I'm having difficulty following the issue...

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

IS it that (2) already seems to evaluate "the cat is on the mat"?

1. Either the cat is on the mat or the cat is in the tree.

2. The cat is not in the tree.

Ergo,

3. The cat is on the mat. [1, 2 Disjunctive Syllogism]

The statement "the cat is on the mat" is logically true i.e. it's an inference.

The statement "the cat is on the mat" is definitionally true by virtue of the meanings of the words in the statement. This too requires logic i.e. it's an inference.

See anything relevant?

how does one account for the existence of irrationality, abstraction, subjectivity and opinion via logical/ reasonable means? They are “true” in that they do exist and exert an effect through the “human condition.” I find it difficult to believe that logic therefore encapsulates the whole truth of things but rather that logic is one facet of the truth.

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

You're addressing the discrepancy with natural language?

Ordinarily we understand "If X then Y" as asserting that if X is true then Y is true because X is true, whereas in formal logic this isn't the case?

So let’s try to fix some of my ambiguity.

The statement "the cat is on the mat" is definitionally true — TheMadFool

This is exactly contrary to the way I am using “true.” 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).

You're addressing the discrepancy with natural language? — Michael

You aren’t completely off here, but I had hoped that my reference to entailment (rather than logical implication) might help avoid focusing on the difficulty of mapping natural language to symbolic logic. I am more focused 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. In order to facilitate this distinction, I will add another term, “lTrue” to mean the conclusion of a valid deductive argument (by use of whatever logical system you desire).

Perhaps we can think of this post as something like, what is the relation between rTrue and lTrue for any proposition, P. Critically, I am not interested in the semantic content of P, whether “P” can stand in relation to rTrue, or what the magic is that causes P to make “P” rTrue (that is, for us to articulate a truth rather than just observe, or potentially observe, one).

As a side note, much like the linked article, I am going to focus on deductive validity rather than inferential validity. Also, this is a maybe helpful quote:

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

I like this question, but I think it is a bit aside from my current focus. This seems like a problem of constructing a useful logic that can account for such things more than investigating the relation of logical consequence to the state of affairs in the first instance (i.e. logic may not have to account for all rTruths but I am asking it can account for some).

IS it that (2) already seems to evaluate "the cat is on the mat"? — Banno

I’m not sure what you mean by “seems to evaluate” in this case. My hope was that by using a mathematical truth as the antecedent, that we could highlight that any true premises plus valid form makes “P” true of logical necessity. Logic is, perhaps, about establishing (discovering?) the rules by which the truth value of one proposition relates to the truth value of another. Classical logic, where any proposition can stand in for any other proposition with the same truth value, leads to many intuitively unsatisfactory “proofs”.

And harkening to your questions about realism, here is another snip from the logical consequences article:

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”

Truthfully, this whole truth bit is complicated, but I don’t find logical proof among the things that make propositions true in articles like this. and yet it so often seems that people say that some proposition is “true” by virtue of some logical argument.

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.

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

That is the backwards part. Yes, rTrue for the world. But can 1 and 2 make 3 rTrue? If we want to know if 3 is rTrue, how is a proof used as a proxy for the world?

Anytime you look to rTrue for the premises, you are focusing on the wrong spot. Pick any argument you want,

If my finger isn’t in my nose then my hand has five fingers.
My finger isn’t in my nose.
Therefore my hand has five fingers.


P then Q
P
Therefore Q

I assure you, my hand has five fingers and my finger isn’t in my nose. Why is Q true?

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

You claim that a proposition is true when it is a "state of affairs". What exactly do you mean by that? I ask because you seem to be implying there's no logic involved in such truths. The only variety of truths I know of that require no arguments for their veracity are so-called self-evident propositions.

Come now to self-evident propositions. Let's take your "cat on the mat" example. You claim that this (the cat is on the mat) is a state of affairs deal. I'll give you that but you have to admit that "the cat is on the mat" can be true only if that sentence is semantically loaded. That's not all, the meanings (of the words) come togther to give the meaning of the sentence and that I believe is an inference (logic).

But can 1 and 2 make 3 rTrue? — Ennui Elucidator

If 1 and 2 are rTrue then 3 is rTrue. But 1 and 2 being rTrue isn't what makes 3 rTrue. Rather 1 and 3 being rTrue is what makes 2 rTrue.

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.

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

But that is what my example shows you, 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. So the proof adds nothing to the rTrue of the consequent and indeed is just a symbols game.

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

We can take 1 as proven to be true, but you haven't proved that 2 is true. How do you prove that if 2 + 2 = 4 then the cat is on the mat?

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.

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.

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".

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

It's not enough to present a valid argument. You have to show that the argument is sound, i.e. that its premises are also true.

My name is Michael
If my name is Michael then I am the King of England
Therefore, I am the King of England

This is a valid argument. The first premise is also true. But I haven't proven that I am the King of England because I haven't proven that the second premise is true (and I can't because it isn't).

So how do you evaluate soundness?

(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

I'll even quote myself in the OP.

So how do you evaluate soundness? — Ennui Elucidator

By proving that the conclusion follows from the premises and that the premises are true. In your case to prove the second premise I must confirm that 2 + 2 = 4 (which I can do by understanding maths) and confirm that a cat is on the mat (which I can do by looking at the world to see if I can see such a thing).

The argument form is already valid and isn't the least bit interesting. The "what makes it sound" part is what I am discussing. 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?

The "what makes it sound" part is what I am discussing. — Ennui Elucidator

I've said; the premises being true are what make it sound.

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?

Nothing. Your argument has no practical use. I have to evaluate the truth of your conclusion (by referring to the world) to evaluate the truth of your premise.

I've said; the premises being true are what make it sound. — Michael

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.

This seems to have the advantage, that we can separate "discussable" statements like if the cat is really on the mat or if it is not because two paws are outside already from the purely deductive part of reasoning.
In the OPs example one can disagree with (2) or even with (1), but if agreement is reached on both points the conclusion should be implied. That or we have a very different way of thinking...

Nothing. Your argument has no practical use. I have to evaluate the truth of your conclusion to evaluate the truth of your premise. — Michael

This feels much like a circle. Yes, there is no point to a logical argument about states of affairs because what makes propositions about states of affairs true are the states of affairs, not our rules of logic. Nevertheless, people want to use logic to dictate how states of affairs can be. What lTruth can you produce that relates to an rTruth? If you are happy to concede from go that logic is just the formalistic manipulation of symbols, I won't complain.

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

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. Logic then is descriptive and not normative, no matter how well the logic (or math) predicts the rTruths. But is this a problem for logic or just for non-logicians?

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.

In such occasions we don’t need to assess the truth of the conclusion to assess the truth of the premise, and so the argument tells us something about the world.

But in your case that 2 + 2 = 4 doesn’t imply that a cat is on the mat, and so we must assess the truth of the conclusion to assess the truth of the premise. Your material “implication” makes for a useless argument.

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

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.

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

You seem stuck in social conventions and definitions. Forget social conventions and focus on mind-independent stuff, you know, the pre-interpreted states-of-affairs. If you are a realist and you wish to avail yourself of the power of logic to determine rTruth, can it do so?

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

They don’t need to. Your questions make no sense. It’s the brick hitting the window that breaks the window, not the argument “if the brick hit the window then the window is broken and the brick hit the window, therefore the window is broken.”

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

This made me think a little.
If for example, the grass is wet, it has to have become wet somehow. If we were in a world where this can only happen by rain, the conclusion clearly is that it must have rained.
Though this is different from "it rains" or "it rained" it is an extension knowledge. Both "rules", seen for themselves, do not have a relation to each other and do not tell you that you have to apply them together. The system is more than the sum of the parts, so to speak. If taking the already implied as "given" we forget that someone still has to do the conclusion.

You will not feel the (past) rain because of such a conclusion, of course.

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

If that is so clear, why do you say "the truth of the premises" and not just "the premises"?