From Is to If: Reasoning with Material Conditionals

0.

Some Worries about the Material Conditional Are Immaterial

In his monumental expository wrkAn Introduction to Non-Classical Logic: From If to Is, Graham Priest (2013) prefaces the discussion of the classical material, indicative, conditional with the worry that it doesn’t belong in natural language reasoning. To drive home the urgency of the problem, and motivate a revisionist solution, he offers classical inferences in natural language as heuristic falsifiers for the rules of classical connectives; particularly, the conditional. These are intended to demonstrate that classical inferences in natural language produce results that are either logically invalid, factually incorrect, or at variance with common sense. But, as will become clear shortly, these falsifiers work only if classical inferences in natural languages are restricted so that only elementary propositions can do duty as premises in any arguments. In this essay I show one could do otherwise without producing invalidities-whether they be logical, factual, or intuitive. Compound variants of Priest’s atomic premises {SEE Heuristic Falsifiers: 1.1; 2.1; 3.1.} are used to deduce classically valid, and intuitively correct results {SEE Heuristics Falsified: 1.2; 2.2; 3.2}. I conclude with the suggestion that worries about the material conditional mentioned are immaterial.

1.

(A → B) & (C → D) ├ (A → D) v (C → B)

Priest offers the following argument to undermine the applicability of the above inference to natural language reasoning.

Heuristic Falsifier 1.1: “If John is in Paris he in France, and if John is in London then he is in England. Hence, it is the case that if John is in Paris he is in England, or that if he is in London he is in France” (2013, 1.7, p. 12).

This argument is set up so that:

John is in Paris = A;
John is in France = B;
John is in London = C;
John is in England = D.

But, of course [1], we can set it up another way:
A’ = If John is in Paris he in France;
B’ = If John isn’t in Paris he isn’t France
C’ = If John is in London then he is in England.
D’ = If John isn’t in London then he isn’t in England.

Voila! The material conditional gives the correct result. Namely, (A’ → B’) & (C’ → D’) ├ (A’ → D’) v (C’ → B’) produces the valid and intuitive result:

Heuristic Falsified 1.2: John is in Paris when in France but not otherwise, and he is in London when in England but not otherwise. Hence, either he is in Paris when in France  and only then, or he is in London when in England and then only.

2.

(A & B) → C ├ (A → C) v (B → C)

Heuristic Falsifier 2.1: “If you close switch x and switch y the light will go on. Hence, it is the case either that if you close switch x the light will go on, or that if you close switch y the light will go on. [Imagine an electrical circuit where switches x and y are in series, so that both are required for the light to go on, and both switches are open]” (2013, 1.9: p. 14).

Again, it appears, a simple tweak in the propositions being reasoned about defangs the bad inference in 2.1.  Consider:

A” = You close both switches x and y.
B” = You close one of switches x and y.
C” = The light will go on if both switches are closed.

Heuristic Falsified 2.2: If you close switches x and y, and also close one of switches x and y, then, the light will go on if both switches are closed. Hence, either the light will go on if both switches are closed when you close switches x and y, or one of x and y, or the light will go on if both switches are closed.

Now, clearly, (A’’ & B’’) → C’’ ├ (A’’ → C’’) v (B’’ → C’’) gives the correct answer! So, we needn’t be in any hurry to be rid of inferences of the form (A & B) → C ├ (A → C) v (B → C).

3.

¬(A → B) ├ A.

Heuristic Falsifier 3.1: “It is not the case that if there is a good god the prayers of evil people will be answered. Hence, there is a god” (2013, 1.9.1p. 15).

This argument is set up so that:

A = There is a good God
B= The prayers of evil people are answered.

But nothing prevents us from formulating the premises like this:

A”’ = If there is a good God the prayers of evil people are not answered.
B”’ = The prayers of evil people are answered.

Heuristic Falsified 3.2: It isn’t that the prayers of evil people are answered when the prayers of evil people are not answered if there is a good God. Hence, if there is a good God the prayers of evil people are not answered.

Clearly, ¬(A’’’ → B’’’) ├ A’’’ is the expected answer.

4.

‘If A then B’ is true iff ‘A ⊃ B’ is true.’

Heuristic Falsifier 4.1: “First, suppose that ‘If A then B’ is true. Either ¬A is true or A is. In this first case, ¬A ∨ B is true. In the second case, B is true by modus ponens. Hence, again, ¬A ∨ B is true. Thus, in either case, ¬A ∨ B is true.”

Now, consider the flowing reformulation of the premises.

A* = ‘If A then B’ is true iff ‘A ⊃ B’ is true.

B* = ‘If A then B’ is false iff ‘A ⊃ B’ is false.

Heuristic Falsified 4.2: Suppose it’s true that if “If A then B’ is true iff ‘A ⊃ B’ is true” then If A then B’ is false iff ‘A ⊃ B’ is false“. Then, either it’s not true “If A then B’ is true iff ‘A ⊃ B’ is true” or it’s true  that “‘If A then B’ is false iff ‘A ⊃ B’ is false” or that “‘If A then B’ is false iff ‘A ⊃ B’ is false“. In the second case, it follows by modus ponens that “If A then B is false iff ‘A ⊃ B’ is false.” So, in all cases, (¬A* ∨ B*) is true.

There’s no danger in retaining (¬A* ∨ B*) ├  (A*  → B*) as a rule of reasoning, clearly.

5.

CONCLUSION

Examples discussed, and falsified, show inferences using the mentioned classical connectives, and classical inference forms, falsify our heuristics. Such heuristic falsifiers really don’t go any way towards showing the problem to lie with the underlying logic, rather than with the complexities of premise formulation and appraisal. Abandoning well-behaved logical connectives, classical or otherwise, isn’t a good fix for filling the gap between logic and intuition. The deduction of true conclusions from non-exhaustively descriptive premises, or premises that need and admit of sensible alternative reformulations, are certainly interesting topics in their own right. But, to tweak logical connectives specifically for failing to deliver deep insight on those issues is to treat the wrong disease.

REFERENCE

Priest, G. (2013). An Introduction to Non-Classical Logic: From If to Is. New York, NY: Cambridge University Press.

Notes:

[1] Provided we are not committed to the idea that only atomic and not compound sentences may be used as premises in an argument.

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One thought on “From Is to If: Reasoning with Material Conditionals

  1. Pingback: Materiality & Relevance: Conditional Reasoning in Practice | POSSIBLE WORLDS SELF-MANAGEMENT

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