375 lines
11 KiB
Markdown
375 lines
11 KiB
Markdown
---
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categories:
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- Logic
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tags: [propositional-logic]
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---
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# Truth-functional connectives
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Propositions generated from other (simple) propositions by means of
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propositional connectives are
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[compound propositions](Atomic_and_molecular_propositions.md).
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We know that
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[logically determinant](Logical_indeterminacy.md)
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propositions express a truth value. When simple propositions are joined with a
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connective to make a compound proposition they also have a truth value. This is
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determined by the nature of the connective and the truth value of the
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constituent propositions. We therefore call connectives of this nature truth
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_functional_ connectives since the **truth value of the compound is a function
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of the truth values of its components**.
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> A propositional connective is used truth-functionally if and only if it is
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> used to generate a compound proposition from one or more propositions in such
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> a way that the truth value of the generated compound is wholly determined by
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> the truth-values of those one or more propositions from which the compound is
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> generated, no matter what the truth values may be.
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Each truth-functional connective has a characteristic **truth table**. This
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discloses the conditions under which the constituent propositions have a given
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truth value when combined with one or more connectives.
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We shall now review each of the truth-functional connectives in detail.
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### Conjunction
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Conjunction is equivalent to the word AND in natural language. We use $\land$ as
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the symbol for this connective.
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A molecular proposition joining two conjuncts P and Q is true iff both conjuncts
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are true and false otherwise:
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```
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P Q P & Q
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T T T
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T F F
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F T F
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F F F
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```
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### Disjunction
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Conjunction is equivalent to the word OR in natural language. We use `v` as the
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symbol of this connective.
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A molecular proposition joining two disjuncts P and Q is true if either disjunct
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is true or if both disjuncts are true and false otherwise. This corresponds to
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the inclusive sense of OR in natural language.
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```
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P Q P ∨ Q
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T T T
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T F T
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F T T
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F F F
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```
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### Negation
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In contrast to the two previous connectives, negation is a unary connective not
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a binary connective. We use `~` to symbolise negation. It does not join two or
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more propositions, it applies to one proposition as a whole. This can be a
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simple proposition or a complex proposition. It simply negates the truth-value
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of whichever proposition it is applied to. Hence applied to P, it is true if P
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is false. And if P is false, it is true when P is true. !
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```
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P ~ P
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T F
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F T
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```
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### Material conditional (a.k.a implication)
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The material conditional approximates the meaning expressed in natural language
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when we say _if_ such-and-such is the case _then_ such-and-such will the case.
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Another way of expressing the sense of the material conditional is to say that
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**P** implies **Q.**
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```
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If it rains today the pavement will be wet.
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```
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We call the proposition that expresses the 'if' proposition the **antecedent**
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and the proposition that expresses the 'then' statement the **consequent**. The
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symbol we use to represent the material conditional is `⊃` although you may see
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`→` used as well.
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The truth table is as follows:
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```
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P Q P ⊃ Q
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T T T
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T F F
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F T T
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F F T
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```
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The material conditional is perhaps the least intuitive of the logical
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connectives. The first case (TT) closely matches what we expect the connective
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to mean: it has rained so the pavement is wet. The antecedent is true and
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therefore the consequent is true. This chimes with what we tend to mean by 'if'
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in natural language. In the second case (TF) it also makes sense: the complex
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proposition is false because it rained and the pavement wasn't wet: this negates
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the truth of the expression. The final case (FF) is also straight forward. It
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didn't rain therefore the pavement wasn't wet, thus the overall assertion that
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rain implies wet pavements is retained.
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FT is less intuitive:
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```
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It did not rain today. The pavement was wet.
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```
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To some degree one just has to take these statements as axioms, whether or not
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they have intuitive sense is a secondary, more philosophical question. The
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semantic issues arise because we tacitly assume the material conditional to be a
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causal connective: there is something about the nature of **P** that _engenders_
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or _brings about_ **Q** but causality is not a logical concern.
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If we instead just focus on the simple propositions that comprise the truth
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value it is more plausible. In the case of FT we can say it didn't rain yet the
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pavement was wet does not stop the pavement being wet when it rains. The fact
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that I can pour a beer on the pavement thereby making it wet doesn't stop or
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render false the idea that the rain can also make the pavement wet. The same
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explanation covers the FF case: it hasn't rained and so the pavement is not wet
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does not contradict the assertion that when it rains the pavement will be wet.
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Things are elucidated when we look at an equivalent expression of P ⊃ Q, ~P v Q:
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```
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P Q ~ P ∨ Q
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T T T
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T F F
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F T T
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F F T
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```
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A disjunction is true whenever either disjunct is true so when both are false
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the overall expression is false, the same as with FT and FF with the material
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conditional.
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### Material biconditional (a.k.a equivalence)
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The material biconditional equates to the English expression 'if and only if',
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as a conditional connective it therefore avoids some of the perplexity aroused
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by its material cousin. In this scenario both antecedent and consequent have to
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be true for the overall expression to be true. If either is false the complex
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proposition is false. Other ways of expressing the semantics of this connective
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is to say that one proposition implies the other or that **P** and **Q** are
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equivalent.
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```
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If and only if James studies every day he will pass the exam.
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```
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There is no possibility in which James passes the exam and has not studied every
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day. If he studies for three out of the seven days leading up to the exam he
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will not pass. Alternatively, there is no possibility that James studied every
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day yet failed the exam. The antecedent and consequent are locked, as indicated
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by the truth-table:
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```
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P Q P ≡ Q
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T T T
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T F F
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F T F
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F F T
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```
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The last condition (FF) maybe requires some explanation: if he has not studied
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every day then he cannot have passed the exam. Therefore, to say that he will
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pass iff he studies every day is rendered true.
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## Combinations of truth-functional connectives
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---
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So far we have applied connectives to simple propositions. In so doing we
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generate complex propositions. However propositions and connectives are
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inherently generative: we can build more complex expressions from less complex
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parts, using more than one type of connective or several different connectives
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to make larger complex propositions and express more detailed logical conditions
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ans statements about the world.
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For example the proposition:
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```
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Socrates was either a philosopher or a drinker but he wasn't a politician.
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```
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Can be expressed with greater logical clarity as:
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```
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Socrates was a philosopher or Socrates was a drinker and Socrates was not a politician.
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```
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Using P for 'Socrates was a philosopher', Q for 'Socrates was a drinker' and R
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for 'Socrates was a politician' we can express this symbolically as:
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```
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(P v Q) & ~R
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```
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Which has the truth table:
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```
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P Q R ( P ∨ Q ) & ~ R
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T T T F
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T T F T
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T F T F
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T F F T
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F T T F
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F T F T
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F F T F
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F F F F
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```
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Let's walk through each case where S stands for the overall proposition.
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1. S is false if Socrates was a philosopher, a drinker and a politician.
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1. **S is true if Socrates was a philosopher, a drinker but not a politician.**
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1. S is false if Socrates was a philosopher, a politician but not a drinker.
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1. **S is true if Socrates was a philosopher but not a drinker or politician.**
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1. S is false if Socrates was not a philosopher but was a drinker and politician
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1. **S is true if Socrates was not a philosopher or politician but was a
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drinker.**
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1. S is false if Socrates was neither a philosopher or drinker but was a
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politician.
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1. S is false if Socrates was neither a philosopher, drinker, or politician.
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If we look just at the true cases for simplicity, it becomes obvious that the
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truth value of the whole is a function of the truth-values of the parts.
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At the highest level of generality the proposition is a conjunction with two
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disjuncts: `P v Q` and `~R` . Therefore, for the proposition to be true both
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conjuncts must be true. The first conjunct is true just if one of the
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subordinate disjuncts is true (Socrates is either a philosopher, a drinker, or
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both). The second conjunct is true just if Socrates is not a politician. Thus
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there is only one variation for the second conjunct (not being a politician) and
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two variations for the first conjunct (being a drinker/being a philosopher)
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hence there are three cases where the overall proposition is true.
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### Logical equivalence
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Once we start working with complex propositions with more than one
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truth-functional connective it becomes clear that the same proposition expressed
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in natural language can be expressed formally more than one way and thus that in
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logical terms, both formal expressions are equivalent. We can prove this
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equivalence by comparing truth tables.
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For example the proposition:
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```
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I am going to the shops and the gym.
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```
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Can obviously be expressed formally as:
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```
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P & Q
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```
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But also as:
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```
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~ (~P v ~Q)
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```
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And we know this because the truth-tables are identical:consistency
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```
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P Q P & Q
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T T T
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T F F
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F T F
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F F F
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```
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```
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P Q ~ ( ~ P ∨ ~ Q )
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T T T
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T F F
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F T F
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F F F
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```
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Another example of equivalent expressions:
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```
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Neither Watson or Sherlock Holmes is fond of criminals
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```
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The first formalisation:
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```
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~P & ~Q
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```
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Equivalent to:
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```
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~(P v Q)
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```
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Again the truth-tables for verification:
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```
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P Q ~ P & ~ Q
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T T F
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T F F
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F T F
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F F T
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```
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`~P & ~Q`
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```
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P Q ~ ( P ∨ Q )
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T T F
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T F F
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F T F
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F F T
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```
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### Important equivalences
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The example above is a key equivalence that you will encounter a lot especially
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when deriving formal proofs. It goes together with another one. We have noted
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them both below for future reference:
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```
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~P & ~Q = ~P v ~Q
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```
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```
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~P v ~Q = ~(P & Q)
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```
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## Enforcing binary connectives through bracketing
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If we had a proposition of the form
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```
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Socrates is man, is mortal and a philosopher.
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```
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We could not write this as:
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```
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P & Q & R
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```
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This would not be a well-formed proposition because at most truth functional
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connectives can only connect two simple propositions. It would not be possible
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to generate truth conditions for this proposition in its current form. Instead
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we introduce brackets to enforce a binary grouping of simple propositions. In
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this instance, the placement of the brackets does not affect the accurate
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interpretation of the truth conditions of the compound, so the following two
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formalisations are equivalent:
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```
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(P & Q) & R
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P & (Q & R)
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```
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