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The Amateur Logician Tutorial:
An Invitation | Arguments | Evaluating Arguments | Laws of Thought | Ontology & Logic | Concepts, Signs, & Names | Categorical Propositions | Negations & Complements | Distribution | Euler’s Circles & Venn Diagrams | Predicables | Categories | Definitions | Square of Opposition | Equivalent “Immediate Inference” | Traditional “Immediate Inference” | Informal “Immediate Inference” | Categorical Syllogism | Syllogisms & Venn Diagrams | Moods & Figures of Syllogisms | Polysyllogisms & the Sorites | Enthymemes | Compound Propositions | Conditional Propositions & Conditional Syllogisms | Conditional Contrapositions, Reductions, & Biconditionals | Conjunctions, Disjunctions, & Disjunctive Syllogisms | Dilemmas | Modal Propositions | Reductio ad Absurdum Arguments | Deduction & Induction | Inductive Abstraction | Inductive Syllogisms | Mill’s Inductive Methods | Science & Hypothesis | Formal Fallacies | Testimony & Unsound Authority | Informal Fallacies & Language | Diversion & Relevancy Fallacies | Presumption Fallacies | Causal & Inductive Fallacies
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Conditional Contrapositions, Reductions, & Biconditionals
(1) Contrapositive Equivalency
— “if p, then q” ≡ “if not q, then not p”
(2) Mixed Hypothetical Syllogisms & Categorical Syllogisms
— can be turned into each other
(3) One person’s modus ponens is another person’s modus tollens
(4) Biconditionals
— “p if and only if q”
Be aware, though, that “converses” and “inverses” are not equivalent.
Equivalences can be derived with conditional propositions. So, equivalent “immediate inference” is possible here, too! Working within an Aristotelian-Scholastic approach to the logic of conditionals, there is one type of immediate inference, however, and that is contraposition.
“If P, then Q” is logically equivalent to “If not Q, then not P.”
Premise 1: “If it is raining, then it is cloudy.”
Conclusion: “If it is not cloudy, then it is not raining.”
Premise 1: “If it is not cloudy, then it is not raining.”
Conclusion: “If it is raining, then it is cloudy.”
A way to verify or prove this equivalency is to engage in a so-called “conditional proof.” With the given premise that “If P, then Q,” we can trace out the consequences of if Q were not the case. Since if Q were not the case, then we would know that P could not be the case. Likewise, with the given premise “If not Q, then not P,” we could trace the consequences of if P were the case. But if P were the case, Q would have to be the case.
Ergo, there is an equivalency.
Modus tollens can be reduced to a modus ponens because of contraposition.
All this takes is a contraposition of the major premise.
Mixed Hypothetical Syllogisms & Categorical Syllogisms
A categorical syllogism can be turned into a mixed hypothetical syllogism, and vice versa.
The “Classic” Categorical Syllogism Example . . .
Premise 1: “All men are mortal.”
Premise 2: “Socrates is a man.”
Conclusion: “Thus, Socrates is mortal.” (Valid as AAA-1 Syllogism)
. . .can be turned into a Mixed Hypothetical Syllogism. . .
Premise 1′: “If Socrates is a man, then Socrates is mortal.“
Premise 2: “Socrates is a man.”
Conclusion: “Thus, Socrates is mortal.” (Valid by Modus Ponens Inference)
One person’s modus ponens is another person’s modus tollens . . .
We can think of a debate between two people.
There’s an argument: “If the premises are true, then such-and-such is true.”
One person agrees with it. He engages in a modus ponens inference.
The premises are true; ergo, such-and-such is true.
A disagreeing person, however, engages in a modus tollens inference.
Such-and-such is false; ergo, the premises are not true.
Here we are assuming that the such-and-such validly followed the premises. So, the disagreeing person recognizes that the only way to reach a false conclusion (a false such-and-such) is through false premises.
Biconditionals
A popular expression in mathematics is “if and only if,” which is often abbreviated as “iff.” While “if P, then Q” states that P is sufficient for Q and Q is necessary for P, “P if and only if Q” states that P is both sufficient and necessary for Q and that Q is both sufficient and necessary for P.
Given that, “P iff Q” is equivalent to the conjunction “if P, then Q; and if Q, then P.”
In other words, P implies Q and Q implies P. This is why mathematicians often define terms with biconditionals. It’s not just the case that if an object is X, then it has such-and-such properties; since, it is also true, that if such-and-such has those properties it is X.
For example, “a triangle has three sides that are equal to each other iff a triangle’s angles are equal to each other.” This biconditional proposition tells us that a triangle with three equal sides will have three equal angles, and vice versa.
Reasoning with a Biconditional. . .
Premise 1: If P, then Q.
Premise 2: If Q, then P.
Conclusion: Thus, P if and only if Q. (Biconditional Law with Premises 1 & 2)
Premise 1: P if and only if Q.
Conclusion 1: Thus, if P then Q. (Biconditional Law with Premise 1)
Conclusion 2: Thus, if Q then P. (Biconditional Law with Premise 1)
Premises: “A triangle has three sides that are equal to each other iff a triangle’s angles are equal to each other. This triangle has its angles equal to each other.”
Conclusion: “Thus, this triangle has three sides that are equal to each other.”
With this example, we can be more explicit with the reasoning process.
Mathematical Propositional Logic is useful here.
Line 1: P ↔ Q (Premise)
Line 2: Q (Premise)
Line 3: Q ➝ P (Biconditional Law with Line 1)
Line 4: P (Modus Ponens with Lines 2 & 3)