Propositional calculus is one of the oldest forms of logic.\u00a0 It lies at the heart of all syllogisms, deductions, inductive inferences, and the like.\u00a0 It is the system that allows us to deal with logic problems such as determining the truth of the following argument:<\/p>\n
or of this argument:<\/p>\n
Note that is doesn\u2019t judge the truth or falsehood of the premises (It is raining or it is sunny) or of the conditionals (If it is\u2026), it only judges the truth content of the conclusion based on these assumptions (premises and conditionals).\u00a0 It won\u2019t know whether or not it is raining, but it will be able to tell me that I am depressed when it is raining (not true \u2013 I like the rain) or that I shouldn\u2019t wear a raincoat when it is sunny.\u00a0 Note also that the system is not equipped to handle syllogism using the quantifiers all, some, or none.\u00a0 Thus the true syllogism<\/p>\n
and the false syllogism<\/p>\n
are outside of its ability to evaluate.<\/p>\n
Despite these limitations, the system is fairly powerful and can be used in successful applications of artificial intelligence to solve real-world, interesting problems.<\/p>\n
Amazingly propositional logic is able to pack a lot of power into a fairly brief amount of symbolism.\u00a0 The system as a whole consists of 2 objects, 5 expressions that define relations between the objects, 10 rules for manipulating the relations, and 3 helper symbols that act as traffic cops to impose order on the steps.<\/p>\n
The simplest object of the system is an atomic proposition, like the statement \u2018It is raining\u2019.\u00a0 This proposition, usually denoted by a single capital letter \u2013 \u2018R\u2019 for \u2018It is raining\u2019.\u00a0 More complex propositions are built out of the atomic propositions using the 5 logical expressions, subject to certain rules.\u00a0 Any primitive proposition and all valid compound propositions are collectively known as well-formed formulae (wffs \u2013 pronounce \u2018woofs\u2019, like the sounds dogs make).\u00a0 Often wffs are denoted by Greek symbols, like $\\phi$ or $\\psi$.<\/p>\n
The 5 logical expressions denote relations associated with the propositions.\u00a0 There is one prefix expression, where the expression symbol operators on only one proposition, and 4 infix expressions that link two propositions together.<\/p>\n
The prefix expression is the \u2018not\u2019 which translates the proposition \u2018It is raining\u2019 into the negative proposition \u2018It is not the case that it is raining\u2019.\u00a0 This somewhat more clunky way of expressing the negation (rather than \u2018It is not raining\u2019) seems to be preferred since it makes adding or removing a negation as simple as adding or removing the phrase \u2018It is not the case that\u2019 to the front of an existing proposition.<\/p>\n
The four infix expressions link two propositions together.\u00a0 These are:<\/p>\n
Since the conjunction, disjunction, and biconditional expressions are symmetric upon interchange of the two propositions (or wffs) there is no special name for the first or second slots.\u00a0 The conditional, however, requires a sense of cause-and-effect and, as result, the first slot is called the antecedent and the second slot the consequent.\u00a0 In the conditional \u2018If it is raining then I am depressed\u2019, \u2018it is raining\u2019 is the antecedent and \u2018I am depressed\u2019 is the consequent.<\/p>\n
The systems objects and expressions can be summarized as<\/p>\n
| Expression<\/th>\n | Name<\/th>\n | Symbol<\/th>\n | Example<\/th>\n<\/tr>\n |
|---|---|---|---|
| It is not the case that<\/td>\n | Negation<\/td>\n | $\\neg$, ~, !<\/td>\n | $\\neg R$<\/td>\n<\/tr>\n |
| \u2026\u00a0 and \u2026<\/td>\n | Conjunction<\/td>\n | $\\land$, &<\/td>\n | $R \\land S$<\/td>\n<\/tr>\n |
| Either \u2026 or \u2026<\/td>\n | Disjunction<\/td>\n | $\\lor$<\/td>\n | $R \\lor S$<\/td>\n<\/tr>\n |
| If \u2026 then \u2026<\/td>\n | Conditional<\/td>\n | $\\rightarrow$<\/td>\n | $R \\rightarrow S$<\/td>\n<\/tr>\n |
| \u2026 if and only if \u2026<\/td>\n | Biconditional<\/td>\n | $\\leftrightarrow$<\/td>\n | $R \\leftrightarrow S$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n In addition to the expression symbols, there are a few additional helper symbols that keep things neat.\u00a0 The first is the \u2018implies\u2019 symbol $\\implies$. It is sometimes called \u2018infer that\u2019 and then is denoted by $\\vdash$. Either basically denotes the final conclusion (the output) of the argument.\u00a0 So the first proposition translates into<\/p>\n $R$ \n$R \\rightarrow D$ \n$\\implies D$<\/div>\n where $R$ is the proposition \u2018It is raining\u2019 and $D$ is the proposition \u2018I am depressed\u2019.\u00a0 The second set of symbols are the parentheses \u2018(\u2018 and \u2018)\u2019 which are used to group terms together to avoid ambiguous expressions such as $A \\land B \\land C$, which could mean \u2018I did A and then I did B and C\u2019 or \u2018I did A and B and then I did C\u2019 or other meanings.<\/p>\n The next piece is the rules of inference that allow proper manipulation of one set of wffs into another.\u00a0 These rules are:<\/p>\n
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