1) 2+2=4
2) if 2+2=4 then the cat is on the mat
Then it follows
3) the cat is on the mat. — Ennui Elucidator
On an ordinary account of logic, if I assume
1) 2+2=4
2) if 2+2=4 then the cat is on the mat
Then it follows
3) the cat is on the mat. — Ennui Elucidator
The statement "the cat is on the mat" is definitionally true — TheMadFool
You're addressing the discrepancy with natural language? — Michael
A closely related analysis for formality is that formal rules are totally abstract. They abstract away from the semantic content of thoughts or claims, to leave only semantic structure. The terms ‘mother’ and ‘cousin’ enter essentially into argument (5). On this view, expressions such as propositional connectives and quantifiers do not add new semantic content to expressions, but instead add only ways to combine and structure semantic content. Expressions like ‘mother’ and ‘cousin’, by contrast, add new semantic content.
[/quote[
. — “SEP on Logical Consequence”
how does one account for the existence of irrationality, abstraction, subjectivity and opinion via logical/ reasonable means? — Benj96
IS it that (2) already seems to evaluate "the cat is on the mat"? — Banno
Perhaps there is a reason to allow the notion of logical consequence to apply even more broadly. In Gentzen’s proof theory for classical logic, a notion of consequence is defined to hold between multiple premises and multiple conclusions. The argument from a set X of premises to a set Y of conclusions is valid if the truth of every member of X guarantees (in the relevant sense) the truth of some member of
Y. There is no doubt that this is formally perspicuous, but the philosophical applicability of the multiple premise—multiple conclusion sense of logical consequence remains an open philosophical issue. In particular, those anti-Realists who take logical consequence to be defined in terms of proof (such as Michael Dummett) reject a multiple conclusion analysis of logical consequence. For an Anti-realist, who takes good inference to be characterised by the way warrant is transmitted from premise to conclusion, it seems that a multiple conclusion analysis of logical consequence is out of the question. In a multiple conclusion argument from A to B, C, any warrant we have for A does not necessarily transmit to B or C: the only conclusion we are warranted to draw is the disjunction B or C, so it seems for an analysis of consequence in terms of warrant we need to understand some logical vocabulary (in this case, disjunction) in order to understand the consequence relation. This is unacceptable if we hope to use logical consequence as a tool to define that logical vocabulary. No such problems appear to arise in a single conclusion setting. (However, see Restall (2005) for a defence of multiple conclusion consequence for Anti-realists; and see Beall (2011) for a defence of certain sub-classical multiple-conclusion logics in the service of non-classical solutions to paradox.)
— “SEP on Logical Consequence”
i.e. what it means for a conclusion to follow from its premise, and trying to contrast it with the requirements of a realist account of truth. — Ennui Elucidator
I'm not sure there's really a problem then. If premise 1 is true then premise 2 is true only if the cat is on the mat. So the realist can say that the realist account of truth is required for premise 2 to be true. — Michael
Something is true when it is the state-of-affairs. When we find the cat is on the mat, we aren’t satisfying definitions, it is simply the case that the cat is on the mat. Part of what this post is about is highlighting the equivocation between true by virtue of state of affairs and true by virtue of definition (but entirely avoiding the analytic/synthetic framework). Maybe we can refer to this version of truth as “rTrue” (for realist truth). — Ennui Elucidator
But can 1 and 2 make 3 rTrue? — Ennui Elucidator
If we want to know if 3 is rTrue, how is a proof used as a proxy for the world? — Ennui Elucidator
It isn't. The argument is only a proof if you can prove 1 and 2 to be true. A deductive argument can have false premises after all. How do you prove 2 to be true? By proving that 2 + 2 = 4 and that a cat is on the mat. You need "the world" to do this. — Michael
I provided what is essentially a tautology in 1 and a claim about the world as a consequent of a conditional. The only way to show that 2 is false (and thereby show the argument is unsound) is to evaluate whether the consequent is true, which is precisely what the proof appears to be proving. — Ennui Elucidator
It is a premise, I don't have to prove it to assume it.You proved that 1 is true, you haven't proved that 2 is true. How do you prove that if 2 + 2 = 4 then the cat is on the mat? — Michael
It is a premise, I don't have to prove it to assume it. — Ennui Elucidator
You have to prove your premises true to prove your conclusion. — Michael
That isn't how logic works. A premise is assumed, not proven. The valid conclusion is of the form "If the premises are true, the conclusion must be true". — Ennui Elucidator
(That is, while the argument form may be valid, the interesting bit - whether the consequence is true - is directly evaluated by reference to the facts. In this respect, soundness is a coincidence of valid form and fact.) — Ennui Elucidator
So how do you evaluate soundness? — Ennui Elucidator
The "what makes it sound" part is what I am discussing. — Ennui Elucidator
If soundness is judged by reference to the world (and that includes evaluating the propositions contained in the conclusion since they must have appeared among the premises), what work is the proof doing for you viz-a-viz the rTruth of the propositions in the conclusion?
I've said; the premises being true are what make it sound. — Michael
Nothing. Your argument has no practical use. I have to evaluate the truth of your conclusion to evaluate the truth of your premise. — Michael
Logic can determine that a set of premises /cannot/ be true or show that the truth of a given conclusion holds under given premises and derive such already implied conlusions. — Heiko
In my mathematical understanding soundness merely means coherence, that is, freedom of contradictions. Logic can determine that a set of premises /cannot/ be true or show that the truth of a given conclusion holds under given premises and derive such already implied conlusions. — Heiko
The difference is that normally we use the material implication as an actual implication, as in the truth of the antecedent implies the consequent. Being a bachelor implies being an unmarried man. Being a man implies being mortal. Winning 270 Electoral College votes implies becoming the next President of the United States. — Michael
If you are a realist and you wish to avail yourself of the power of logic to determine rTruth, can it do so? — Ennui Elucidator
That is the bit that most people lose, I think, that all conclusions are of necessity assumed in the premises. Logic is useful revealing novel relations given a particular set of rules, but cannot reveal rTruths. — Ennui Elucidator
In the type of arguments we deal with, an argument is valid if the truth of the premises entails the truth of the conclusion, and an argument is sound if it is valid and its premises are true. — Michael
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