This chapter is a collection of situations that are brought up as problems to be solved by various philosophers.
There are only three ways of completing a proof:
The circular argument, in which the proof of some proposition presupposes the truth of that very proposition
The regressive argument, in which each proof requires a further proof, ad infinitum
The dogmatic argument, which rests on accepted precepts which are merely asserted rather than defended
This is a dilemma for classical logic in particular; it could be addressed by introducing nonmonotonic logic (default and challenge). Some claims (e.g. first person observations) come with a default justification (which is not based on the justification of other claims). Yet it must be defended when challenged.
Original argument appearring in Chapters II and III of [1] (1893), and S.E.P. commentary here:
Bradley notes that there appears to be such a thing as ‘a lump of sugar’, and this thing appears to have qualities such as whiteness, sweetness, and hardness.
But what is this “thing” that bears properties?
On the one hand, it’s odd to assume that there is something to the lump of sugar beside its several qualities
So, postulating a property-less bearer of properties is incoherent.
On the other hand, he notes that the lump cannot merely be its qualities either, since the latter must somehow be united.
For Bradley, unity or “coexistence” of qualities presupposes relations.
But the ontology of relations is prone to the following infinite regress:
We postulate the relation \(C\) that relates properties \(A\) and \(B\)
We then need a relation to relate \(C\) to \(A\) (and to \(B\)).
An easy way to tell if someone saying \(P\) means \(\phi\) rather than \(\psi\) by some observable concept is to show them \(\phi\)’s (witness them say \(P\)) and \(\psi\)’s (and witness them not say \(P\)). Call this witnessing the distinction via dispositions of the \(P\)-speaker. Disjunctivitis is a thought experiment that says this is not a sufficient tool for us to distinguish \(\phi\)’s and \(\psi\)’s generally. This is a Kripkensteinian problem.
We’re worried about whether the word ‘porcupine’ really means porcupine, or whether it means porcupine or echidna. Because most of us porcupine users can’t tell porcupines from Australian echidnas. And, although everything I’ve ever seen in called a ‘porcupine’ was a porcupine, everything I’ve ever seen called a ‘porcupine’ was also a porcupine or echidna. And the question is, well, what what fact is it in virtue of which ‘porcupine’ means porcupine, and not porcupine or echidna?
You can’t tell it from my dispositions on the output side, and on the input side, though, it might have been equally true that they’ve all been porcupines they’ve also been porcupines or echidnas. (Furthermore, for all I know, they all happened to have been been male porcupines, in which case we have to worry that my word porcupine means male, porcupines and again, I can’t tell them from the female ones.
Solution from Sellars: the people who taught you the word porcupine, what pattern were they trying to inculcate in you? Was it to say ‘porcupine’ when confronted with a porcupine or echidna? This is a crucial piece of how the noise ‘porcupine’ coming out of my mouth means porcupine.
A problem metaethical theories belonging that are emotivist/ expressivist. Also sometimes called the embedding problem.
Theories in the noncognitivist tradition share the view that the distinctive meaning of moral words does not concern what they are about, and it either does not require or is not exhausted by any answer to what makes moral sentences true. For example, according to A. J. Ayer, the word ‘wrong’ works more like ‘dammit’ than like ‘common’, so that ‘stealing money is wrong’ means something more like, ‘dammit, stealing money!’ than like ‘stealing money is common’. But standard ways of understanding the meanings of complex sentences, and of understanding the logical relationships between sentences, depend on an answer to what those sentences are about, or what would make them true. So noncognitivists need a different, nonstandard, answer to how the meanings of simple sentences give rise to the meanings of complex sentences. The problem of how to do so, and of whether it can even be done, has come to be known as the Frege-Geach problem.
To follow a rule (e.g. ‘under circumstances \(C\), do \(A\)’, or ‘Whenever you see a cat, raise your hand’), we have to understand the concepts that are involved within (i.e. \(C\) or cat, or raising one’s hand).
However, to grasp a concept (such that the original rule can be followed) requires a further rule.
Put another way, because rules have many interpretations, for any rule \(R\) we need another rule which tells us whether or not we correctly applied \(R\).
If a law says ‘Every man must serve in the army’, then it will naturally require a law for determining who qualifies as ‘man’. That law (say, ‘A man is whatever the scientific experts label with the word man’) will naturally require a law for determining who qualifies as ‘scientific expert’… the regress will continue if we try to adjudicate ‘scientific expert’ with another rule.
This skepticism about rules is really a skepticism about a certain theory of meaning (i.e. semantic skepticism).
There are many flavors, but consider the brain in a vat thought experiment:
You are sleeping one night, when the scientist takes out your brain, puts it in a vat, and connects its neurons to a machine that gives/receives electrical inputs such that the brain activity is identical to certain possible real life scenarios if the brain was still in the body.
The brain in the vat could be having the same experiences I am having right now, so why can I be confident that I am not a brain in a vat?
It seems there is no problem thinking about what triangles and white things are, but what of triangularity and whiteness?
This seems to be an instance of the tooth pain issue. Noun-ness was for ordinary objects (with obvious ontological status) in an earlier game, but our creativity with language led us to nounify many other words, leading to ‘objects’ with unclear ontological status.
Realism asserts that universals are real things, that triangularity does refer to something in the world, although perhaps some metaphysical realm.
Plato is an example.
Berkeley rejected Locke’s view of abstract ideas on empiricist grounds. For him triangularity is just a name that refers to all triangles, rather than referring to something in particular.
There are no abstract ideas, just particulars.
One can accept that universals refer to things in the world but be clear that they are mental things (concepts). Locke is an example.
By Lewis Caroll in 1895. [2]
The Tortoise assumes a proposition \(p\) and a material conditional \(p \implies q\).
The exact \(p\) and \(q\) aren’t important to the moral of the story, though it’s something like “If \(A=B\) and \(B=C\) (\(p\)), then \(A=C\) (\(q\))”
The Tortoise is playing a game: I’ll do anything you tell me to do, so long as you make explicit the rule you’re asking me to follow.
Achilles tries to convince the Tortoise to accept \(q\).
He says that logic obliges you to acknowledge \(q\) in this case.
The Tortoise is willing to go along with this but demands that this rule be made explicit:
Achilles adds an extra axiom: \(p \land (p \implies q) \implies q\).
Achilles says that, now, you really have to accept \(q\), given that you’re committed to:
\(p\)
\(p \implies q\)
\(p \land (p \implies q) \implies q\).
But the Tortoise notes that, if taking those three propositions and concluding \(q\) is really something logic obliges one to do, then it bears writing down:
\(p \land (p \implies q) \land (p \land (p \implies q)) \implies q\)
This can go ad infinitum; the Tortoise wins.
The most influential pragmatist work in the philosophy of logic.
The lesson:
in any particular case, you can substitute a rule (that tells you you can go from this to that) with an axiom.
But there have got to be some moves you can make without having to explicitly license them by a principle.
I.e. you’ve got to distinguish between 1.) premises from which to reason 2.) principles in accordance with which to reason.
This teaches an un-get-over-able lesson about the necessity for an implicit practical background of making some moves that are just okay. Things that would be put in a logical system, not in the forms of axioms, but in the form of rules.
(This is from one of his Sellars lectures)
This seems analogous to descriptivism vs expressivism.
It illustrates what we lose when we reduce all discourse to descriptive discourse. When we choose our definitions such that natural laws are facts in the world, just like any other ordinary empirical fact, we lose both:
Their role in reasoning
A story for our knowledge/justification of them.
The story shows how treating a rule as a fact strips it of its normative force. It is the problem of conflating description in the narrow sense with description in the wider sense: see here.
The last line of this commentary makes me think this can also be used to counter some forms of radical skepticism, i.e. to recover an air of dignity to making working assumptions.