Explanation
Lewis Carroll was showing that there's a regress problem that arises from modus ponens deductions.
- (1) P ⇒ Q
- (2) P
- ---------------
- Therefore, Q.
The regress problem arises, because, in order to explain the logical principle, we have to then propose a prior principle. And, once we explain that principle, then we have to introduce another principle to explain that principle. Thus, if the causal chain is to continue, we are to fall into infinite regress. However, if we introduce a formal system where modus ponens is simply an axiom, then we are to abide by it simply, because it is so. For example, in a chess game there are particular rules, and the rules simply go without question. As players of the chess game, we are to simply follow the rules. Likewise, if we are engaging in a formal system of logic, then we are to simply follow the rules without question. Hence, introducing the formal system of logic stops the infinite regression—that is, because the regress would stop at the axioms or rules, per se, of the given game, system, etc. Though, it does also state that there are problems with this as well, because, within the system, no proposition or variable carries with it any semantic content. So, the moment you add to any proposition or variable semantic content, the problem arises again, because the propositions and variables with semantic content run outside the system. Thus, if the solution is to be said to work, then it is to be said to work solely within the given formal system, and not otherwise.
Some logicians (Kenneth Ross, Charles Wright) draw a firm distinction between the conditional connective (the syntactic sign "→"), and the implication relation (the formal object denoted by the double arrow symbol "⇒"). These logicians use the phrase not p or q for the conditional connective and the term implies for the implication relation. Some explain the difference by saying that the conditional is the contemplated relation while the implication is the asserted relation. In most fields of mathematics, it is treated as a variation in the usage of the single sign "⇒," not requiring two separate signs. Not all of those who use the sign "→" for the conditional connective regard it as a sign that denotes any kind of object, but treat it as a so-called syncategorematic sign, that is, a sign with a purely syntactic function. For the sake of clarity and simplicity in the present introduction, it is convenient to use the two-sign notation, but allow the sign "→" to denote the boolean function that is associated with the truth table of the material conditional.
These considerations result in the following scheme of notation.
The paradox ceases to exist the moment we replace informal logic with propositional logic. The Turtle and Achilles don't agree on any definition of logical implication. In propositional logic the logical implication is defined as follows:
P ⇒ Q if and only if the proposition P → Q is a tautology
hence de modus ponens ⇒ Q, is a valid logical implication according to the definition of logical implication just stated. There is no need to recurse since the logical implication can be translated into symbols, and propositional operators such as →. Demonstrating the logical implication simply translates into verifying that the compound truth table is producing a tautology.
Read more about this topic: What The Tortoise Said To Achilles
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