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I am a beginner in symbolic logic and still mostly using intution to solve the problems. After spending some time trying to figure out why (p -> q) is true if p is false I've realized that this also means (p -> q) & ( p -> ~q) is not a contradiction.

How come statements like "if it rains the road will be wet" and "if it rains the road will not be wet" don't contradict each other?

When we write the truth table we can see that the both statements are true when p is false. So according to this both of those intuitively contradictory statements are true when it doesn't rain. Am I missing something or is symbolic logic supposed to depict something other than everyday usage of logic?

WVrock
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    No, they do not. The first one is False only when p is T and q is F, while the second one is F only when both are F. Both are "compatible" with the fact that it doers not rain. – Mauro ALLEGRANZA Jan 27 '21 at 07:03
  • @MauroALLEGRANZA Can we say that in symbolic logic "true" does not mean it is actually "true" it just means it can not be disproven?

    it seems p->q being true does not make a statement about the laws of universe, it only says that the current situation does not disprove it? It is the only way I can make sense of it. If we can't disprove it we call it true(for whatever reason) If it doesn't rain we can not disprove any statement about what happens when it rains, so we say all of them are true.

     – WVrock Jan 27 '21 at 07:25
  • NO; in classical logic True means True. – Mauro ALLEGRANZA Jan 27 '21 at 07:26
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    As per infinitely many similar posts on this site, the symbolic translation of "if p, then q" does not mean: "p is the cause of q" and neither "q follows logically from p". – Mauro ALLEGRANZA Jan 27 '21 at 07:39
  • The answer is because in deductive reasoning the term contradiction means something specific. In slang or street variations people use the word contradiction different which is what you are doing now. You don't show you understand the relationships between propositions. Aristotle came up with the SQUARE OF OPPOSITION which indicates relationships between propositions. Mathematical logic does not use the original square of Opposition but a modern version where only contradiction exist. So you won't learn about the original proposition relationships in Mathematical logic. – Logikal Jan 27 '21 at 17:01
  • Contradiction means there is a relationship between two propositions where both cannot be true & simultaneously both cannot be false. That is, one must necessarily be true while exactly one must be false always . The slang definition of contradiction only refers to one being true while the other is false. This slang definition ignores the possibility of both propositions being false. This is likely why you think the propositions are contradictory. You would need to place the negation outside of the conditional proposition. Notice your negation is in the consequent portion of the conditional. – Logikal Jan 27 '21 at 17:07

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You are correct to observe that with many conditionals, such as the one you quote, "if P then Q" is contrary to "if P then not Q". It is true of conditionals that express causal relationships, and those that describe dispositional properties, and it is true of counterfactual conditionals, among others. Fundamentally, this is because a conditional is not so much a sentential connective but a device for introducing a suppositional context. "if P then Q" can be glossed as, "Suppose P... In that case, Q". If you assert, "if P then Q" and I agree to accept your supposition P and disagree with you about whether Q is true under that supposition, then our beliefs are contrary.

There are conditional logics built around the principle that "if P then Q" is contrary to "if P then not Q", such as connexive logic. It represents a strand of thought in the history of logic running from Aristotle through Boethius to Abelard. It is also worth noting that conditional probabilities have this connexive property: if P(B|A) is high then P(¬B|A) is not high.

However, this does not hold true of all uses of conditionals. Sometimes we wish to advance a supposition P that may or may not actually be true, because we wish to consider what would follow if it were. Reasoning by reductio follows this pattern. If we suppose P and it leads to a contradiction then we may conclude that P is false. In such cases, "if P then Q" and "if P then not Q" are both correct and they conjointly imply not P. Conditionals like this are very useful in mathematics and formal logic, because often we want to start from a bunch of hypothetical assumptions and 'solve' for P.

The simplest conditional that has the latter property is material implication (→). From P → Q and P → ¬Q we can prove ¬P. Negating a material implication gives P ^ ¬Q, so we might say that the semantics of material implication is that P, ¬Q are not both true. By contrast, the negation of a connexive "if P then Q" is "if P then ¬Q" which suggests a stronger relation of incompatibility between P and not Q.

When you begin to study logic, material implication is the first conditional you are introduced to, because it is the simplest and the only one that is a truth function. It also has a special place in classical logic, because by the deduction theorem it is the sentential connective that corresponds to the classical relation of entailment. But it is important to remember that it is not the only conditional. Indeed, I would go so far as to say that the great majority of uses of 'if' in English cannot be correctly translated by material implication.

Bumble
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