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? Syntactic? Or is it purely logical (whatever that means)?
What precisely is meant by the word “guarantee” (i.e., 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., syntactical)?
Similarly, how do we recognise the soundness and validity of arguments? Through semantic connections or through logical (syntactic) form alone? More interestingly, does the logical (or syntactic) run entirely free of the semantic?
Modal logic is implied by propositional logic and predicate logic (or first-order logic). That is, with words such as “necessarily” and “possible”, aren’t we moving beyond propositional and predicate logic?
Let’s consider an example. If I say —
It isn’t possible that the premises are true and the conclusion false.
That introduces possibility. 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) Implicit/Hidden Premises
Some arguments may only have one premise — meaning we move from that single premise to a conclusion. This seems to be the case with Descartes’ Cogito: “I think. Therefore I am.” It seems to follow, then, that in order for the argument to valid (if not true), the single premise may — or must — contain 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.
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.
This is because, again, there are implicit premises involved. Thus in the following
i) Jim is a bachelor. ii) Therefore Jim is 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, true 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 —
i) All corbetts are bricks. ii) All bricks can solve equations. iii) 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.