OLI - 4. Derivations

The Basics

• By using the deductive method, we can apply rules of inference that correspond to valid patterns of inference, we derive the conclusion of the argument and therefore establish its validity.
  • This is to avoid doing long truth tables.
    ![[Screenshot 2023-08-16 at 10.59.02 AM.png]]
• On the right, there's the justification for the formula's presence in the derivation.
• The last line is the conclusion.
• This style of derivation is called a Fitch diagram.
• The argument that it shows needs to be valid and translated into sentential logic formula.

Introduction rules

• Introduction rules: allow us to introduce new conjunctions, disjunctions and conditionals.

Conjunction introduction

![[Screenshot 2023-08-16 at 11.06.01 AM.png]]

• conventions used:
  • We use the letters p for premises, a for assumptions and c for the conclusion.
  • the order in which we cite the lines matches the order in which the conjuncts appear in the inference line.
  • the expressions appearing in the rule definition contain no specific formulae, but contain metavariables ranging over all formulae.
• Examples:
  • ![[Screenshot 2023-08-22 at 7.53.15 AM.png]]
  • ![[Screenshot 2023-08-22 at 7.53.58 AM.png]]

Disjunction introduction

![[Screenshot 2023-08-22 at 7.55.21 AM.png]]

• Two different forms of the rule
  • ∨IR, with which we add the new disjunct on the right.
  • ∨IL, with which we add the new disjunct on the left.
• Example
  • ![[Screenshot 2023-08-22 at 7.59.38 AM.png]]
• We introduced our first two rules of inference by relating them to the truth-conditions of the connectives they introduce.

Conditional introduction

![[Screenshot 2023-08-22 at 8.04.52 AM.png]]

• Rather than focus on the truth-conditions here, it helps to think about the falsity conditions, instead.
  • A conditional happens to be false just in case its antecedent is true and its consequent false.
  • If we want to introduce a conditional, then, all we need to do is rule out this situation; then we can safely infer the truth of the conditional.
• We do something new: we assume the antecedent, then within the scope of that assumption, we derive the consequent.
• Considering the example:
  • ![[Screenshot 2023-08-22 at 8.02.07 AM.png]]
  • We assumed a new justification.
  • The box is the scope of the assumption. This helps us not use the assumption in other places where it doesn't have an effect.
• Subderivation: a derivation inside of the derivation, when we make an assumption we make a miniature derivation.
• We can also apply a rule that discharges the assumption, so we avoid making mistakes.
• We can use a rule called reiteration to copy a premise to be able to include it inside the scope:
  • ![[Screenshot 2023-08-22 at 8.07.29 AM.png]]

Other considerations of introduction

• Introduction rules can be applied forward to premises and/or other available (i.e., already justified) lines in the derivation, and they can be applied backward to the conclusion or any other goal with the appropriate main connective.
• Keep in mind:
  • Apply introduction rules backward.
  • Only apply an introduction rule backward if it will produce subgoal(s) that are possible to derive

Elimination rules

• Elimination rules: allow us to extract certain subformulae from conjunction, disjunctions and conditionals.

Conditional elimination

• The rule →E, traditionally known as Modus Ponens, captures this pattern of inference as follows:
  • If we know that a conditional (φ→ψ) is true, and we also know that the antecedent φ is true, then we can safely infer from these two tidbits that the consequent ψ must also be true.
  • ![[Screenshot 2023-08-22 at 8.11.36 AM.png]]
• Example:
  • ![[Screenshot 2023-08-22 at 8.13.20 AM.png]]

Conjunction elimination

![[Screenshot 2023-08-22 at 8.13.41 AM.png]]

• Example:
  • ![[Screenshot 2023-08-22 at 8.14.14 AM.png]]

Disjunction elimination

![[Screenshot 2023-08-22 at 8.19.53 AM.png]]

• If we know that some disjunction is true, then we know that at least one of the disjuncts is true, but we can't be certain which on.
• Argument by cases:
• Example:
  • ![[Screenshot 2023-08-22 at 8.31.55 AM.png]]

Summary

• The basic structure of derivations is introduced.
• The introduction rules for the binary connectives are introduced.
• Conjunction introduction produces a conjunction from its two conjuncts.
• Disjunction introduction produces a disjunction from one of its disjuncts.
• Conditional introduction produces a conditional from its consequent, possibly within the scope of the assumption of the antecedent.
• Introduction rules are usually most effective when applied backward to a particular goal.
• The elimination rules for the binary connectives are introduced.
• Conditional elimination extracts the consequent from a conditional and its antecedent.
• Conjunction elimination extracts either conjunct from a conjunction.
• Disjunction elimination produces a formula that follows from each disjunct individually.
• Elimination rules are usually most effective when applied forward to obtain (or get closer to obtaining) a particular goal.
• Finally, we give some advice on how to approach more complicated derivations.

New terms

• conjunction introduction
• disjunction introduction
• conditional introduction
• conjunction elimination
• disjunction elimination
• conditional elimination
• positive subformulae