For some time now I have considered the axioms of logic to be (or correspond to) the only non-syllogistic inferences.
p, therefore p
not-p, therefore not p
p, therefore not not-p
The axioms of logic cannot be confirmed by Reason, only by Revelation (if there be such a thing).
Now recently I have been thinking about synthetic and analytic judgments; and I have come to suspect that synthetic judgments correspond to syllogisms, and analytic judgments to the axioms of logic.
p and q, therefore p
“Bachelors are unmarried men, therefore all bachelors are unmarried.”
p, therefore r
“All liquids I know are poisonous, therefore all liquids are poisonous.”
In the second example, the necessary second premise is missing:
Premise A: “All liquids I know are poisonous.”
Premise B: “All other liquids are also poisonous.”
Conclusion: “All liquids are poisonous.”
Kant apparently imagined a “faculty” by which we could provide ourselves with the second premise without recourse to experience.
But it’s true that the actual axioms cannot be proven, nor are they reached by reason. They are accepted as self-evident, or are merely asserted as conventions.
If you want to know logic, stay the fuck away from Kant. He knew nothing of it.
syllogism = syn-logism = multiple statements having been put together
synthesis = syn-thesis = multiple things’ being put together
analysis = ana-lysis = a single thing’s being dissolved into elements
Therefore syllogisms are not analytic, as multiple statements have been put together and a conclusion has been drawn from the multiple statements put together. An “analytic syllogism” on the other hand would be something like this:
Premise A: “Harry is a bachelor.”
Premise B: “Harry goes shopping every Saturday.”
Conclusion: “Harry is unmarried.”
By the way, I never meant to say syllogisms were synthetic; that’s just the title. What I meant, and said, was that synthetic judgments are (implicitly) syllogistic.
“Bachelor” is simply another word for “an unmarried man”. But I think I have not been clear on the correspondence between the axioms of logic and non-syllogistic inferences.
“A equals A” is a statement. We may symbolise this statement by “p”.
“p, therefore p” is a non-syllogistic inference. But it is really a form of the axiom of logic known as the law of identity: for p follows from p because p equals p. Thus:
“p equals p”, that is:
“{A equals A} equals {A equals A}”.
“A equals A, therefore A equals A.”
“p follows from p, therefore p follows from p.”
As you can see clearly in this last form, the inference itself depends on the premise. So how do you think reason can confirm the conclusion? In order to do so, it would have to confirm the premise. But the premise is essentially one of the axioms of logic. Therefore reason cannot rely on those very axioms to confirm them (for that would be circular reasoning). So it must rely on something prior to those axioms? But there’s nothing prior to those axioms in logic!
p therefore p means nothing at all. It’s not an inference. That’s just a misuse of the word.
This is not valid, and so is not a syllogism. I don’t care about the etymology.
The syllogism is an argument form, having nothing to do with any analytic/synthetic dichotomy.
I beg you, stay away from Kant.
All logic is analytic.
In your example, p does not “follow” from p. “If I am thirsty then I a thirsty” is not an inference because, for example, the P–>q form can be true even if the antecedent is false but the consequent true, which cannot be the case if they are the same statement.
Your definitions are not logical definitions they are pop culture.
In logic (and I don’t mean Star Trek logic) Aristotelian syllogisms are 100% analytical.
I accept the significance of the claim that you didn’t mean to say syllogisms were synthetic (however I’m not sure I believe the actual claim given the pop culture cover up, which is beside the point, whatever that should now be).
Reason confirms the truth value of axioms.
To say otherwise is just plain silly.
Reason confirms that all bachelors are unmarried.
Reason confirms the definitional equivalence and therefore the truth value of the initial statement.
Axioms are of course testable by scientific experiment, and if not falsified are accepted as being true.
Reason confirms that all men are humans without recourse to such scientific endeavours.
It is not true that the truth value of axioms cannot be tested.
Of course deductive and inductive logical process is self perpetuating.
New first principles are synthesised through induction, and in turn are accepted as axioms, to then be re-entered into the deductive syllogistic process, and repeat.
This is of course the revolution of Bacon, Galileo et al to reapply Aristotelian method in an ever increasing circle of scientific usefulness and is probably the main reason so many monkeys have free access to posting unreasonable ‘axioms’ all over the internet.
Alphie - it’s actually silly to say that reason does confirm the axioms of logic. I have read a lot of logicians, and I don’t think any professional logician would ever say that.
Convention “confirms” that all unmarried men are bachelors. This is not an epistemic statement, it’s just a definition.
Okay, but then I did put “syllogism” between quotation marks.
That makes sense, because truth, i.e., experience, is pre-logical, right? So all logic is analytic a priori.
I must admit I am, and have been, struggling to understand this. I regret that you have not put the P–>q form in common language, like “If I am thirsty then I am thirsty”. I will make an attempt:
“If I am sweating then I am thirsty”. It cannot be the case that I am sweating but not thirsty. It can be the case that I am thirsty but not sweating.
But if I insist that “p–>p” is an “axiomatic” inference, that means there are inferences for which the traditional truth tables do not hold. Then again, that will only be the case if the axioms of logic are true. It could be the case that I am thirsty but not thirsty. The only “reason” why A cannot be unequal to A is that A is equal to A…
Well, Saully, logic is a technical field, and so this is one instance where it pays to be precise.
You are saying that p is a sufficient condition of q and that q is a necessary condition of p. So, yes.
Bivalent logic does not address this. More importantly, it does not address truth-in-the-world. So you can’t use logic to disprove itself. You may demonstrate - in the world - that a does not equal a, but logic is not designed for that task, and it cannot help you.
But I suppose that you could try to show us a formal proof for a–>a, using formal logic. Good luck.
In terms of scientific method and the practical applications of logic, reason is generally accepted to be ‘worth its weight in gold’, so to speak, and to discount its general role in all things reasonable, would be unfortunately beyond my ability.
However I can appreciate where you are coming from, and from an entirely propositional ‘point of view’, discussions of ‘reason’ (as we have chosen to call it) could be ‘left to one side’ in favour of a purely restricted ‘logical’, ‘discussion’, and as I am not one overly convinced of the merits of ‘metaphysical splittings of hairs’ for the sake of argument I have no problem with claims that bachelors being unmarried is not epistemologically true, whether this is in actual fact, true, or in fact, otherwise.
It took me a little while to understand this, because your two mentions of p refer to different things:
My being thirsty is a necessary condition of my sweating.
My not sweating is a sufficient condition of my being thirsty.
But Faust, before I even try that (and I don’t think I will), could you tell me if it makes sense to you to regard a–>a as the if/then equivalent (I don’t know the technical term) of the law of identity?
Well, the “application of logic” and “reason” are synonymous. What I am saying is that axioms are unprovable. This is necessarily so, for they are the starting points of proofs. They are unprovable by definition. In other words, if they were provable, then they wouldn’t be axioms.
A = A, for instance, is considered self-evident. (A → A is not, by the definition of implication itself, which is why implication requires a proof.)
