{"id":12973,"date":"2025-09-04T13:53:52","date_gmt":"2025-09-04T12:53:52","guid":{"rendered":"https:\/\/www.blopig.com\/blog\/?p=12973"},"modified":"2025-10-03T23:01:05","modified_gmt":"2025-10-03T22:01:05","slug":"exploding-barbers-paradoxes-part-i","status":"publish","type":"post","link":"https:\/\/www.blopig.com\/blog\/2025\/09\/exploding-barbers-paradoxes-part-i\/","title":{"rendered":"Exploding Barbers (Paradoxes \u2014 Part I)"},"content":{"rendered":"\n

Prelude<\/h2>\n\n\n\n

I came upon a traveller on a dust-swept road at dusk.<\/em>
Along the cliff’s high edge it ran, where seabirds rode the gust;
Upon a stone he rested still, with gaze toward the deep,
As though the sea held secrets vast that mortals may not keep.
Behind us wound the ancient way through heather wild and wood,
To where a castle, firm and fair, upon the hilltop stood.<\/em><\/p>\n\n\n\n\n\n\n\n

Then spake the man, his voice a husk: \u201cNow whither goest thou?\u201d
\u201cTo yonder keep,\u201d I gave reply, \u201cthat crowns the ridgehead brow.\u201d
He nodded once, as if he knew the place as well as I,
And rose as slow as starlight wakes amid a darkening sky.
\u201cThen let us tread,\u201d said softly he, “this path so long and steep\u2014
for those who climb such roads alone may find their footing weak.\u201d

We trod a while in silence ’til I asked him for his name;
He smiled but said, \u201cWhat’s lost to time is seldom worth reclaim.\u201d
But I shall tell thee of my kin\u2014my brother, eldest born,
A barber dwelling in a vale, both dutiful and sworn.
He shaves the men within his town who shave not for their own\u2014
Yet shaves he not those gentlemen who tend their beards alone.\u201d

I paused, then frowned, and shook my head. \u201cBut how can this be true?<\/em>
If he should shave himself, then lo!\u2014he breaks his rule in two.
Yet if he shave not his own chin, then by his law he must.
The man you speak of cannot be\u2014he’s naught but wind and dust.\u201d
The traveller smiled with cryptic mien, and met my searching eye:
\u201cPerhaps,\u201d he said, \u201cyet still he lived\u2014at least as well as I.\u201d<\/em><\/p>\n\n\n\n

Basics<\/h2>\n\n\n\n

This ill-judged little lyrical prelude furnishes our discussion for some elementary remarks on classical logic, starting with propositions<\/em>. In a very loose sense, propositions are simply statements of (potential) fact about the world\u2014some world, at least. Not everything we might call a \u201cstatement\u201d in everyday speech is a proposition, but a lot of those we really care about are<\/em>: “water boils at 100\u00b0C”; \u201cthe Earth is not flat\u201d; \u201cOdy may be overthinking his blogpost\u201d.<\/p>\n\n\n\n

Propositions may be true (\"T\") or false (\"F\")\u2014the above are all true, in case you\u2019re wondering\u2014and these concepts are mutually exclusive. Something cannot be true and false at the same time; that would be oxymoronic<\/em>. It may be that the truth value of a proposition is unknown, for lack of information, or it may be that a statement is ambiguous with unclear definitions (in which case it may correspond to multiple possible propositions), but once you nail your definitions down firmly enough, the underlying truth value, known or not, is generally taken to exist (a metaphysically loaded term) \u2018out there\u2019 in the void and it is one of these two: \"T\" or \"F\". So far, so simple.<\/p>\n

Propositions may be composed into new ones by the familiar logical operators (\"\lnot,\land,\lor,\rightarrow,) also known as connectives. Given known truth values for the (atomic) components, they yield the truth value of the composite by mindless, mechanical evaluation. This is what we have built the modern world upon. Such a composite proposition is therefore a truth function<\/em> (typically defined via a truth table<\/em>) with truth values<\/em> as inputs and outputs. Implications<\/em>, i.e. conditional if-then<\/code> propositions like \"p\rightarrow, are particularly important logical connectives. Just like \"p\land, an implication is a truth function: if \"p\" but \"\lnot then \"p\rightarrow is \"F\", for example (in fact only<\/em> then). Some of us like to additionally adorn our propositions with quantifiers (\"\forall,\exists\"), which make the whole formalism quite a bit more expressive and complicate things a bit, but we shall not go into the details here.<\/p>\n

Implications are not be confused with inferences<\/em> which are statements that connect premises (propositions) and a conclusion (also a proposition) into a logical move, such as \"p,\, also written<\/p>\n<\/div>\n\n\n\n

\"\displaystyle<\/p>\n\n\n\n

and itself not a proposition. Inferences are not<\/em> truth functions\u2014they are statements of an argument (a structural relationship<\/em> between propositions, i.e. potential facts), rather than statements of fact themselves\u2014but they do have a binary property: they can be valid<\/em> or invalid<\/em>, and these concepts are mutually exclusive. An inference is (deductively) valid if the conclusion necessarily follows from the premises, i.e. there is no situation in which all<\/em> the premises are \"T\", but the conclusion is not \"T\".<\/p>\n\n\n\n

Implications and inferences alike have some counterintuitive edge cases. The relevant truth table will inform you that \"F\rightarrow evaluates to \"T\", for example; this is known as a vacuously true<\/em> implication. The statement ‘all cell phones in the room are turned off’ is vacuously true<\/em> in this way if there are no cell phones in the room; the converse is also (vacuously) true\u2014they’re also turned off\u2014which means the conjunction is true, as well (they’re both on and off)\u2014this is no contradiction so long as the phones don’t exist1<\/a><\/sup>. Similarly there are vacuously valid<\/em> inferences. To illustrate, let’s consider two propositions: “(O)Pigs can fly.” (\"p\") and “The Queen was2<\/a><\/sup> rich.” (\"q\"). Two independent (non-vacuously) valid inferences constructed from these propositions would be \"(i)\ and \"(ii)\ (each conclusion is always true when all of its premises are; you can go through the truth table algebra, if you don’t believe me). Chaining two valid inferences together \"(i should give us another valid inference, so we plug in \"q\" at the start and get out \"p\". In other words, pigs can fly.<\/p>\n\n\n\n

No, wait. What went wrong? We first assumed \"q\" in \"(i)\" and later assumed \"\lnot in \"(ii)\" \u2014 a contradiction. Therefore there is no situation in which all the premises are true and the conclusion isn’t\u2014in fact, there is no situation in which all the premises are true at all. So the inference is (vacuously) valid<\/em> by definition, but as a conditional statement whose antecedent is always false, it can never lead us to conclude the conclusion. It gives us no information about the world. And I was so looking forward to my flying lessons\u2026<\/p>\n\n\n\n

For this reason vacuous validity does not threaten the integrity of the logical machinery as a means to reason about the world and derive novel facts from known ones. It does demonstrate, however, why contradictions like \"q must<\/em> be defined to be \"F\". If so much as a single contradiction were ever \"T\", you could exploit it by exactly the above inference to derive \u2026 anything (ex contradictione quodlibet<\/em>) and blow up the entire edifice. Appropriately, this is known as the principle of explosion.<\/p>\n\n\n\n

The Man That Must Not Be<\/h2>\n\n\n\n

Which brings us back to the traveller’s elder brother, the barber who shaves all those\u2014and only those\u2014who do not shave themselves. Does he shave himself? Well, if he does, he mustn’t; and if he doesn’t, he must3<\/a><\/sup>. A contradiction. We can show this symbolically, too: let’s define \"P(x)\" as “\"x\" is a person that needs regular shaving” and \"S(x,y)\" as “\"x\" shaves \"y\"“. We then describe the barber thus: \"\exists. Now, \"y\" is universally quantified, so there comes a time when we must evaluate the case \"y=x\", in which case we get \"\lnot, which is a contradiction (and hence \"F\"). So we end up with \"\exists, which, believe it or not, is \"F\". No such barber can exist. He must<\/em> not exist. <\/p>\n\n\n\n

… right?<\/p>\n\n\n\n

The plot thickens in Part II (coming soon)\u2026<\/em><\/p>\n\n\n\n

Footnotes<\/h2>\n\n\n
  1. Defining \"F\rightarrow (an implication\u2014i.e. conditional\u2014with an antecedent that is false or fails to refer to anything) as \"T\" rather than \"F\" is actually quite important for the syntax to work as one would expect. We want to be able to make statements like \"\forall and have them be true even if there are \"x\" for which \"P(x)\" fails and \"Q(x)\" happens to be true. \"F\rightarrow evaluates to \"T\" for the same reason. \u21a9\ufe0e<\/a><\/li>
  2. RIP \u21a9\ufe0e<\/a><\/li>
  3. At least if we assume he needs shaving in the first place\u2014work with me here. \u21a9\ufe0e<\/a><\/li><\/ol>","protected":false},"excerpt":{"rendered":"

    Prelude I came upon a traveller on a dust-swept road at dusk.Along the cliff’s high edge it ran, where seabirds rode the gust;Upon a stone he rested still, with gaze toward the deep,As though the sea held secrets vast that mortals may not keep.Behind us wound the ancient way through heather wild and wood,To where […]<\/p>\n","protected":false},"author":125,"featured_media":13012,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"nf_dc_page":"","wikipediapreview_detectlinks":true,"_monsterinsights_skip_tracking":false,"ngg_post_thumbnail":0,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"[{\"id\":\"865a771c-bb4c-4ee4-8d30-a9a049be2b27\",\"content\":\"Defining $latex F\\\\rightarrow T$ (an implication\\u2014i.e. conditional\\u2014with an antecedent that is false or fails to refer to anything) as $latex T$ rather than $latex F$ is actually quite important for the syntax to work as one would expect. We want to be able to make statements like $latex \\\\forall x:\\\\ P(x)\\\\rightarrow Q(x)$ and have them be true even if there are $latex x$ for which $latex P(x)$ fails and $latex Q(x)$ happens to be true. $latex F\\\\rightarrow F$ evaluates to $latex T$ for the same reason.\"},{\"id\":\"7fdaf532-7c7d-49b7-bea8-5d0970fe3785\",\"content\":\"RIP\"},{\"id\":\"97b6fe8a-0fd3-4de6-ad97-6e4a9f0fb6ea\",\"content\":\"At least if we assume he needs shaving in the first place\\u2014work with me here.\"}]","jetpack_post_was_ever_published":false,"_ppma_block_editor_authors":""},"categories":[123],"tags":[441],"ppma_author":[783],"class_list":["post-12973","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-commentary","tag-logic"],"jetpack_sharing_enabled":true,"authors":[{"term_id":783,"user_id":125,"is_guest":0,"slug":"ody","display_name":"Odysseas Vavourakis","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/b74030bdaef5f39ec32be3ae7bb5af054cbcb0b431b1cc51ba1b41d723ecee48?s=96&d=mm&r=g","author_category":"","user_url":"","last_name":"Vavourakis","first_name":"Odysseas","job_title":"","description":""}],"jetpack_featured_media_url":"https:\/\/i0.wp.com\/www.blopig.com\/blog\/wp-content\/uploads\/2025\/09\/image.png?fit=1536%2C1024&ssl=1","_links":{"self":[{"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/posts\/12973","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/users\/125"}],"replies":[{"embeddable":true,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/comments?post=12973"}],"version-history":[{"count":5,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/posts\/12973\/revisions"}],"predecessor-version":[{"id":13013,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/posts\/12973\/revisions\/13013"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/media\/13012"}],"wp:attachment":[{"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/media?parent=12973"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/categories?post=12973"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/tags?post=12973"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/ppma_author?post=12973"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}