Language

(I) Symbols: Ø = contradiction, → = conditional, and [] = Modal Operator

(II) Variables: p, q, r, p’, q’, r’. (Variables lower case)

Well Formed Formula for Language

(i) Ø and any variable is a modal sentence.

(ii) If A is a modal sentence, then []A is a modal sentence.

(iii) If A is a modal sentence and B is a modal sentence, then A implies B (A→B) is a modal sentence.

* A, B, and C are modal sentences, i.e. upper case letters are modal sentences. These upper case letters are “variables as well”. They represent the lower case variables in conjunction with contradiction, conditional, or modal operator.

So A may possibly stand for p, or q, or r. It may also possibly stand for a compound of variables and symbols. So A may stand for q, or A may stand for p→Ø, and etc.

Negation (~) = A→Ø

Conjunction (&) = ~(A→B)

Disjunction (v) = ~A→B

Biconditional (↔) = (A→B) & (B→A)

Because Ø indicates contradiction, Ø is always false. But by the truth table of material implication, A → Ø is true if and only if either A is false or Ø is true. But Ø can’t be true. So A → Ø is true if and only if A is false.

This symbol ∞ will stand for something being proved.

(1) Hypothesis (HY) : A new hypothesis may be added to a proof anytime, but the hypothesis begins a new sub-proof.

(2) Modus Ponens (MP) : If A implies B and A, then B must lie in exactly the same sub-proof.

(3) Conditional Proof (CP): When proof of B is derived from the hypothesis A, it follows that A implies B, where A implies B lies outside hypothesis A.

(4) Double Negation (DN): Removal of double negation ~~A & A lie in the same same sub-proof.

(5) Reiteration (R): Sentence A may be copied into a new sub-proof.

Proof of Modus Tollens: Given the conditional claim that the consequent is true if the antecedent is true, and given that the consequent is false, we can infer that the antecedent is also false.

(If p implies q & ~q, then necessarily true that ~p)

Premise (1) p implies q (Hypothesis)

Premise (2) ~q (Hypothesis)

(3) q implies Ø ((2) and of Definition ~)

(4) p (Hypothesis)

(5) p implies q (Reiteration of (1))

(6) q (Modus Ponens by (4) and (5))

(7) q implies Ø (Reiteration of (3))

(8) Ø (Modus Ponens by (6) and (7))

(9) p implies Ø ( Conditional Proof by (5) through (8))

Conclusion (10) ~p ((9) and Definition of ~)

Shortened version, with some steps omitted, would go as follows.

P (1) p implies q

P (2) ~q

(3) q implies Ø ((2) and Definition of ~)

(4) p (Hypothesis)

(5) q (Modus Ponens by (1) and (4))

(6) Ø (Modus Ponens by (3) and (5))

(7) p implies Ø (Conditional Proof by (3) through (6))

C (8) ~p ((7) and Definition ~)

Here is an even shorter proof of Modus Tollens, and it only requires the rule of inference of Hypothetical Syllogism:

(1) p implies q (Hypothesis)

(2) q implies Ø (Hypothesis)

(3) p implies Ø (Hypothetical Syllogism by (1) and (2))

(4) ~p (Reiteration of (3) by Definition of ~)

So we have proved that If p implies q and ~q, then ~p is necessarily true.