Logic

The prevailing view in North American philosophical writing seems to be that the phrase ‘just in case’ can be translated into the phrase ‘if and only if’. Consequently, this view holds that the phrase ‘just in case’ is best symbolized by the logical connective known as the biconditional (\leftrightarrow).

Now, this seems wrong to me for two reasons. One is the difference between ‘just in case’ in this sense and the sense it has in British English, as noted by Geoffrey K. Pullum:

  • British English: “We’ll bring an umbrella just in case it rains.”
  • American philosophers: “A formula is a tautology just in case it is true on all valuations.”

That’s a fine difference to note, but I also have a hard time grasping why ‘just in case’ should count as ‘if’ and ‘only if’ at all. That is, to me, ‘just in case’ sounds more like ‘only if’. It seems that it spells out a necessary condition but not necessarily a sufficient one. Consider:

  • Something is a tree just in case it is a plant.

Now, according to what seems to be the standard view, this is a false statement because something can be a plant and not a tree. That is, ‘if something is a plant then it is a tree’ is false, so this sentence, just like ‘something is a tree if and only if it is a plant’ is false.

But it seems to me that this sentence actually means ‘something is a tree only in the case that it is a plant’. That is, I’m more inclined to translate ‘just in case’ as ‘only if’. Under such a translation, the above sentence is true, because being a plant is a necessary condition for treehood.

The problem is that the lexical definition of ‘just’, as an adverb, spells out multiple meanings. One is ‘exactly’ or ‘precisely’, which supports the prevailing intuition that ‘if and only if’ best captures the meaning of ‘just in case’–that it means ‘exactly in the cases that’. But there is also the meaning ‘only’ or ‘simply’. This is the source of my intuition.

Meanwhile, it seems that a number of students in elementary logic classes agree with me, since I often see them translating ‘P just in case Q’ into something like ‘P\to Q‘. Officially their textbook and notes equate ‘just in case’ with ‘if and only if’, so I’m not meant to give them the marks for this, but I do empathize.

Logic

To write natural deduction proofs in LaTeX, I use a package called fitch.sty. The package was written by Johan W. Klüwer and offers a nice clean way to typeset Fitch-style proofs. He provides a nice example:

Lovely. However, in some of my proofs, I wanted to have lines without numbers because they featured information that was not strictly part of the proof. For instance, like others, I commonly add a line that indicates the formula we’re out to prove after the list of premises. This is especially useful in teaching proofs. That line, I don’t want numbered — instead I want the counter to skip that line and continue after it, like so:

I had to dig around in the fitch.sty file itself to figure out how to do this, since there’s not really any documentation outside of it. I figured I’d share what I did for anyone facing the same issue.

Here’s what you do. Instead of beginning a line with “\fa” or something like that, add a line like this:

\ftag{~}{\vline\hspace{\fitchindent} CONTENT } \\

Where CONTENT is replaced by whatever you want to have on that line. The exact code for the my ‘∴ B’ line, for example, is:

\ftag{~}{\vline\hspace{\fitchindent} \fbox{$\therefore~ B$}} \\

And that’s all there is to it. I hope this helps someone looking to do the same thing as I was.

Happy typesetting!

Logic, Philosophy

Here’s a tautology in propositional logic:

⊨(P → Q) ∨ (Q → R)

Try throwing that into English. Here’s a reading using some propositions I just came up with:

“I’ll die if I’m immortal, or I’ll live if I die.”

Obviously, neither of those are the case. But this formula, (P → Q) ∨ (Q → R), is both provable and self-implied in classical propositional logic.

Here’s a syntactic proof by means of natural deduction using some basic rules of inference:

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Epistemology, Logic

In his essay “Empiricism and the Philosophy of Mind” (1955-56), Wilfrid Sellars launches an attack against sense-datum theorists. I don’t want to defend sense-data because I think that’s a flawed concept, but I do want to point out a misstep that Sellars makes in this paper.

This objection has probably been made before, but for the sake of my notes I’m spelling it out as I read it.

In the paper, Sellars runs through a set of three propositions that he believes that sense-datum theorists must hold, but that Sellars believes is inconsistent.
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Logic

Story time!

In Logic I at uOttawa last year, I was short on time on an exam and wrote “Magic!” as a justification for a step in natural deduction. I was going for part-marks. Fast-forward to Logic II the following semester, and my professor mentioned in class that you “can’t just write ‘magic’ as a rule of inference” if you’re stuck, as “one student did”. Ha! I promised him that day that I would either prove magic, which, I’m happy to say, I did not, or provide him with an explanation of the rules I used, which I did. My Logic II final exam ended with a page introducing a new operator, magically P, into logic.

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