What is the difference between Conditional and Logical consequence in everyday language?

Question

I am confused about the distinction in everyday language and in propositional logic. I even understand what is logical implication in a complete argument. But in a propositional logic, I do not understand. I developed a way of thinking, I'd like help to identify gaps and clarify the definitions.

Conditional

I usually apply a causal relationship.

If A then B
If I clap (under normal conditions), then there will be a clapping noise.

So if I clap (TRUE) and there is a clapping noise (TRUE), then I have a TRUE conditional.

If I clap (TRUE) and I do not hear a clapping noise (FALSE), then the conditional is FALSE (there is no possibility, in normal conditions, that I clap and the noise does not come out)

If I do not clap (FALSE) and hear clapping, the condition is TRUE (I may have just played an audio on my cell phone).

If I do not clap (FALSE) and there is no clapping (FALSE), I have a TRUE condition. No cause, no effect.

Summing up:

A → B
T T = T
T F = F
F T = T
F F = T

Logical consequence

Here where my doubt is.

If A then B
If Earth is blue, then Earth is round.

Both are true. On conditional, this would be true. But it is not a logical implication.

So, my doubts are:

1. Create a causal relationship to the conditional (like I did), is it correct?
2. How to create a way to understand the logical implication in everyday language?

Answer 1

The conditional you describe in your question is the material conditional. It is a truth functional connective with the truth table T/F/T/T. It is just a truth function, which is to say that its truth depends only on the truth values of its arguments and not on their propositional content. This kind of conditional is not well suited to expressing causal relations, since "A causes B" is not a truth function of A and B. Causal judgements usually support a corresponding counterfactual conditional. If A causes B then we can usually say "if A were to happen B would happen", or "if A had happened B would have happened," and these are not trivially true merely because A is false. Counterfactual conditionals, like causal claims, are not truth functions and so cannot be represented by a simple material conditional.

Several attempts have been made to explicate causal relations using a logical formalism. John Mackie analysed causal relations as an INUS condition, i.e. that the antecedent A is an Insufficient but Non-redundant part of an Unnecessary but Sufficient condition for B. David Lewis has an account of causation in terms of relations between possible worlds. Judea Pearl has an account of causation that can be expressed using a probabilistic do-calculus. None has achieved any general consensus.

Answer 2

Logic might be thought of as the study of the equivalence truth-preserving structures. In order to accomplish this, it is necessary for the structural elements of logic to be valid independently from empirical fact. Logical implication is one of these structural elements, and it stands in distinction from causal implication. The difficulty which you have observed lies in the fact that causal implication is ultimately rooted in empirical fact and cannot be abstracted from it.

When we assert that A causes B, we are asserting more than a simple temporal sequence. We are asserting that there is a causal nexus between A and B which is somehow made possible by the nature of their ontology. That idea can't be captured with a truth table because the content of a truth table is nothing more than truth values. There is no explanatory content which shows the relations between those values. The best you can do is assert the existence of the causal nexus as a separate clause in your proposition. For example, you might try to capture the idea by doing something like the following:

• Cxy = x causes y
• Ox = x occurs
• Nxy = x is related to y by some causal nexus

∀xy[Cxy ↔ ((Ox → Oy) & Nxy)]

The logical implication Ox → Oy by itself is not enough, because the rule of logic provide no means of distinguishing it from other accidental events. However, when you add the additional clause, Nxy, your proposition no longer depends on the rules of logic, but upon empirical fact.

Immanuel Kant long ago observed the impossibility of trying to establish a firmer criteria for logical relations based on empirical determinations:

"Now an universal criterion of truth would be that which is valid for all cognitions, without distinction of their objects. But it is evident that since, in the case of such a criterion, we make abstraction of all the content of a cognition (that is, of all relation to its object), and truth relates precisely to this content, it must be utterly absurd to ask for a mark of the truth of this content of cognition; and that, accordingly, a sufficient, and at the same time universal, test of truth cannot possibly be found. As we have already termed the content of a cognition its matter, we shall say: 'Of the truth of our cognitions in respect of their matter, no universal test can be demanded, because such a demand is self-contradictory.' [...] Farther than this logic cannot go, and the error which depends not on the form, but on the content of the cognition, it has no test to discover." (Critique of Pure Reason A57/B82)

The contradiction to which he is referring is seeking that which is necessary or non-contingent in the contingent.

Answer 3

Contrary to the basic notions of Classical thought, what logic you need really does depend on what you are doing.

Classical logic uses the material conditional, your first option, because it is relatively safe to do so. It is part of a simplified model, and in that model, this way of accounting truth does not lead to contradictions we do not like. However, it does not correctly model deduction, and it is particularly unrealistic in modeling situations where deductions are made with limited information.

If you use the axiom A => (B -> A) directly in real life, you are making no sense. The true fact that most elephants are not yellow is not a consequence, in any natural way, of the fact that some monkeys like sugar. But the former implies the latter in Classical logic, or in a computer circuit or a decision in Newtonian physics. It is an axiom of most logical systems.

It summarizes the idea that we don't really know the exact deductive path from one true fact to another true fact, but we don't feel we need to. We trust fields like math and basic science, and even large chunks of philosophy, to a certain degree to be timeless and complete in a way that makes all true things equally true.

In those contexts, which make up most contexts, the answer to (1) is that it is true except in bizarre circumstances. There are ways, see Curry's paradox, in which we can get weird results using these axioms. But we would notice those right off. We are safe using this logic in deductions about everyday things.

In contexts like law, where judges do not know what decisions other judges are making at the same time, or have made in distant jurisdictions, this principle does not hold. There can ultimately be a contradiction between their decisions in the future. You cannot simply depend on the material conditional to tell you whether something is true.

Otherwise, you could not lose now and win on appeal, unless the logic of the first case were found provably incorrect. But we see it happen -- decisions change, for good reason, that were, at the time they were made, right. There is just limited information that requires real arguments, using a more 'real' notion of implication, to flow together in a more literal way. So for instances like that, we need ways of reasoning that are less abstract than the ones in ordinary mathematics.

The same problem occurs in fields of physics like quantum dynamics, and in reasoning about other people psychologically, where things can only be found consistent to a limited degree. Sometimes the simplified rules just don't cut it.

So the answer to (2) is that you can really follow the threads of an argument as closely as you need to, when you need to. We spend the time to do it in these domains, and we are confident of the results. But if there were a short, constructive answer to (2), we would never have invented Classical logic to begin with.