How paradoxical is this argument?

[Garry G]

These implications should be familiar.

(1) ⊨ (¬p ⋁ q) → (p → q)
(2) ⊨ (p → q) → (¬p ⋁ q)

Together (1) and (2) define material implication. They are theorems of propositional logic and not usually included amongst the paradoxes of material implication. On closer inspection that is surprising.

(2) is passable but (1) is paradoxical. Here's the counter example.

p: "the new unidentified species is an amphibian"
q: "the new unidentified species is a mammal"

(¬p ⋁ q) → (p → q) then reads:

"the new unidentified species is not an amphibian or it is a mammal, then: if it is an amphibian then its a mammal"

There are possibilities for which the initial implication "it is not an amphibian or it is a mammal" may be true. If for example the new species is a mammal, or if it is not an amphibian.

The conclusion "if it is an amphibian then its a mammal" is false.

Material implication allows sets of possibilities that are true to lead to a false conclusion.

If you feel like defending (1) ...how? ....why?

[wtf]

These implications should be familiar.

(1) ⊨ (¬p ⋁ q) → (p → q)
(2) ⊨ (p → q) → (¬p ⋁ q)

Together (1) and (2) define material implication. They are theorems of propositional logic and not usually included amongst the paradoxes of material implication. On closer inspection that is surprising.

(2) is passable but (1) is paradoxical. Here's the counter example.

p: "the new unidentified species is an amphibian"
q: "the new unidentified species is a mammal"

(¬p ⋁ q) → (p → q) then reads:

"the new unidentified species is not an amphibian or it is a mammal, then: if it is an amphibian then its a mammal"

There are possibilities for which the initial implication "it is not an amphibian or it is a mammal" may be true. If for example the new species is a mammal, or if it is not an amphibian.

The conclusion "if it is an amphibian then its a mammal" is false.

Material implication allows sets of possibilities that are true to lead to a false conclusion.

If you feel like defending (1) ...how? ....why?

You know, this isn't hard to figure out. There are 16 possible logical connectives for two input variables. That is, for a truth table consisting of rows TT, TF, FT, and FF, we can have output columns TTTT, TTTF, TTFT, ..., FFFF. Sixteen in all if you work it out.

Of those 16 possible connectives, only material implication TFTT comes even remotely close to the everyday natural language sense of "if ... then." Every other idea is worse. So we just get used to it and get on with our lives. Most beginning students of sentential logic take from a few minutes to a few hours to grok this.

There is universal agreement (ie you did not just discover something new) that material implication is a poor fit for the causal interpretation of "if ... then." In natural language we distinguish between "If I kick the football it will fly across the field," and "If 2 + 2 = 5 then I am the Pope." Material implication does not capture this distinction.

These concerns are summarized here.

https://en.wikipedia.org/wiki/Paradoxes ... mplication

There's an extensive literature on causality and on various philosophical alternatives to material implication.

https://en.wikipedia.org/wiki/Causality

Your concerns are not new nor are they much of a problem for anyone past the beginning student phase. The bottom line is that we want to define implication as SOME truth connective; and of the 16 possible choices, only one of them, material implication, comes even remotely close. So we accept it and move on to more substantive problems.

You might be interested in the indicative conditional and the counterfactual conditional.

https://en.wikipedia.org/wiki/Indicative_conditional

https://en.wikipedia.org/wiki/Counterfa ... onditional

If I'm missing your point, perhaps you could state it more clearly and succinctly for simple minds like my own.

[Garry G]

If I'm missing your point, perhaps you could state it more clearly and succinctly for simple minds like my own.

The counter examples shows (1) is more than a bit paradoxical or counter intuitive or fails to capture "if...then...", the thread was started because (1) allows true premises to lead to false conclusions. A logic that allows truth to lead to false has a substantive problem does it not?

Yes the subject is over a hundred years old. It appears the problems of material implication have been normalised.

"If I kick the football it will fly across the field," and "If 2 + 2 = 5 then I am the Pope."

This example makes material implication look a bit paradoxical but nothing really fatal going on. It is just an oddity. Move along nothing to see here.

But this example is a more serious challenge: "(A) it is not an amphibian or it is a mammal, then: (B) if it is an amphibian then its a mammal"

It shows a consequent (A) that may reasonably be assumed true, leads to argument (B) that is false.

[Logik]

But this example is a more serious challenge: "(A) it is not an amphibian or it is a mammal

https://en.wikipedia.org/wiki/Fallacy_o ... ted_middle

[Speakpigeon]

No one has to accept that what mathematical logic calls "propositional logic" is at all a model of logic proper. By logic proper, I mean logic as understood by most logicians since Aristotle and up to Boole. That is to say, broadly, what Boole himself called "the laws of thought".
Propositional logic is certainly a logical theory, in the usual sense of the term, in that it is logically consistent. I hope you agree with that. However, that a theory is logical doesn't make it ipso facto a formalisation of logic. Of course, this is what mathematicians like Boole and Frege hoped to achieve. And they really meant to produce a good formalisation of logic, understood as the laws of thought. They tried to do that but they failed. Russell sort of did the next best thing and settled for material implication. The manifest dishonesty was in still calling that "logic". They should at least have qualified it as "material": material logic. The expression "Material logic" wouldn't have to be understood as referring to a part of logic at all, just as a dwarf planet is not a planet properly so called. And the following generations of mathematicians just learnt their logic from the textbook. Material logic is now the de facto standard in mathematics, analytical philosophy and computer sciences. Who is left to care?
EB