Four Brief Thoughts on Validity & Soundness

1) Modal Logic From the Beginning?

An argument is valid if “the truth of the premises guarantees the truth of the conclusion”.

So how does that actually work?

Regardless of the truth (or otherwise) of the premises and conclusion, what is the relation (in a valid argument) between premises and conclusion? Is it a necessary connection? Is it semantic? Is it syntactic? Or is it logical – full stop?

Moreover, what precisely is meant by the word “guarantee” (it doesn't seem like a word from logic)?

Similarly with the word “impossible” (as in “it's impossible for the premises to be true and the conclusion false”)? What does the modal word “impossible” mean in this context? Is it natural impossibility? Or, again, purely logical (i.e., syntactic)?

Similarly, how do we recognise the soundness and validity of arguments? Again, through semantic connections or through logical (syntactic) form alone? More interestingly, does the logical (syntactic) run entirely free of the semantic?

Modal logic is implied by propositional logic and predicate logic (or first-oder logic). That is, with words such as “necessarily” and “possible”, aren't we moving beyond propositional and predicate logic?

For example, if I say,

It couldn't be possible for the premises to be true and the conclusion false.

that introduces possibility. (Indeed even the world “couldn't” has modal import.)

Similarly,

If the premises are true, then necessarily the conclusion must be true.

That introduces necessity.

Finally, I can say:

It is impossible for the premises to be true and the conclusion to be false.

That, again, introduces possibility.

2) The Cogito and Implicit/Hidden Premises

Some arguments may only have one premise. Thus we move from that single premise to a conclusion.

This seems to be the case with Descartes' Cogito: “I think. Therefore I am.”

It follows, then, that in order for the argument to valid (if not true), the single premise may - or must - have hidden content. That's certainly the case with the Cogito.

The “I think” leads to the conclusion “Therefore I am” because that “I think” has implicit/hidden/co premises (or hidden content). So what is its hidden content?

It's this: “Anything that thinks, must exist.” Then it can be said that “I think. Therefore I am” is effectively a tautology in that the “I think” itself contains the notion of the speaker's (or thinker's) necessary existence. In other words, existence is implied in the premise - “I think”. Thus:

i) I, a living and existing being, think.

ii) Therefore I am.

Or the implicit premise can be even more detailed or broad. Thus:

i) If a thing thinks,
ii) then it must exist.iii) I think.
iv) Therefore I am.

Thus we have two conditionals (or one conditional within another conditional). Thus:

i) If a thing thinks,

ii) then it must exist.

and then:

i) I think.

ii) Therefore I exist.

There are other examples of a one-premise argument.

For example,

i) The world is flat.

ii) Therefore the world is not mountainous.

Or:

i) Jim is a gay.

ii) Therefore, Jim's not heterosexual.

This is because, again, there are implicit premises involved. Thus in the following

I) Jim is a bachelor.

ii) Therefore Jim's an unmarried man.

the implicit premise is:

No bachelor can also be married.

Similarly with 'gay' and 'heterosexual', as well as with 'flat' and 'mountainous'.

Whereas 'bachelor' and 'unmarried man' can be deemed synonyms, that's not the case with 'flat' and 'mountainous'. In this case we have antonyms rather than synonyms. However, it isn't really the case the 'mountainous' is the antonym of 'flat'. A more accurate antonym of 'flat' would be, say, 'bumpy'. Or, more logically, the purest antonym of 'flat' is, in fact, 'not flat' (except, of course, that antonyms don't usually simply negate the source of the antonym).

3) Validity Without Soundness

An invalid argument can have a true conclusion.

To put it simply: if the conclusion doesn't follow from the premises, then it doesn't matter if it's true or false because, well, it doesn't follow from the premises.

That argument itself works as a conditional. Thus:

i) If a conclusion doesn't follow from the premises of an argument,

ii) then it doesn't matter – logically - if the conclusion is either true or false.

If a conclusion genuinely follows from false premises, then the conclusion can come out false. Again, that would only be the case if the logical moves from the premises to the conclusion are valid. In other words, in this scenario falsity is passed on from premises to conclusion.

What about the case in which the premises are true yet the argument is invalid? In that case, false premises can lead to a true conclusion if the argument is invalid because any conclusion (as already stated) can follow an invalid argument.

The obvious point to make is that because content (or even truth) is unimportant when it comes to recognising a logical form, you can create bizarre arguments which are nevertheless valid (though not sound).

For example,

All corbetts are bricks.

All bricks can solve equations.

Therefore all corbetts can solve equations.

The importance of this lack of a connection between premises and conclusion (or between validity and soundness) can be shown with the example of a true conclusion which follows an invalid argument. Or, more likely, one may not immediately believe that the conclusion is true because of the invalid argument. Thus one may look for a flaw in the argument which led to it. However, even if the argument is invalid, the conclusion can still be true.

4) Either/Or Arguments

The following argument is valid because it's impossible for the premises to be true and the conclusion to be false:

i) Either Corbett eats Cornflakes or he eats Ready Brek.

ii) Corbett doesn't eat Cornflakes.

iii) Therefore Corbett eats Ready Brek.

Of course the obvious question is: Why is this an either/or case? Couldn't Corbett eat neither Cornflakes nor Ready Brek? Sure. However, that would be a factual matter and not the concern of logic. Corbett may eat neither Cornflakes nor any other cereal. Again, that would be irrelevant from a logical point of view. What matters here isn't content or fact, but logical form. More precisely, it's the relation between a disjunctive premise (as in “...or...”) , a premise which is a existential negation (“... does not...”) and a conclusion (“Therefore...).

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A → B

“The words ‘to imply’ in the English language mean the same as ‘to contain’ or ‘to involve’…[note the] confusion of implication with the consequence relation.”

— Rudolf Carnap (1937), The Logical Syntax of Language.

[Note: this is an essay in the philosophy of logic, rather than in (pure) logic.]

Does the Symbolism A → B Always Have a Semantics?

If the symbol A has no meaning (or intension), then how can it imply or entail B? If A is just an inscription (or is self-referential — i.e., an autonym), then what connection could it have to B?

On one reading, A implies B (or B is a consequence of A) because B “contains” (or “involves”) something that’s already also “in” A. This is the standard Kantian view of conceptual analyticity. B, on the other hand, can be a consequence of A without it containing (or involving) something that’s common to A.

What exactly does it mean to say that “B is contained within A”? The American philosopher W.V.O. Quine accused Immanuel Kant of speaking at a metaphorical level when he used the word “containment”. However, what non-metaphorical way of describing what’s at issue here have we got left? If A is simply an inscription (or syntactic form), then of course it can’t contain B — it can’t contain anything except itself.

Symbol A will require semantic content if it’s to to imply B. In that case, it all depends on what the symbol A stands for. Does it stand for a concept, sentence, statement/proposition, fact or state of affairs? All these possibilities have content. (Arguably “content” is a problematic term when it comes to facts or states of affairs.)

For example, if the symbol A stands for the concept [politician], then what content will it have? We can say that contained within the concept [politician] are the macro-concepts [human being], [person], etc; as well as the micro-concepts [professional], [Member of Parliament], etc.

In a certain sense, it’s quite arbitrary to categorise certain concepts as micro-concepts and others as macro-concepts because it all depends on context. However, we can say that in one (or this) context we can categorise the concept [politician] as a micro-concept.

There’s a simple way to decide what’s what. We can ask this question:

Is it necessary for a politician to be a person or a human being?

The answer is (admittedly dismissing contemporary artificial intelligence and aliens) yes.

Then we can ask this question:

Is it necessary for a human being or person to be a politician?

The answer is no.

Thus, in this simple sense, the macro-concepts encompass the micro-concepts. Of course there are also yet higher levels of concept.

Take the concept [animal]. This includes the concepts [human being] and [person]. And there are yet higher-order concepts than that: e.g., [living thing]. This could go on until we reach the concepts [object], [thing], [entity], [a spatiotemporal part] and so on.

If A is taken to be a concept, then it may well have a large amount of content. Thus it could imply all sorts of things. Indeed it’s a strange thing to take A as simply standing for a single concept. (It’s hard to make sense of a concept all on its own.) Thus we need to fill in the dots ourselves.

On the other hand, if the symbol A stands for a sentence, then things become a little clearer and not as broad-ranging. (Usually, however, we have the symbols p and q for sentences or statements.) The sentence may (or will) of course include concepts; though concepts within a sentential framework will probably be more finely delineated and circumscribed. Something will be said about the concepts contained and they’ll also be contextualised.

To state that “the concept [politician] implies the concepts [human being] and [person]” sounds strange. In a sense, the bare concept [politician] isn’t actually saying or implying anything. Thus the idea of containment must be taken more literally in the case of concepts standing for A than if A stands for a statement (or a sentence).

How would B, therefore, be a consequence of A if the relation between A and B weren’t that of conceptual containment? In nature, B can be caused by A without sharing anything at all with A. Perhaps B can also be deduced from A without sharing anything with A. If this is so, then how would that deduction actually come about?

Note:

There are various kinds of conditional in philosophy, mathematics and logic. Among them are the following:

(1) Material conditional = This is found in propositional calculus and in mathematics.
(2) Conditional probability = The probability of an event B given that another event A has occurred.
(3) Causal conditional = If X then Y — where X is a/the cause of Y.
(4) Conditional proof = In logic this is a proof that asserts a conditional, and proves that the antecedent leads to the consequent.
(5) Strict conditional = In philosophy and logic there we often have symbols for propositions (such as p → q); rather than the symbolism A → B. In this case, despite the p’s and q’s rather than A’s and B’s, it’s still not always entirely clear whether state of affairs (or facts) are being discussed or propositions (or statements).
(6) Relevance conditional = In relevance logic there’s a demand that the consequent be “relevant” to the antecedent — or vice versa.

[I can be found on Twitter here.]

A Miniature: Panpsychism & Monism

What is the argument for panpsychism? Is it this? -

1. Since there is consciousness in the complex parts of the world, and
2. the complex is only a sum of simple parts, it must follow that
3. there must also be consciousness in the simple parts of the world.

Premises i and ii seem intuitively true. However, is iii a correct conclusion to premises i and ii? Is it, in fact, true?

There is indeed “consciousness in the complex parts of the world”. For instance, in the brains of human beings (if ‘in’ is the correct word here). And many people agree that consciousness is a result of some level of complexity.

How complex must something be in order to bring about consciousness? Human beings, for example, have billions of neurons in their brains. Ants have, say, tens - or hundreds - of thousands of neurons in their brains. Is this enough to bring about consciousness? Or is there more to this story than simply counting how many neurons a creature has?

If a creature had trillions of neurons, would it be ultra-conscious (or have ultra-consciousness)? And need that complexity only be the complexity of neurons and other aspects of the human brain?

What if we connected a thousand computers up together and programmed them to work as one? If that could be done, then would that large-scale and complex integration automatically bring about consciousness? Alternatively, is it an a priori truth that something simple (like an earthworm or even a pencil) can’t be conscious?

As for premise ii). Yes, it would seem to be the case that “the complex is only a sum of simple parts”. (Indeed the complex must be a sum of simple parts otherwise it wouldn’t be, well, complex.) However, what about the word “only” - as in “only the sum”? Perhaps there's more to a complex machine, creature or thing than simply the addition of simple parts to one another in order to create a complex whole. Perhaps something emerges from the addition of simple parts. Is a nation state, for example, only the sum of its parts? Is a class or set only the sum of its members? Is mind or consciousness only the sum of the parts of the brain (as well as, perhaps, other physical things)?

Was Wittgenstein a Language-games Relativist?

In very simple terms, the philosopher Ludwig Wittgenstein (1889–1951) was concerned with whether concepts are useful or not.

At first glance, Wittgenstein’s position on the introduction of new concepts into different (what he called) language-games (or Sprachspiel) seems very pragmatist or even instrumentalist in nature. (See ‘Wittgenstein and Pragmatism’ and ‘Wittgensteinian instrumentalism’.)

So one may now ask how can such concepts be useful if the entities they refer to don’t in fact exist or have any reality. In other words, doesn’t the usefulness of such concepts depend on — or presuppose — the existence or reality of what it is they refer to? (Wittgenstein specifically wrote about Sigmund Freud’s “subconscious thoughts” and Georg Cantor’s “different infinities”.)

So does “use” rely — or depend — on reality?

[It can be seen in the quote above — from Philosophical Grammar — that Wittgenstein never believed that this issue was only about “use”.]

Isn’t it the case that, say, the concepts ROUND SQUARE and FLAT NUMBER aren’t much use if the entities they refer to don’t exist or aren’t real?

[The word “refer” as just been used. It’s usually words or terms which are said to refer, rather than concepts. Concepts are said to have extensions instead.]

Take the possible worlds of analytic philosophers.

Possible worlds may not be real, However, they have proved to be useful and fruitful when it comes to formalising and clarifying our modal notions. Thus the concepts PHILOSOPHICAL DONKEY and GOLDEN MOUNTAIN may prove useful in some respects. That said, these two examples aren’t exactly equivalent to the concepts ROUND SQUARE and FLAT NUMBER. That’s primarily because there are mountains, there is gold, there are donkeys and there is the phenomenon of being philosophical. And gold and mountain can be juxtaposed without contradiction. So too with donkey and being philosophical. However, roundness and squareness, as well as flatness and number, can’t likewise be juxtaposed.

The American philosopher W.V.O. Quine’s position on (abstract) numbers wasn’t so distant from Wittgenstein’s position on the introduction of new concepts. Quine basically believed that numbers have instrumental use value. Quine also believed that it’s dishonest to deny the existence or reality of numbers when during one’s practice (i.e., in mathematics and physics) one effectively assumes that they do in fact exist or are real. (See ‘Quine–Putnam indispensability’.)

So is it the case that because numbers have instrumental value, then they must also exist or be real?

Why use the concept (to use Wittgenstein’s own example) SUBCONSCIOUS THOUGHTS (or plain SUBCONCIOUS) in our grammar if there are no such things as subconscious thoughts? What possible use could such a concept have if that usage doesn’t depend on the reality or existence of subconscious thoughts?

[Unconscious or non-conscious mental activities shouldn’t be confused with Freud’s and other theorists’ subconscious thoughts.]

Of course we can now debate the concept EXISTENCE itself; which has been used — or hinted at — a couple of times above. However, that’s doesn’t seem to be the point that Wittgenstein was making. In his view, we could introduce just about any concept into our grammar if we deem it useful to do so.

In a sense Wittgenstein was absolutely right.

The concept SUBCONSCIOUS THOUGHTS no doubt does have its uses. In other words, that concept explains many mental and behavioural phenomena.

For example, why does subject S behave in such a self-contradictory manner? It can be supposed that he does so because his subconscious thoughts are having some kind of effect on his conscious mental life and behaviour…

… Yet is it actually the case that S has such subconscious thoughts?

After all, there may be many other explanations of S’s behaviour which don’t include references to subconscious thoughts or to the subconscious generally.

Perhaps Wittgenstein might have responded: Then why not introduce new concepts into your grammar instead?

The only thing that matters, according to this reading of Wittgenstein, is whether or not these new concepts work within the overall domain of our grammar (or within a particular language-game).

Additionally, if old concepts no longer work, then get rid of them.

Some will argue, however, that certain old concepts didn’t work precisely because they had no basis in reality (e.g., PHLOGISTON, ETHER, etc.). That said, the characters and events in works of fiction aren’t actual or real; though they nevertheless prove to be (if in a loose sense) useful. They’re useful in that they display to readers, say, general kinds of situation and general kinds of character. That said, fictional events and characters are often (or always) parasitical on events and characters in the real (or actual) world. Thus such works of fiction work precisely because they indirectly refer (in the minds of the readers) to existents. Indeed even the most extreme works of fictional irrealism must depend on these kinds of indirect reference otherwise their readers wouldn’t relate to the works or even be able to make sense of them.

If we return to Wittgenstein’s general thesis.

Wouldn’t such a fast-and-loose attitude to concept-formation result in a multiplicity of contradictory concepts?

Relativism, Rules and Laws

If practices, customs, “forms of life” (see Wittgenstein’s position here) or language-games are truly autonomous, then perhaps the notion of conceptual contradiction doesn’t really have much purchase. That’s primarily because it presumes the existence of a correct (or true) language-game (or even a meta-practice) which somehow stands above all other language-games to pass its supreme judgement on them.

So if we take Wittgenstein’s (possible) position to its extreme limit, then any group can formulate any concept it wishes. That is, there will be no meta-language-game (or meta-practice) to tell them what concepts they can or can’t formulate. Thus everything everybody says would make sense in the context of the language-game it’s embedded in.

Again, Wittgenstein’s position automatically rules out any meta-language (or meta-language-game) which would try to make sense of the conceptual flux and chaos around it. Thus we would simply need to accept that it’s literally the case that anything goes. And that would primarily be because each individual language-game would be formulating its own rules.

In any case and according to the late Wittgenstein, such rules aren’t (as Ray Monk puts Wittgenstein’s early position) “fixed by immutable laws of logical form”. They are, instead, fixed by custom, practice or forms of life. Therefore, on this reading, rules couldn’t possibly be laws (certainly not “immutable laws”). That’s because laws are (usually) deemed to be universally applicable — i.e., applicable across the board. The rules of a language-game, on the other hand, may only apply to two individual (as it were) players.

Rules are therefore simply contingent conveniences instigated to serve the particular purpose at hand. Indeed they can be disregarded (or amended) according to new situations or purposes.

In may even be the case that — at least some — laws are laws even though they simply and only abide by arbitrary and contingent rules which are themselves custom- or practice-relative. Thus on this reading such laws are custom-built entities which have often (or usually) come to be seen as belonging to something beyond the station of mere rules. In other words, at least some laws are taken to be universal. However, such laws may be no more universal than poached eggs or flared trousers.

Main Reference

Wittgenstein, Ludwig, Philosophical Investigations (1945–1949).

[I can be found on Twitter here.]

Comments on Wittgenstein's Beetle-in-a-Box Argument

Mental events and states were traditionally deemed to be private. Public language, on the other hand, is intersubjective. Yet we use a public language to communicate purportedly private mental events, items or states. If mental events, items or states were genuinely private, then how would we know that other people were referring to the same things we refer to? To us (if not them), other people’s private mental items, states and events would be like possible “beetles in a box”. That's partly why Ludwig Wittgenstein concluded that our language for sensations isn't only public: it must also refer to public items. (This is all part of Wittgenstein's Private Language argument.)

For example, when someone says, “I'm in pain”, he's not describing a private pain-state. The utterance is (as it were) the pain state. Our behaviour (including linguistic behaviour) is literally everything. If someone were to refer to something only they had seen, then we wouldn’t get the whole import of their reference. Thus when we use a language for pain, we must be referring to essentially public items. We can't therefore believe that pain-words refer to private pain-things. This is the case with other names or words too. They refer to entities which are essentially publicly accessible.

Pain-words must refer to things which are publicly accessible: not private. Thus pain can't be a beetle in a permanently-closed box. If pain is in any kind of box, then it's in an open and publicly-accessible one. The named object, if private, must “drop out of consideration as irrelevant”. Other people can't and don't see private mental states or events. Public items are needed for a public language. Language is a shared phenomenon primarily because there are shared words and the shared objects of these words. Without the shared objects of our names and words, there'd be no shared language. The shared objects largely determine the shared language.

If a language isn't public (or shared), then it's not, in fact, a language. If we were just communicating with ourselves, then perhaps we wouldn’t need a descriptive language at all. We would simply know that we are in pain and, therefore, pain-descriptions wouldn't be necessary.

It follows that if there are private mental states, events or items in S's mind, and S were to name these items, then he'd have to introduce his own names or words rather than use names or words he'd acquired from other people. After all, how could public names refer to truly private items? There couldn’t be a public (or intersubjective) ostensive definition of S’s private mental items, states or events. Indeed it was Wittgenstein’s argument that there couldn’t even be a private ostensive definition of one’s own private mental items, states or events. How would S know that he was describing (or defining) a mental item, state or event correctly or incorrectly? The correctness or otherwise of an ostensibly private ostensive definition would depend on a third party determining whether or not such an attribution is correct or incorrect. As Wittgenstein put it (to paraphrase):

If S doesn’t know if he’s going right, then how would he know that he’s going wrong (in his self-attributions)?

“Yes, but what about pain itself?”

Despite all the above, there must still be a sense that certain mental items, states or events are indeed private - such as pain. This seems to be self-evident.

For example, if I don't communicate my pain with either linguistic or physical behaviour, then no one else would know that I'm in pain. But I am in pain. Only I know that I'm in pain. Therefore my pain is private.

It's true that if I were to express my pain (either in a language or even by physical behaviour), then, yes, I would be using a public language. And even if I use such words to talk to myself about my own pain, then I'd still be using public words and public expressions. Thus we could say that the epistemic status of the belief (how it's known or the way it's known) depends on the external world and public language.

However, there's more to a pain than our knowledge of it. It has both an ontological and an experiential status. We can also accept the fact that any ontological and experiential status the pain does have will itself be coloured by public language. (For one, those parts of public language which have given us the tools and concepts to think about a pain’s ontological and experiential status!) Though, yet again, there's still something about pain that's above and beyond its epistemic position and its ontological and experiential status. There's a state - a pain - that's the subject of all these public expressions. These public expressions are about something other than themselves. They're about pain.

Even though the very concept PAIN is a public concept, what the concept itself is about is more than a concept: it's about something which falls under the concept PAIN (or under the word “pain”). Thus even if it's the case that everything we say (or even think) about a pain is somehow polluted by public language, there'd still be something about the pain that has nothing to do with public discourse. More to the point, there's something about a pain that's indeed private.

The problem is that (as Wittgenstein himself argued) as soon as we say anything about this private something, it becomes (for other persons and even for oneself) a public something. What's said about this something (its experiential nature, etc.) is public. Though what all this public talk is about is something private. We can’t, however, get to this private something without polluting it with public language. And even when we get to it ourselves, it's still thus polluted. The private pain, then, is like a Kantian “thing-in-itself”. Its true reality or status can never be known if such a status is deemed to be how the pain is regardless of public language.

Of course many philosophers will deem the very idea of a pain-in-itself suspect. Does that idea even make sense? Does it serve any purpose?

Reference

Wittgenstein, Ludwig. (1945 to 1949) Philosophical Investigations.