paradox, but I should concede from the start that it is somewhat unintuitive. In fact, that is part of the reason I give the Bomber’s Paradox a high-ish paradoxicality grade. I think it teaches us about a way in which our intuitions go awry.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">My response starts with the observation the argument involved in the Bomber’s Paradox is, in fact, valid. I believe that the reasoning in the preceding page is correct, and that there is no way of getting around that fact.<br /></span></p>
<p><span style="font-family: 'book antiqua', palatino;">It follows that it is impossible for there to be a sequence of bombs set up as the case describes. It seems to me, moreover, that it is not just medical or physical impossibility. I think the relevant setup entails an absurdity, and is therefore <em>logically</em> impossible.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">There is no denying that there is something disconcerting about this claim. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">One way to bring this out is to imagine an infinite being, who attempts to set up the relevant configuration of bombs. If the setup is logically impossible, something will go wrong. But what? </span></p>
<p><span style="font-family: 'book antiqua', palatino;">I’m inclined to think that this question deserves a flat-footed answer. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">Consider an analogy. Suppose that our infinite being tries to bring about a more straightforward absurdity. Perhaps she wishes to build objects <span class="math inline">\(A\)</span>, <span class="math inline">\(B\)</span> and <span class="math inline">\(C\)</span> such that: <span class="math display">\[\text{Mass}(A) < \text{Mass}(B) < \text{Mass}(C) < \text{Mass}(A)\]</span> We know that something will go wrong, and that she won’t succeed. But what? The answer is not determined by the story so far, but note that it could turn out to be something rather mundane. Perhaps she creates objects with masses of <span class="math inline">\(1kg\)</span>, <span class="math inline">\(2kg\)</span> and <span class="math inline">\(3kg\)</span>, respectively, and thereby fails to conform to the required setup. Or perhaps she discovers the futility of the project and loses heart. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">I claim that the reason our infinite being fails to put together the relevant configuration of bombs could turn out to be similarly mundane. Perhaps she messes up the timers, and thereby fails to conform the paradoxical setup. Or perhaps she discovers the futility of the project and looses heart.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">To my mind, at least, this feels initially unsatisfying. But I think this is because I am subject to a certain kind of mistake. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">When I consider case of the three objects I get a strong feeling that the constraints of the problem are at odds with one another: that the only way to satisfy some of the constraints is to fail to satisfy others. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">In contrast, when I think about the Bomber’s case it feels to me like the constraints of the case are not at odds with one another, and that our infinite being should be able to build a suitable sequence of bombs. ("She first builds bomb <span class="math inline">\(B_0\)"</span>, I think to myself, "she then builds bomb <span class="math inline">\(B_1\)</span> …".) </span></p>
<p><span style="font-family: 'book antiqua', palatino;">But this is actually an illusion: the constraints of the case are, in fact, at odds with one another. If that does not seem intuitively obvious to me, it is because I’m not smart enough to see the problem at a glance, as I am in the case of <span class="math inline">\(A\)</span>, <span class="math inline">\(B\)</span> and <span class="math inline">\(C\)</span>. I can only see it by going through a non-trivial piece of reasoning.</span></p>
<p><span style="font-family: 'book antiqua', palatino;">I think I am also misled by the fact that no inconsistency arises when it comes to finite subsets of the paradoxical configuration of bombs. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">Consider, for example, the case with just bomb <span class="math inline">\(B_0\)</span>, which is set to go off at noon. Since there are no bombs set up to go off before then, <span class="math inline">\(B_0\)</span> will go off. No paradox so far! Now suppose we add <span class="math inline">\(B_1\)</span>, which is set to go off at 11:30. Since there are no bombs set up to go off before then, <span class="math inline">\(B_1\)</span> will go off. And in this scenario <span class="math inline">\(B_0\)</span> fails to go off because <span class="math inline">\(B_1\)</span> goes off before noon. Again, no paradox! We can keep adding bombs, one by one. And as long as only finitely many of them have been added there is no paradox. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">This means that we have a discontinuity: although our setup is perfectly consistent when there are only finitely many bombs, it becomes inconsistent when there are infinitely many bombers. In other words, there is a inconsistency that arises as a global feature of an infinite scenario, but not for any of its finite parts.</span></p>
<p></p>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+24.118x+2T2020+type@video+block@6d0f613bdf1043619dc05ec46f490840">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-course-id="course-v1:MITx+24.118x+2T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+24.118x+2T2020+type@video+block@6d0f613bdf1043619dc05ec46f490840" data-has-score="False" data-runtime-version="1" data-block-type="video" data-request-token="583cde46edb311eeb45c1299a322540b" data-graded="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Video: The Walls</h3>
<div
id="video_6d0f613bdf1043619dc05ec46f490840"
class="video closed"
data-metadata='{"ytTestTimeout": 1500, "autohideHtml5": false, "generalSpeed": 1.0, "sources": [], "end": 0.0, "saveStateUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@6d0f613bdf1043619dc05ec46f490840/handler/xmodule_handler/save_user_state", "speed": null, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@6d0f613bdf1043619dc05ec46f490840/handler/transcript/available_translations", "completionEnabled": false, "transcriptLanguage": "en", "streams": "1.00:Sgs48M691Rg", "captionDataDir": null, "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "showCaptions": "true", "ytMetadataEndpoint": "", "autoplay": false, "recordedYoutubeIsAvailable": true, "savedVideoPosition": 0.0, "transcriptLanguages": {"en": "English"}, "poster": null, "autoAdvance": false, "prioritizeHls": false, "start": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@6d0f613bdf1043619dc05ec46f490840/handler/transcript/translation/__lang__", "completionPercentage": 0.95, "duration": 0.0, "saveStateEnabled": false, "publishCompletionUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@6d0f613bdf1043619dc05ec46f490840/handler/publish_completion"}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="6d0f613bdf1043619dc05ec46f490840"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_6d0f613bdf1043619dc05ec46f490840">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_6d0f613bdf1043619dc05ec46f490840">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@6d0f613bdf1043619dc05ec46f490840/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@6d0f613bdf1043619dc05ec46f490840/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+24.118x+2T2020" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+24.118x+2T2020+type@vertical+block@d94aec68eeb24ae0a50f4bc1cb2314b6" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-request-token="583cde46edb311eeb45c1299a322540b" data-graded="False">
<h2 class="hd hd-2 unit-title">Yablo's Paradox</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+24.118x+2T2020+type@html+block@2cbdd821c437419193f13c2bf742f0c1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-course-id="course-v1:MITx+24.118x+2T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+24.118x+2T2020+type@html+block@2cbdd821c437419193f13c2bf742f0c1" data-has-score="False" data-runtime-version="1" data-block-type="html" data-request-token="583cde46edb311eeb45c1299a322540b" data-graded="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><span style="font-family: 'book antiqua', palatino;">Here’s another reverse-omega-sequence paradox. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">This one does not involve time, but is structurally very similar to the Bomber Paradox, which does. The paradox was discovered by <a href="http://www.mit.edu/~yablo/home/Hello.html" target="_blank">Steve Yablo</a>, who is a famous philosophy professor at MIT (and a member of my dissertation committee, many years ago).</span></p>
<p><span style="font-family: 'book antiqua', palatino;">Suppose you have infinitely many sentences, one for each natural number. We label the sentences <span class="math inline">\(S_0,S_1,S_2\dots\)</span>, and characterize them as follows:</span></p>
<p><span style="font-family: 'book antiqua', palatino;"><span class="math display">\[\begin{array}{cc} \text{ Label} & \text{ Sentence} \\ \text{ $S_0$} & \text{“For each \(i>0\), sentence \(S_i\) is false"} \\ \text{ $S_1$} & \text{“For each \(i>1\), sentence \(S_i\) is false"} \\ \text{ $S_2$} & \text{“For each \(i>2\), sentence \(S_i\) is false"} \\ \text{$\vdots$} & \text{$\vdots$}\\ \text{ $S_k$} & \text{“For each \(i>k\), sentence \(S_i\) is false"} \\ \text{$\vdots$} & \text{$\vdots$}\\ \end{array}\]</span> We therefore know that each of the following statements must be true:</span></p>
<p><span class="math display" style="font-family: 'book antiqua', palatino;">\[\begin{array}{cc} (0) & \text{\(S_0\) is true if and only if, for every \(n>0\), \(S_n\) is false.} \\ (1) & \text{\(S_1\) is true if and only if, for every \(n>1\), \(S_n\) is false.} \\ (2) & \text{\(S_2\) is true if and only if, for every \(n>2\), \(S_n\) is false.} \\ \vdots & \vdots \\ \mathbf{(k)} & \text{\(S_k\) is true if and only if, for every \(n>k\), \(S_n\) is false.} \\ \mathbf{(k+1)} & \text{\(S_{k+1}\) is true if and only if, for every \(n>k+1\), \(S_n\) is false.} \\ \end{array}\]</span></p>
<p><span style="font-family: 'book antiqua', palatino;">Which sentences are true and which ones are false? </span></p>
<p><span style="font-family: 'book antiqua', palatino;">Here is a proof that <span class="math inline">\(S_k\)</span> can’t be true:</span></p>
<blockquote>
<p><span style="font-family: 'book antiqua', palatino;">Suppose that sentence <span class="math inline">\(S_k\)</span> is true. It follows from statement <span class="math inline">\((k)\)</span> above that <span class="math inline">\(S_n\)</span> must be false for each <span class="math inline">\(n > k\)</span>. This means, in particular, that <span class="math inline">\(S_{k+1}\)</span> must be false. But it follows from statement (<span class="math inline">\(k+1\)</span>) that the only way for that to happen is for <span class="math inline">\(S_m\)</span> to be true for some <span class="math inline">\(m>k+1\)</span>. And that’s impossible: we concluded earlier that <span class="math inline">\(S_n\)</span> must be false for each <span class="math inline">\(n > k\)</span>.</span></p>
</blockquote>
<p><span style="font-family: 'book antiqua', palatino;">But wait! Here’s a proof that sentence <span class="math inline">\(S_k\)</span> must be true:</span></p>
<blockquote>
<p><span style="font-family: 'book antiqua', palatino;">The previous argument shows that sentence <span class="math inline">\(S_m\)</span> is false for arbitrary <span class="math inline">\(m\)</span>. This means in particular, that sentence <span class="math inline">\(S_m\)</span> is false for every <span class="math inline">\(m>k\)</span>. So it follows from statement <span class="math inline">\((k)\)</span> above that <span class="math inline">\(S_k\)</span> must be true.</span></p>
</blockquote>
<p><span style="font-family: 'book antiqua', palatino;">What’s going on?</span></p>
<p></p>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+24.118x+2T2020+type@video+block@7874d9f6c3d54bb594d76c946ffd7079">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-course-id="course-v1:MITx+24.118x+2T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+24.118x+2T2020+type@video+block@7874d9f6c3d54bb594d76c946ffd7079" data-has-score="False" data-runtime-version="1" data-block-type="video" data-request-token="583cde46edb311eeb45c1299a322540b" data-graded="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Video: Yablo's Paradox</h3>
<div
id="video_7874d9f6c3d54bb594d76c946ffd7079"
class="video closed"
data-metadata='{"ytTestTimeout": 1500, "autohideHtml5": false, "generalSpeed": 1.0, "sources": [], "end": 0.0, "saveStateUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@7874d9f6c3d54bb594d76c946ffd7079/handler/xmodule_handler/save_user_state", "speed": null, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@7874d9f6c3d54bb594d76c946ffd7079/handler/transcript/available_translations", "completionEnabled": false, "transcriptLanguage": "en", "streams": "1.00:gd2EQ5XcwFE", "captionDataDir": null, "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "showCaptions": "true", "ytMetadataEndpoint": "", "autoplay": false, "recordedYoutubeIsAvailable": true, "savedVideoPosition": 0.0, "transcriptLanguages": {"en": "English"}, "poster": null, "autoAdvance": false, "prioritizeHls": false, "start": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@7874d9f6c3d54bb594d76c946ffd7079/handler/transcript/translation/__lang__", "completionPercentage": 0.95, "duration": 0.0, "saveStateEnabled": false, "publishCompletionUrl": "/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@7874d9f6c3d54bb594d76c946ffd7079/handler/publish_completion"}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="7874d9f6c3d54bb594d76c946ffd7079"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_7874d9f6c3d54bb594d76c946ffd7079">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_7874d9f6c3d54bb594d76c946ffd7079">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@7874d9f6c3d54bb594d76c946ffd7079/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@video+block@7874d9f6c3d54bb594d76c946ffd7079/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+24.118x+2T2020" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+24.118x+2T2020+type@vertical+block@c917bc995baa4819b80d176957da2a25" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-request-token="583cde46edb311eeb45c1299a322540b" data-graded="False">
<h2 class="hd hd-2 unit-title">The Liar Paradox</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+24.118x+2T2020+type@html+block@6e1d0c150e3c419392ccf8bd5fb8c4a1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-course-id="course-v1:MITx+24.118x+2T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+24.118x+2T2020+type@html+block@6e1d0c150e3c419392ccf8bd5fb8c4a1" data-has-score="False" data-runtime-version="1" data-block-type="html" data-request-token="583cde46edb311eeb45c1299a322540b" data-graded="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><span style="font-family: book antiqua, palatino;">Yablo's Paradox is closely analogous to the Bomber's Paradox, since one paradox can be turned into the other by substituting talk of bombs going off for talk of sentences being true. But there is also a respect in which the two paradoxes might seem to come apart, at least initially.</span><br /><br /><span style="font-family: book antiqua, palatino;">My response to the Bomber's Paradox was based on the claim that no bombs could ever be configured in the way that the paradox requires. It is not initially clear, however, that an analogous response is available in the case Yablo's Paradox.</span></p>
<p><span style="font-family: book antiqua, palatino;">Whereas in the case of the Bomber's Paradox we were speaking of hypothetical bombs, in the case of Yablo's Paradox we are talking about sentences of <em>English</em> (or, more, precisely sentences of the language that results from enriching English with unproblematic logical notation). And one might think that unlike the bombs, which don't actually exist, the relevant sentences do, in fact, exist -- not in the sense of being represented by actual blobs of ink or spoken sounds, but in the sense of being well-defined and available for use by English speakers.</span><br /><br /><span style="font-family: book antiqua, palatino;">That is one reason Yablo's paradox is so interesting. But there is another...</span></p>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+24.118x+2T2020+type@html+block@98e230f5aca143ef839a4e27143fedd4">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-course-id="course-v1:MITx+24.118x+2T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+24.118x+2T2020+type@html+block@98e230f5aca143ef839a4e27143fedd4" data-has-score="False" data-runtime-version="1" data-block-type="html" data-request-token="583cde46edb311eeb45c1299a322540b" data-graded="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><span style="font-family: book antiqua, palatino;">The Liar Paradox<br />Paradox Grade: 10</span></p>
<p><span style="font-family: book antiqua, palatino;">Yablo's Paradox teaches us something important about the most famous of all the semantic paradoxes:</span></p>
<p style="padding-left: 30px;"><span style="font-family: book antiqua, palatino;"><strong>The Liar Paradox</strong></span><br /><span style="font-family: book antiqua, palatino;">Consider the starred sentence: <br /></span><br /><span style="font-family: book antiqua, palatino;">\((\star)\) The starred sentence is false.<br /></span><br /><span style="font-family: book antiqua, palatino;">Is the starred sentence true or false? If it is true then what it says is true, so it is false. If it is false, what it says is false, so it is true.</span></p>
<p><span style="font-family: book antiqua, palatino;">It is tempting to think that the Liar Paradox is, at root, a puzzle about self-reference, and that the way out of the paradox is to find a way of disallowing sentences that make reference to themselves in "vicious'' ways. </span></p>
<p><span style="font-family: book antiqua, palatino;">But Yablo's Paradox suggests that self-reference isn't really the root of the problem. For none of the sentences involved in Yablo's Paradox makes reference to itself. So any solution to the Liar Paradox that is general enough to apply to Yablo's Paradox must not rely essentially on considerations of self-reference.</span><br /><br /><span style="font-family: book antiqua, palatino;">Here I'll describe my own view about the Liar Paradox. As you read on, please keep in mind that my view is not orthodoxy. The paradox has generated an enormous literature over the years, and there is no real consensus about how it ought to be addressed. (If you'd like to learn more, have a look at the readings I suggest at the end of this lecture.)</span><br /><br /><span style="font-family: book antiqua, palatino;">My views on the Liar Paradox starts with a fairly radical claim about language. I believe it is a mistake to think that sentences have meanings independently of the interpretations that are agreed upon by language users in the context of particular assertions. </span></p>
<p><span style="font-family: book antiqua, palatino;">Accordingly, linguistic communication is best thought of as a coordination game between Speaker and Hearer. The goal of the game is for Hearer to interpret Speaker's assertion as Speaker intends. When Speaker and Hearer successfully coordinate on an interpretation for Speaker's assertion, the asserted sentence is thereby rendered meaningful in the relevant context. When they fail to coordinate, the sentence remains meaningless.</span><br /><br /><span style="font-family: book antiqua, palatino;">What resources do Speaker and Hearer use to coordinate on an interpretation for Speaker's assertion? They use what they know (and assume one another to know) about past linguistic usage. But they also use what they know (and assume one another to know) about the context in which the assertion takes place.</span><br /><br /></p>
</div>
</div>
<div class="vert vert-2" data-id="block-v1:MITx+24.118x+2T2020+type@html+block@636d9a7ca6734ed688b68aab60856a24">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-course-id="course-v1:MITx+24.118x+2T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+24.118x+2T2020+type@html+block@636d9a7ca6734ed688b68aab60856a24" data-has-score="False" data-runtime-version="1" data-block-type="html" data-request-token="583cde46edb311eeb45c1299a322540b" data-graded="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><span style="font-family: book antiqua, palatino;">Here is an example to help illustrate the role of context in successful coordination.</span></p>
<p><span style="font-family: book antiqua, palatino;">Suppose that I am Speaker and you are Hearer. I make an assertion. Your knowledge of past linguistic usage is enough to establish that what I said is either ''the last word in Zoyman's assertion is obscene" or ''the last word in Zoyman's assertion is 'obscene' ''. </span></p>
<p><span style="font-family: book antiqua, palatino;">But you cannot immediately decide between these two interpretations because quotation marks are not pronounced in English. (Note that in the first case, I am referring to an obscenity-laced assertion, and in the second, I am referring to an assertion that ends with the word "obscene''.) So you turn to context for help. Here are three different scenarios: </span></p>
<ul>
<li><span style="font-family: book antiqua, palatino;">Suppose, first, that Zoyman is a drunken sailor who speaks no English and is notoriously vulgar. You and I are in a bar with Zoyman, and you can't understand what he's saying. <br /><br />When my assertion is made in this context, you can reasonably interpret me as saying ''the last word in Zoyman's assertion is obscene''. Since I can reasonably assume that my assertion will be so interpreted, I wouldn't have made it in that context unless I intended it to be so interpreted. So we have coordinated successfully.<br /><br /></span></li>
<li><span style="font-family: book antiqua, palatino;">Next, suppose that Zoyman is not a drunken sailor but an Oxford don. We are all sitting at high table and he makes an assertion in the context of a conversation about Victorian standards of lewdness. Someone sneezed at a crucial moment, and you and I can see that you couldn't quite catch Zoyman's last word. <br /><br />When my assertion is made in this context, you can reasonably interpret me as saying ''the last word in Zoyman's assertion is "obscene" ''. Since I can reasonably assume that my assertion will be so interpreted, I wouldn't have made it in that context unless I intended it to be so interpreted. So, again, we have coordinated successfully.<br /><br /></span></li>
<li><span style="font-family: book antiqua, palatino;">Now consider a third scenario. Suppose that <em>I</em> am Zoyman, and that it is common knowledge that this is so. It is also common knowledge that the only relevant assertion I have made is the very assertion you are trying to interpret. Then your ability to find a stable interpretation for my assertion will break down. <br /><br />If you assume that the last word of my assertion is "obscene'', then you will be forced to interpret me as asserting an obvious falsehood (since ''obscene'' is not an obscene word). And if you assume that the last world in my assertion is the result of putting quotes around "obscene", then again you will be forced to interpret me as asserting an obvious falsehood (since the word " "obscene" " does not refer to itself: it refers to the word "obscene''). <br /><br /></span></li>
</ul>
<p><span style="font-family: book antiqua, palatino;">We have seen that when it is common knowledge that I am Zoyman, you loose the ability to find a stable interpretation for my assertion. But note that there is nothing mysterious or paradoxical going on. </span></p>
<p><span style="font-family: book antiqua, palatino;">Here is an analogy. Suppose we decide to play a game in which one wins if one does something Martha is not doing. Most people are in a position to do well in such a game, but poor Martha is not, since in her case the strategy of doing something Martha is not doing becomes unstable. A game has been set up, and it is built into the rules that the game cannot be won by Martha. </span></p>
<p><span style="font-family: book antiqua, palatino;">Something similar is going on when it comes to my assertion about Zoyman. A coordination game has been set up. In most ordinary contexts, the game can be won because the assertion is readily interpreted. But when things are set up so that I am Zoyman and the only assertion I have made is the very assertion you are trying to interpret, ordinary interpretative strategies become unstable.</span></p>
</div>
</div>
<div class="vert vert-3" data-id="block-v1:MITx+24.118x+2T2020+type@html+block@ae1c72497f21496b99db157929b54a53">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-course-id="course-v1:MITx+24.118x+2T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+24.118x+2T2020+type@html+block@ae1c72497f21496b99db157929b54a53" data-has-score="False" data-runtime-version="1" data-block-type="html" data-request-token="583cde46edb311eeb45c1299a322540b" data-graded="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><span style="font-family: book antiqua, palatino;">I claim that something similar is going on in the Liar Paradox. </span></p>
<p><span style="font-family: book antiqua, palatino;">In most ordinary contexts, an assertion of "the starred sentence is false'' is easily interpretable. For example, in a context in which the starred sentenced is "walruses are tuskless'' one would have no difficulty interpreting an assertion of "the starred sentence is false''. </span></p>
<p><span style="font-family: book antiqua, palatino;">But a context in which the starred sentence is "the starred sentence is false'' is like the context in which I am Zoyman: efforts to interpret the relevant assertion become unstable, and coordination between Speaker and Hearer becomes impossible. As before, nothing mysterious or paradoxical is going on: a coordination game has been set up, and our ordinary interpretative strategies are such that the game cannot be won in the relevant context. </span><br /><br /><span style="font-family: book antiqua, palatino;">I have argued that an assertion of "the starred sentence is false'' cannot be interpreted in a context in which the starred sentence is "the starred sentence is false''. But recall that, on my view, it is a mistake to think that sentences have meanings independently of particular contexts of use. Instead, Speaker and Hearer render a sentence meaningful by coordinating on a suitable interpretation in a particular conversational context. Since "the starred sentence is false'' resists such coordination in the relevant context, it fails to rendered meaningful and is neither true nor false. </span></p>
<p><span style="font-family: book antiqua, palatino;">(This is all very rough, of course, but I develop these ideas further in one of the articles listed at the end of this lecture.)</span></p>
<p><span style="font-family: book antiqua,palatino;">If I had to assign the Liar Paradox a paradoxicality grade I'd assign it a perfect 10. This is partly to do with the tenacity of the problem: even though versions of the Paradox have been known since antiquity, philosophers have yet to settle on a answer that generates broad agreement. It is also to do with my own assessment of the problem: I think the Liar Paradox calls for a fairly radical rethinking of the way language works.</span> <br /><span style="font-family: book antiqua, palatino;"><br /></span></p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+24.118x+2T2020" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+24.118x+2T2020+type@vertical+block@7cf29b9be60a45b69846549d69841fa6" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-request-token="583cde46edb311eeb45c1299a322540b" data-graded="False">
<h2 class="hd hd-2 unit-title">Assessment: Yablo's Paradox</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+24.118x+2T2020+type@html+block@4b4d45b1465044d488ed4053202cbc1f">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-course-id="course-v1:MITx+24.118x+2T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+24.118x+2T2020+type@html+block@4b4d45b1465044d488ed4053202cbc1f" data-has-score="False" data-runtime-version="1" data-block-type="html" data-request-token="583cde46edb311eeb45c1299a322540b" data-graded="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<h3><span style="font-family: book antiqua, palatino;">Yablo’s Paradox</span><br /><span style="font-family: book antiqua, palatino;">Paradox Grade: 8</span></h3>
<p><span style="font-family: book antiqua, palatino;">It seems to me that Yablo's Paradox ought to be addressed in the same sort of way as the Liar Paradox. </span></p>
<p><span style="font-family: book antiqua, palatino;">Rather than assuming that \(S_0,S_1,\dots\) have meanings independently of particular contexts of use, one should think that Speaker and Hearer must attempt to render these sentences meaningful in conversation, by coordinating on suitable interpretations. But as in the case of the Liar Paradox, there is no stable way of doing so. So the sentences must remain uninterpreted and are neither true nor false.</span><br /><br /><span style="font-family: book antiqua, palatino;">Note that this response to Yablo's Paradox is in some ways analogous to my response to the Bomber's Paradox. For in saying that \(S_0,S_1,\dots\) must remain uninterpreted, I am claiming that one couldn't really set up a configuration of (meaningful) sentences that gives rise to the paradox, which is analogous to saying that one couldn't really set up a configuration of bombs that gives rise to the Bomber's Paradox.</span></p>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+24.118x+2T2020+type@problem+block@959338c5d18c4654a48ccfff7d82ceac">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+24.118x+2T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+24.118x+2T2020+type@problem+block@959338c5d18c4654a48ccfff7d82ceac" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-request-token="583cde46edb311eeb45c1299a322540b" data-graded="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_959338c5d18c4654a48ccfff7d82ceac" class="problems-wrapper" role="group"
aria-labelledby="959338c5d18c4654a48ccfff7d82ceac-problem-title"
data-problem-id="block-v1:MITx+24.118x+2T2020+type@problem+block@959338c5d18c4654a48ccfff7d82ceac" data-url="/courses/course-v1:MITx+24.118x+2T2020/xblock/block-v1:MITx+24.118x+2T2020+type@problem+block@959338c5d18c4654a48ccfff7d82ceac/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="959338c5d18c4654a48ccfff7d82ceac-problem-title" aria-describedby="block-v1:MITx+24.118x+2T2020+type@problem+block@959338c5d18c4654a48ccfff7d82ceac-problem-progress" tabindex="-1">
Problem 1
</h3>
<div class="problem-progress" id="block-v1:MITx+24.118x+2T2020+type@problem+block@959338c5d18c4654a48ccfff7d82ceac-problem-progress"></div>
<div class="problem">
<div>
<p>
<span style="font-family: 'book antiqua', palatino;">Suppose we modify the setup of Yablo&#8217;s Paradox, so that <span class="math inline">\(S_0,S_1,S_2\dots\)</span> are instead characterized as follows:</span>
</p>
<p>
<span style="font-family: 'book antiqua', palatino;">
<span class="math display">\[\begin{array}{cc}
\text{ Label} &amp; \text{ Sentence} \\
\text{ $S_0$} &amp; \text{ &#8220;For some \(i&gt;0\), sentence \(S_i\) is false''} \\
\text{ $S_1$} &amp; \text{ &#8220;For some \(i&gt;1\), sentence \(S_i\) is false''} \\
\text{ $S_2$} &amp; \text{ &#8220;For some \(i&gt;2\), sentence \(S_i\) is false''} \\
\text{$\vdots$} &amp; \text{$\vdots$}\\
\text{ $S_k$} &amp; \text{ &#8220;For some \(i&gt;k\), sentence \(S_i\) is false''} \\
\text{$\vdots$} &amp; \text{$\vdots$}\\
\end{array}\]</span>
</span>
</p>
<p>
<span style="font-family: 'book antiqua', palatino;">Is it possible to come up with a stable assignment of truth and falsity to <span class="math inline">\(S_0,S_1,S_2\dots\)</span>?</span>
</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_959338c5d18c4654a48ccfff7d82ceac_2_1">
<fieldset aria-describedby="status_959338c5d18c4654a48ccfff7d82ceac_2_1">
<div class="field">
<input type="radio" name="input_959338c5d18c4654a48ccfff7d82ceac_2_1" id="input_959338c5d18c4654a48ccfff7d82ceac_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="959338c5d18c4654a48ccfff7d82ceac_2_1-choice_0-label" for="input_959338c5d18c4654a48ccfff7d82ceac_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_959338c5d18c4654a48ccfff7d82ceac_2_1">
<span style="font-family: 'book antiqua', palatino;">Yes.</span>
</label>
</div>
<div class="field">
<input type="radio" name="input_959338c5d18c4654a48ccfff7d82ceac_2_1" id="input_959338c5d18c4654a48ccfff7d82ceac_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="959338c5d18c4654a48ccfff7d82ceac_2_1-choice_1-label" for="input_959338c5d18c4654a48ccfff7d82ceac_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_959338c5d18c4654a48ccfff7d82ceac_2_1">
<span style="font-family: 'book antiqua', palatino;">No.</span>
</label>
</div>
<span id="answer_959338c5d18c4654a48ccfff7d82ceac_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_959338c5d18c4654a48ccfff7d82ceac_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div><div class="solution-span">
<span id="solution_959338c5d18c4654a48ccfff7d82ceac_solution_1"/>
</div></div>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 1" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_959338c5d18c4654a48ccfff7d82ceac" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_959338c5d18c4654a48ccfff7d82ceac">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="959338c5d18c4654a48ccfff7d82ceac-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="959338c5d18c4654a48ccfff7d82ceac-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="959338c5d18c4654a48ccfff7d82ceac-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+24.118x+2T2020" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+24.118x+2T2020+type@vertical+block@f9ae1ca1eb8349719a8e5d61be3ba156" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-request-token="583cde46edb311eeb45c1299a322540b" data-graded="False">
<h2 class="hd hd-2 unit-title">[Bonus: Yablo in Person]</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+24.118x+2T2020+type@html+block@cf37b91f8fbd4952b38288d612c68e6d">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-course-id="course-v1:MITx+24.118x+2T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+24.118x+2T2020+type@html+block@cf37b91f8fbd4952b38288d612c68e6d" data-has-score="False" data-runtime-version="1" data-block-type="html" data-request-token="583cde46edb311eeb45c1299a322540b" data-graded="False">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<h3><span style="font-family: 'book antiqua', palatino;">Yablo in Person</span></h3>
<p><span style="font-family: 'book antiqua', palatino;">Stephen Yablo is the inventor of Yablo’s Paradox, and one of the most respected contemporary philosophers in the world. He is also a professor at MIT, so I convinced him to join my class and give a guest lecture about \(\omega\)-sequences. You can watch the lecture <a href="/courses/course-v1:MITx+24.118x+2T2020/jump_to_id/fc1caa34aee146c098b22076510dbeaa" target="[object Object]">here</a>. (The lecture is partly about the Hat-Problem of the next section, so you might consider checking out that material first.)<br /></span></p>
<p></p>
<p></p>
</div>
</div>
</div>
</div>
© All Rights Reserved