Definition vs Logic

Question

Logical consequence means that, if, P implies Q, then Q is True not because (or when) we interpret (or find) it to be True, but it is set to True as soon as P is set to True. This means that Q is nothing more than mere exploration of P. And we did that by resorting to logical analysis. So, P, in a way, means Q.

Is this what Wittgenstein meant when he said that:

2.012 : In logic nothing is accidental: if a thing can occur in a state of affairs, the possibility of the state of affairs must be written into the thing itself.

and

6.1251 : Hence there can never be surprises in logic.

What I find possible is that logic and full definition are one and the same. How does an extensive definition (or maximum possible analysis) of P differ from logical consequence of P.

Answer 1

I would take issue with several of the things you say about logic.

1. Logical implication does not typically involve setting propositions to be true. Logic in its most straightforward form is usually understood to be concerned with propositions that have truth values. In classical logic, such propositions are always either true or false, and this is so whether or not we know them to be true or false. To speak of setting propositions to be true suggests that we are moving into a different realm: that of belief revision. I can learn that a proposition P is true, realise that it implies Q, and then acquire the belief that Q is true, but describing this process requires an account of belief revision, which goes well beyond basic logic. The logical relation of implication between P and Q holds timelessly whatever my beliefs about P and Q.

2. As Conifold said in his comment, the idea that the relation of logical consequence is one of containment, or mere explication of what is already present in a concept, is no longer popular. It just does not fit with the complexities of modern logics.

3. For my part at least, I am willing to say that Wittgenstein was simply mistaken about the nature of logic. We cannot know all the possibilities that attend a particular concept or proposition. And in any case, as our knowledge progresses, new possibilities arise that we could not have anticipated. The logical atomism embraced by Wittgenstein is too static: it suggests that we are able by pure analysis to work out all logical possibilities and then resort to experiential data merely to tell us whether each proposition is true or false. This is not how scientific knowledge progresses. Also, to say that there can be no surprises in logic is dubious at best. Logic is actually very useful in telling us things we did not know before. It is particularly good at combining information from different sources.

4. Appealing to definitions is of little use in understanding logic. Even within technical disciplines where we have stipulative definitions, these definitions are open to revision in the light of empirical discoveries. For example, in classical mechanics momentum was defined to be mass times velocity. In relativistic mechanics this is revised to incorporate the Lorentz contraction factor. The reasons to accept special relativity and hence the reasons to adopt a revised definition are ultimately empirical. Moreover, outside of technical disciplines, definitions are not stipulated, they merely describe the way that speakers of a language use words. Compilers of dictionaries do not stipulate definitions; they document conventional usage. These definitions are frequently vague, imprecise, and often tend to drift over time. More importantly, they reflect the current state of our knowledge and will change in the light of new knowledge. Definitions are tools that help us organize information, but they are not the source of logic, nor of a priori knowledge.

Answer 2

How does an extensive definition (or maximum possible analysis) of P differ from logical consequence of P.

Logic, as a performance of human beings, does not operate on "full definitions" or on Wittgensteinian "state of affairs". It operates on the form, or ostensive structure, of the argument.

a is F;

All y that are F are G;

Therefore, a is G.

This argument is valid even though there are no definitions provided for a, y, F and G, outside the type of argument that they are. And logic effectively stops once you are satisfied that the argument is valid.

Thus, logic does not require definitions.

Actual definitions will be required only when you apply logic. Application of logic is not logic.

Application of logic doesn't operate on the real things that may be referred to in arguments and formulas. It operates on the beliefs that you have about the world and beliefs can be very approximate models of the states of affair of the real world.

The notion of logical consequence does not require you to understand or know anything about the real world outside your head.

Application of logic, on the contrary, requires you to believe something about the real world to justify that you take the premises to be true.

Still no definitions required, though, and certainly no knowledge required of the "possibility of the state of affairs written into the thing itself".

All definitions have to stop somewhere. You can only analyse concepts up to the point where you can't go any further, always far short of the reality of the states of affair of the real world.

I don't believe that this has ever been a problem for the survival of homo sapiens. All humans do deductive inferences, routinely and without even being aware that they do. Homo sapiens already did that 200,000 years ago without obviously the possible benefit of Wittgenstein's counsel.

6.1251 : Hence there can never be surprises in logic.

What is true is that all that is possibly in the conclusion is already somehow in the premises, so there is nothing new in the conclusion.

However, to say that there is "no surprise in logic" is a very unfortunate expression.

Surprise is a mental state and we can definitely be very, very surprised upon discovering that a logical formula is valid.

Our ability to assess the validity of logical formulas using only the power of our own brain to do so is limited to very simple formulas and each formula is a special case. Many thinkers in history thought like Wittgenstein does in the extract provided, but they essentially talked from ignorance.

Even for simple arguments, what is new is the conclusion itself. Socrates being mortal follows from Socrates being a man and all men being mortal. Yet, it is effectively not written in the premises that Socrates is mortal.

The situation is very similar to that of arithmetic: 5 + 3 = 8 ... There is nothing new in the result 8 compared to the addition of 3 and 5. Yet, it is a fact that "8" does not appear at all in "3 + 5". So, what is new is the result of the operation 3 + 5. Isn't that something? Something new?

Answer 3

I would say, it's important to have a philosophical position staked out prior to answering your question. Concepts like truth, logic, and definition mean very different things to different people. For instance, the difference between these fundamental concepts between philosophers of objectivism and embodied philosophies have very different takes. It should also be noted that early and late Wittgenstein are also very different creatures. Bertrand Russell and Wittgenstein shared views in Tractatus but diverged so that by Investigation, BR was ill at ease at best with LW's philosophy. That admonishment being said I'll stake out a general response speaking from Borchert's.

BR and LW argued for the correspondence theory (as opposed to coherence, pragmatic, or semantic theories of truth) and developed logical atomism, which roughly says that truth is defined as a truth-bearer such as a proposition predicated true if and only if it corresponds to a fact, that is, it's a relational property between the internal truth-bearer (expressed as meaningful language) and external reality (an atomic fact). Thus, to answer your question, if P implies Q, then the truth bearers P and Q are true together in reality, such that the language of P and Q being true are universally found to be in correspondence with the facts of P and Q. An example might help to clear this up.

"If a man is in the kitchen of his house, then he is in the house itself" is both a truth-bearer (a logical proposition) and a physical reality whenever some member of Homo sapiens with an XY pair of chromosomes is physically present in a room where food is generally prepared and often eaten in. Hence, the current affairs, whereby our man is standing in the kitchen (an atomic fact) is also a different atomic fact (he's present in the house), and that second atomic fact is written into the first atomic fact (a scientific fact of how volumes contained in volumes exhibit transitivity, specifically). So, LW then raises the point that the truth-bearer itself, which corresponds to our experience of the physical reality, 'Q is true because of P' isn't accidental, he is merely commenting on how the logical proposition mirrors or corresponds to the more fundamental fact about the world we observe.

This would be my understanding of your question as presented.

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Answer 4

Wittgenstein himself did say that all the results of logic are tautologies. This is probably what you are after.

But in fact, the full definition of a real thing is always more than mere logic. Were this not the case, we would not be able to discuss ambiguity or paradox -- things that logic does not adequately address, and cannot properly address because of its form.

And these issues are automatically part of every kind of definition that is not purposely abstracted away from the full definition into the realm of logic, as demonstrated by Quine in his discussions of vagueness and natural kinds.

As a kind of silly example, we can meaningfully argue over the definition of a fish and a half -- unless we have abstracted away the physical fish from the concept of counting fish.

In some senses, a fish and a half plus a fish and a half might be more than two fish. In other senses, it won't. It will be two fish and some waste, if the fish are pets (which may be less than two fish, because it might make them ill and we might fear they will die, or it may reduce our aesthetic experience of the fish) and three whole fish, if the fish are food.

Logic removes such details from whatever it is applied to. So it surely cannot contain any full definitions. And it cannot, therefore, be identical with the full definitions of all the things it covers.