<p>There isn't really a single formalization of the kind of thing you are asking. There are many, many aspects to truth, trust, lies, and fallible reasoning, and this leads to an enormous variety of logical formalisms, each handling different aspects of this problem. </p>

<ol>
<li><p>If you want to account for uncertainty about your hypotheses, the traditional route is via <em>Bayesian probability theory</em>. See E.T. Jaynes' (sadly incomplete) <em>Probability Theory: the Logic of Science</em> for a good exposition of this point of view.</p></li>
<li><p>One difficulty with probabilistic methods is that it is difficult to interpret quantifiers, essentially because limits do not necessarily exist. That is, a proposition may hold for all finite approximations, but fail to hold in the infinite limit.</p>

<p>Taking this into account leads you to view propositions topologically, which (a) leads you the Beth semantics of <em>intuitionistic logic</em>, and (b) also leads you to <em>geometric logic</em>. See Steve Vickers' <em>Topology via Logic</em> for an introductory exposition, and Peter Johnstone's <em>Stone Spaces</em> to jump into the deep end of the pool. </p>

<p>(However, to my knowledge there is not yet any satisfactory constructive account of probability theory.) </p></li>
<li><p>If you are interested in the difference between <em>truth</em> and <em>belief</em>, then you need to look at <em>epistemic logic</em>. The idea is to extend traditional logic with a modality $B_X(A)$, which is intended to read "$X$ believes $A$". See <a href="http://plato.stanford.edu/entries/logic-epistemic/">the SEP entry on epistemic logic</a> for an introduction, and follow the bibliographic links in the article for more details. </p></li>
<li><p>However, when you have multiple agents, you also have to think about the difference between truth and <em>assertion</em>. This is particularly important in applications like authorization (e.g., I can look at a file, because the owner of the file has the right to delegate the right to look at it, and says that I can look at it). There is a huge amount of work on this; a good entry point into this literature is in Deepak Garg's PhD thesis, <a href="http://www.mpi-sws.org/~dg/papers/thesis.pdf"><em>Proof Theory for Authorization Logic and its Application to a Practical File System</em></a>. </p></li>
<li><p>Next, one of the traditional principles of logic is <em>ex falso quodlibet</em>: that is, $\bot \to A$ holds for all propositions $A$ — if you assume false, everything follows. If you are thinking about the possibility of believing false things (for example, someone has lied to you, or you have a database with errors in it), this is potentially disastrous. In practical reasoning, you don't want to derive a contradiction and then happily believe everything — you want to find a contradiction, and deduce that you have made an error. </p>

<p>The study of what happens to logic when you drop ex falso is called <em>relevance logic</em>, so named because the idea is that you should only make inferences from hypotheses that are relevant to the conclusion. Again, see <a href="http://plato.stanford.edu/entries/logic-relevance/">the SEP article on relevance logic</a> for more. Also, you might wish to tolerate contradictions in your logic system. In this case, the thing to look at is <a href="http://plato.stanford.edu/entries/logic-paraconsistent/"><em>paraconsistent logic</em></a>. </p></li>
<li><p>Next, you mentioned a worry that individual inferential steps might not be completely reliable. In traditional logic we accept the unrestricted use of modus ponens. That is, inside a proof, if we know $A \to B$ and $A$ hold, we can conclude $B$ holds, any number of times. If you start to consider the idea that individual inferential steps might not be completely reliable, then you might think that proofs depending on long chains of inferential steps are "less reliable" than short ones. </p>

<p>It's a bit tricky to formalize this, but it is one of the (several) motivations for <em>ultrafinitism</em>. See Mannucci and Cherubin's draft <a href="http://arxiv.org/abs/cs/0611100"><em>Model Theory of Ultrafinitism I: Fuzzy Initial Segments of Arithmetic</em></a>, for an exploration of this idea (and some explanation of its connection to fuzzy logic). </p></li>
</ol>

<p>Finally, note that that <em>none</em> of these approaches actually talk about Frankfurt's idea of bullshit as an assertion which is made with an indifference to its truth value. You probably want to look at J.L. Austin's theory of speech acts (e.g., his book <em>How to Do Things With Words</em>) to help organize your thinking on this, and if you try to formalize it, you will probably find Per Martin-Löf's judgemental methodology (see for example <a href="http://docenti.lett.unisi.it/files/4/1/1/6/martinlof4.pdf">his Siena lectures</a>) helpful. </p>

CC BY-SA 3.0
2013-03-01T09:46:09.753Z
2013-03-05T14:12:34.273Z
2013-03-05T14:12:34.273Z
Neel Krishnaswami
<p>I was reading Harry Frankfurt's <a href="http://rads.stackoverflow.com/amzn/click/0691122946" rel="nofollow">On Bulls*t</a>, a 1986 philosophical essay about this blurry notion between truth and falsity.</p>

<p>This is not a gratuitous exercise.  This may have applications to computer science, since we are always <a href="https://github.com/spotify/luigi" rel="nofollow">piping data-sets into each other</a>.  Some of these data sources may be specious, the piping process can break down, or the <a href="http://www.wired.com/opinion/2013/02/big-data-means-big-errors-people/" rel="nofollow">conclusions we draw from them</a> can be wrong as well.</p>

<p>One way of approaching Frankfurt's theory may be to express in terms of logical circuits, where the integrity of the gates or the inputs may be in question.</p>

<p>On pencil and paper, we mostly use boolean logic with values $T,F$ and gates $\mathbf{not},\vee,\wedge$.  Maybe it is possible to perturb boolean logic slightly to model how circuits are robus or break down with respect to noise.</p>

<p>Do logical theories exist accounting for doubt and uncertainty? Can we measure how much a lie hurts the integrity of a conclusion?</p>

<p>I'm sure that even with a collection of verifiably true or false statements, it's possible to write arguments (and conclusions) whose values are in the middle.  Or even to decide if one argument is "more" valid than another.</p>

<hr>

<p>I apologize ahead of time, if there's no single question here.</p>

<p><strong>COMMENTS</strong></p>

<p>Logic is a very broad subject, but I'm not a logician so I'm not sure how to be more specific.  Ease of use is a priority, which is why I consider just bootstrapping Boolean logic. </p>

<p>I think that when we "call out" a proposition... the conclusion may be true, but the thought process may be wrong, as VijayD suggests in the comments.  </p>

<p>It's not clear if bulls**t is the same as uncertainty -- we may be quite sure the proof is wrong.</p>

<p>I think it would be nice to see an extension of boolean logic, which assigns a value to <strong>proofs</strong> rather than <strong>statements</strong>.  A proof where all the steps are valid gets assigned a value of <strong>T</strong>, if steps are faulty, we'd like to measure to what extent the conclusion doesn't follow from the premises.</p>

<p>This idea must have been tried before.  A Google search comes up with the notions like <a href="http://en.wikipedia.org/wiki/Heyting_algebra" rel="nofollow">heyting algebra</a>, <a href="http://en.wikipedia.org/wiki/Topos" rel="nofollow">topos</a>, <a href="http://en.wikipedia.org/wiki/Many-valued_logic" rel="nofollow">multi-valued logic</a> and even more sources in the comments and answer.</p>

CC BY-SA 3.0
2013-02-28T14:38:28.583Z
This is a really broad question. There are many studies about multivalued logics, which can incorporate values like "unknown". Fuzzy logics deal with a continuous degree of truth, other logics include [knowledge](https://www.google.co.il/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&ved=0CDUQFjAA&url=http%3A%2F%2Fwww.cs.rice.edu%2F~vardi%2Fpapers%2Fbook.pdf&ei=mGsvUaPwEJGVswb4k4CwBQ&usg=AFQjCNE8pC-PqWEa0HRzUmWxoSM61ca1aA&sig2=ba5VA3OD-5sCLe2cNi0xsg&bvm=bv.42965579,d.Yms) or [uncertaintly](http://mitpress.mit.edu/books/reasoning-about-uncertainty). In short - lots of work on this subject.
Shaull
CC BY-SA 3.0
2013-02-28T15:25:57.177Z
How can I make this more specific?  I'm particularly interested in notions of "bullsh*t" and lying, and making the precise and usable.  There are many situations (in boolean logic) where we question the validity of an argument or conclusion, yet some arguments are more valid than others.
john mangual
CC BY-SA 3.0
2013-02-28T15:31:33.483Z
Perhaps somewhat unrelated, but lying is well studied in strategy-proof classification (e.g. [Reshef Meir's M.Sc thesis](http://www.cs.huji.ac.il/~reshef24/spc.thesis.pdf)). But it's not in the logic perspective of things.
Shaull
CC BY-SA 3.0
2013-02-28T15:45:03.787Z
The question is very open-ended, but I like Harry Frankfurt and I like the idea of trying to formalize what he's doing. (This almost sounds like a question that Manuel Blum would ask!) Still I think it is generally difficult to give appropriate answers in this kind of forum. Someone might point you to existing literature on uncertainty in logic, but it's unlikely we will be able to help you formalize bullsh*t.
Ryan Williams
CC BY-SA 3.0
2013-02-28T19:23:44.343Z
The word uncertainty is used in different senses in the logical literature. I suggest Joseph Halpern's Reasoning about Uncertainty as one starting point.
Vijay D
CC BY-SA 3.0
2013-02-28T19:26:10.303Z
I believe the statements used alone are not sufficient and the structure of the thought process should be considered. Mathematically it seems as if the question is whether given a proof, you can perturb some of the hypotheses and still have a proof.
Vijay D
CC BY-SA 3.0
2013-02-28T20:58:53.980Z
Maybe what you want is a computational model allowing for random errors in the process of computing. One can then ask which problems admit solution by an algorithm which is fault-tolerant in some sense
Vanessa
CC BY-SA 3.0
2013-03-01T03:10:36.523Z
Also check the handbook of logic in computer science and AI. There are various many valued logics, and some of them do focus on uncertainty. There is also work in KRR on decision making under uncertainty. I agree with others that the question is too broad and vague.
Kaveh
CC BY-SA 3.0
2013-03-01T18:52:34.147Z
?!? is it just me or does this whole Q/A feel like verging on an early april fools joke? its a question for another forum probably. trying to take this seriously as everyone else. frankfurt seemed to be alarmed by postmodernisms focus on subjectivity-at-expense-of-objectivity and this forum is dedicated to a premise quite the polar opposite of that. it seems fundamentally sociological/abstract/non-formal(izable). however something famous/notorious/vaguely related to consider is the [sokal hoax](http://en.wikipedia.org/wiki/Sokal_affair)
vzn
CC BY-SA 3.0
2013-03-02T07:18:18.033Z
Assigning values to proofs is more complicated for various reasons, but there are still works on similar ideas. However I don't wee why you want that from your question. Assigning trueness to statements seems more like what you need. For example some many valued logic like fuzzy logics are designed to capture uncertainity. However as long as you don't define what you exactly mean by uncertainity and how the trueness of a statement is measured you won't get a specific answer.
Kaveh
CC BY-SA 3.0
2013-03-02T07:21:04.710Z
Your question is not in a form that can be answered right now other than pointing you to references or giving an overview as Neel did. We can't read your mind or your intentions or what you want to do. Neel's answer is a good overview of the possibilities, and as I said above, a good place to look is the handbook of logic in CS and AI. Have a look at it.
Kaveh
CC BY-SA 3.0
2013-03-02T16:07:07.233Z
continuing the thought, arguably the correct analog to Frankfurts ideas about "bs" in technical fields are "crank proofs" that make claims on famous open problems, see eg [woeginger P vs NP page](http://www.win.tue.nl/~gwoegi/P-versus-NP.htm). in CS its captured by the saying "garbage in, garbage out". see also rj lipton on ["mathematical diseases"](http://rjlipton.wordpress.com/2009/11/04/on-mathematical-diseases/). tao also has a blog on the [3n+1 problem](http://terrytao.wordpress.com/2011/08/25/the-collatz-conjecture-littlewood-offord-theory-and-powers-of-2-and-3/) as a case of it.
vzn
CC BY-SA 3.0
2013-03-02T17:04:19.997Z
these are good suggestions.  i will probably just neel a check.
john mangual
<lo.logic><ds.data-structures><circuit-complexity><randomized-algorithms><db.databases>
logic in the presence of doubt, uncertainty, lies
john mangual
# logic in the presence of doubt, uncertainty, lies

(Asked by: username_0 on 2013-02-28)

I was reading Harry Frankfurt's [On Bulls\*t](http://rads.stackoverflow.com/amzn/click/0691122946), a 1986 philosophical essay about this blurry notion between truth and falsity.

This is not a gratuitous exercise. This may have applications to computer science, since we are always [piping data-sets into each other](https://github.com/spotify/luigi). Some of these data sources may be specious, the piping process can break down, or the [conclusions we draw from them](http://www.wired.com/opinion/2013/02/big-data-means-big-errors-people/) can be wrong as well.

One way of approaching Frankfurt's theory may be to express in terms of logical circuits, where the integrity of the gates or the inputs may be in question.

On pencil and paper, we mostly use boolean logic with values $T,F$ and gates $\mathbf{not},\vee,\wedge$. Maybe it is possible to perturb boolean logic slightly to model how circuits are robus or break down with respect to noise.

Do logical theories exist accounting for doubt and uncertainty? Can we measure how much a lie hurts the integrity of a conclusion?

I'm sure that even with a collection of verifiably true or false statements, it's possible to write arguments (and conclusions) whose values are in the middle. Or even to decide if one argument is "more" valid than another.

---

I apologize ahead of time, if there's no single question here.

**COMMENTS**

Logic is a very broad subject, but I'm not a logician so I'm not sure how to be more specific. Ease of use is a priority, which is why I consider just bootstrapping Boolean logic.

I think that when we "call out" a proposition... the conclusion may be true, but the thought process may be wrong, as VijayD suggests in the comments.

It's not clear if bulls\*\*t is the same as uncertainty -- we may be quite sure the proof is wrong.

I think it would be nice to see an extension of boolean logic, which assigns a value to **proofs** rather than **statements**. A proof where all the steps are valid gets assigned a value of **T**, if steps are faulty, we'd like to measure to what extent the conclusion doesn't follow from the premises.

This idea must have been tried before. A Google search comes up with the notions like [heyting algebra](http://en.wikipedia.org/wiki/Heyting_algebra), [topos](http://en.wikipedia.org/wiki/Topos), [multi-valued logic](http://en.wikipedia.org/wiki/Many-valued_logic) and even more sources in the comments and answer.

**Tags:** lo.logic, ds.data-structures, circuit-complexity, randomized-algorithms, db.databases

> **Comments:**
> **username_1**: This is a really broad question. There are many studies about multivalued logics, which can incorporate values like "unknown". Fuzzy logics deal with a continuous degree of truth, other logics include [knowledge](https://www.google.co.il/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&ved=0CDUQFjAA&url=http%3A%2F%2Fwww.cs.rice.edu%2F~vardi%2Fpapers%2Fbook.pdf&ei=mGsvUaPwEJGVswb4k4CwBQ&usg=AFQjCNE8pC-PqWEa0HRzUmWxoSM61ca1aA&sig2=ba5VA3OD-5sCLe2cNi0xsg&bvm=bv.42965579,d.Yms) or [uncertaintly](http://mitpress.mit.edu/books/reasoning-about-uncertainty). In short - lots of work on this subject.  
> **username_0**: How can I make this more specific?  I'm particularly interested in notions of "bullsh*t" and lying, and making the precise and usable.  There are many situations (in boolean logic) where we question the validity of an argument or conclusion, yet some arguments are more valid than others.  
> **username_1**: Perhaps somewhat unrelated, but lying is well studied in strategy-proof classification (e.g. [Reshef Meir's M.Sc thesis](http://www.cs.huji.ac.il/~reshef24/spc.thesis.pdf)). But it's not in the logic perspective of things.  
> **username_2**: The question is very open-ended, but I like Harry Frankfurt and I like the idea of trying to formalize what he's doing. (This almost sounds like a question that Manuel Blum would ask!) Still I think it is generally difficult to give appropriate answers in this kind of forum. Someone might point you to existing literature on uncertainty in logic, but it's unlikely we will be able to help you formalize bullsh*t.  
> **username_3**: The word uncertainty is used in different senses in the logical literature. I suggest Joseph Halpern's Reasoning about Uncertainty as one starting point.  
> **username_3**: I believe the statements used alone are not sufficient and the structure of the thought process should be considered. Mathematically it seems as if the question is whether given a proof, you can perturb some of the hypotheses and still have a proof.  
> **username_4**: Maybe what you want is a computational model allowing for random errors in the process of computing. One can then ask which problems admit solution by an algorithm which is fault-tolerant in some sense  
> **username_5**: Also check the handbook of logic in computer science and AI. There are various many valued logics, and some of them do focus on uncertainty. There is also work in KRR on decision making under uncertainty. I agree with others that the question is too broad and vague.  
> **username_6**: ?!? is it just me or does this whole Q/A feel like verging on an early april fools joke? its a question for another forum probably. trying to take this seriously as everyone else. frankfurt seemed to be alarmed by postmodernisms focus on subjectivity-at-expense-of-objectivity and this forum is dedicated to a premise quite the polar opposite of that. it seems fundamentally sociological/abstract/non-formal(izable). however something famous/notorious/vaguely related to consider is the [sokal hoax](http://en.wikipedia.org/wiki/Sokal_affair)  
> **username_5**: Assigning values to proofs is more complicated for various reasons, but there are still works on similar ideas. However I don't wee why you want that from your question. Assigning trueness to statements seems more like what you need. For example some many valued logic like fuzzy logics are designed to capture uncertainity. However as long as you don't define what you exactly mean by uncertainity and how the trueness of a statement is measured you won't get a specific answer.  
> **username_5**: Your question is not in a form that can be answered right now other than pointing you to references or giving an overview as Neel did. We can't read your mind or your intentions or what you want to do. Neel's answer is a good overview of the possibilities, and as I said above, a good place to look is the handbook of logic in CS and AI. Have a look at it.  
> **username_6**: continuing the thought, arguably the correct analog to Frankfurts ideas about "bs" in technical fields are "crank proofs" that make claims on famous open problems, see eg [woeginger P vs NP page](http://www.win.tue.nl/~gwoegi/P-versus-NP.htm). in CS its captured by the saying "garbage in, garbage out". see also rj lipton on ["mathematical diseases"](http://rjlipton.wordpress.com/2009/11/04/on-mathematical-diseases/). tao also has a blog on the [3n+1 problem](http://terrytao.wordpress.com/2011/08/25/the-collatz-conjecture-littlewood-offord-theory-and-powers-of-2-and-3/) as a case of it.  
> **username_0**: these are good suggestions.  i will probably just neel a check.  

---

## 1 answers

### Answer by username_7 on 2013-03-01. Score: 12 | Accepted answer

There isn't really a single formalization of the kind of thing you are asking. There are many, many aspects to truth, trust, lies, and fallible reasoning, and this leads to an enormous variety of logical formalisms, each handling different aspects of this problem.

1. If you want to account for uncertainty about your hypotheses, the traditional route is via *Bayesian probability theory*. See E.T. Jaynes' (sadly incomplete) *Probability Theory: the Logic of Science* for a good exposition of this point of view.
2. One difficulty with probabilistic methods is that it is difficult to interpret quantifiers, essentially because limits do not necessarily exist. That is, a proposition may hold for all finite approximations, but fail to hold in the infinite limit.

   Taking this into account leads you to view propositions topologically, which (a) leads you the Beth semantics of *intuitionistic logic*, and (b) also leads you to *geometric logic*. See Steve Vickers' *Topology via Logic* for an introductory exposition, and Peter Johnstone's *Stone Spaces* to jump into the deep end of the pool.

   (However, to my knowledge there is not yet any satisfactory constructive account of probability theory.)
3. If you are interested in the difference between *truth* and *belief*, then you need to look at *epistemic logic*. The idea is to extend traditional logic with a modality $B\_X(A)$, which is intended to read "$X$ believes $A$". See [the SEP entry on epistemic logic](http://plato.stanford.edu/entries/logic-epistemic/) for an introduction, and follow the bibliographic links in the article for more details.
4. However, when you have multiple agents, you also have to think about the difference between truth and *assertion*. This is particularly important in applications like authorization (e.g., I can look at a file, because the owner of the file has the right to delegate the right to look at it, and says that I can look at it). There is a huge amount of work on this; a good entry point into this literature is in Deepak Garg's PhD thesis, [*Proof Theory for Authorization Logic and its Application to a Practical File System*](http://www.mpi-sws.org/~dg/papers/thesis.pdf).
5. Next, one of the traditional principles of logic is *ex falso quodlibet*: that is, $\bot \to A$ holds for all propositions $A$ — if you assume false, everything follows. If you are thinking about the possibility of believing false things (for example, someone has lied to you, or you have a database with errors in it), this is potentially disastrous. In practical reasoning, you don't want to derive a contradiction and then happily believe everything — you want to find a contradiction, and deduce that you have made an error.

   The study of what happens to logic when you drop ex falso is called *relevance logic*, so named because the idea is that you should only make inferences from hypotheses that are relevant to the conclusion. Again, see [the SEP article on relevance logic](http://plato.stanford.edu/entries/logic-relevance/) for more. Also, you might wish to tolerate contradictions in your logic system. In this case, the thing to look at is [*paraconsistent logic*](http://plato.stanford.edu/entries/logic-paraconsistent/).
6. Next, you mentioned a worry that individual inferential steps might not be completely reliable. In traditional logic we accept the unrestricted use of modus ponens. That is, inside a proof, if we know $A \to B$ and $A$ hold, we can conclude $B$ holds, any number of times. If you start to consider the idea that individual inferential steps might not be completely reliable, then you might think that proofs depending on long chains of inferential steps are "less reliable" than short ones.

   It's a bit tricky to formalize this, but it is one of the (several) motivations for *ultrafinitism*. See Mannucci and Cherubin's draft [*Model Theory of Ultrafinitism I: Fuzzy Initial Segments of Arithmetic*](http://arxiv.org/abs/cs/0611100), for an exploration of this idea (and some explanation of its connection to fuzzy logic).

Finally, note that that *none* of these approaches actually talk about Frankfurt's idea of bullshit as an assertion which is made with an indifference to its truth value. You probably want to look at J.L. Austin's theory of speech acts (e.g., his book *How to Do Things With Words*) to help organize your thinking on this, and if you try to formalize it, you will probably find Per Martin-Löf's judgemental methodology (see for example [his Siena lectures](http://docenti.lett.unisi.it/files/4/1/1/6/martinlof4.pdf)) helpful.


---

