2. Rules of Implication - Modus Tollens (MT) Natural deduction (also called the proof method) allows you to prove that a

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2. Rules of Implication - Modus Tollens(MT)
Natural deduction (also called the proof method)allows you to prove that a conclusion follows from a set ofassumptions (premises) by applying rules that tell you whatconclusions follow from statements of certain forms. Each new linein a proof must follow from one or more lines above it according toone of the rules of natural deduction. Each added proof lineconsists of a new statement that follows from the lines above, therule by which the new statement follows, and the line or lines towhich the rule was applied. If you can show that the conclusion ofan argument follows from the set of premises in a series of naturaldeduction steps, then you will have shown that the argumentis valid. A correct proof is one in which each step is acorrect application of a rule of inference, and the conclusion isthe last line of the proof.
The given numbered statements are the premises of the argumentto be proved valid, and the line beginning with a single slash isthe argument's conclusion. The conclusion is not part ofthe set of initial premises, and you should not use theconclusion line as justification for any new lines in the proof.The conclusion line identifies the goal of the proof or thestatement you are trying to obtain on its own line as you work tocomplete the proof.
The next rule you will learn to apply is the modustollens (MT) rule. Modus tollens is defined asfollows:
Modus tollens is a rule of implication, which meansthat its conclusion can be inferred from its premises but may notbe logically equivalent to those premises. All rules of implicationcan be applied only to the main operator of thestatements matching the form of the rule. Modustollens states that if you have aconditional p ⊃ q on its own line, and if youhave the negation of the consequent of that conditional ~q onanother line, then you can conclude the negation of the antecedent~p on a new line.
You can apply the modus tollens rule only if theconditional premise is on its own line with the horseshoe as themain operator. In other words, you cannot apply modustollens to a part of a line. Remember that p and q can standfor any statement and that they may stand for compound statementswith other operators (even other horseshoes) within. It does notmatter whether the conditional statement is listed on a line beforeor after the line listing the conditional's antecedent. It onlymatters that you have two previous proof lines that match the formof the two statements in the form of modus tollens, regardlessof the order in which they appear.
Consider the natural deduction proof given below. Using yourknowledge of the natural deduction proof method and the optionsprovided in the drop-down menus, fill in the blanks to identify themissing information (premises, inferences, or justifications) thatcompletes the given application of the modus tollens (MT) rule.