In Predicate Logic there are two new logical connectives, the Universal Quantifier (∀x) and the Existential Quantifier (∃x). These are used for symbolizing certain English constructions (they also have their own rules of inference and their own semantics, which we will learn about later).
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In predicate logic, many different styles of expression in English get cast into the same 'property-is-had-by-entity' form. For example,
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The symbolization of English statements or propositions can be done using instruments of varying degrees of logical sophistication. The presentation here is restricted to propositional and predicate logic. Researchers in linguistics, philosophical logic, advanced computer science, or artificial intelligence would likely use some more advanced form of logic. However, predicate logic, encompassing propositional logic, is not introductory and it is entirely adequate for many purposes. Being familiar with it is worthwhile intellectually (and it is non-trivial to learn).
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Propositional logic, while good for many purposes, is not adequate for everything. One of the central uses of logic is to judge which arguments are valid and which are not. But there are many valid arguments which cannot be shown to be valid using propositional logic alone. For example,
Beryl is a philosopher.
All philosophers are wise.
Therefore
Beryl is wise.
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In English there is usually more than one way to say the same thing. For example, the sentences 'Forests are widespread or grass is plentiful.' and 'Either forests are widespread or grass is plentiful.' assert the same compound proposition-- the new word 'either' at the beginning of the second sentence does not alter the underlying logical structure. Both these sentences should be symbolized to (F∨G).
One symbolic formula can represent the logical structure of a proposition asserted by several different English sentences (this is one reason why we symbolize).
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We will start with propositional logic, then move on to the more advanced predicate logic.
Starting on propositional logic ...
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To become familiar with the notions of argument, valid, invalid, premise, and conclusion. To learn how to symbolize atomic propositions.
The main role of logic is to assess arguments-- to say whether an individual argument is valid or whether it is invalid. In logic, arguments are taken to consist of two components--premises, and a conclusion.
For example,
If it rains, I get wet.
It rains.Therefore,
I get wet.
Indicative sentences in a natural language, English, for instance, are either true or false. For example, 'There are 35 State Governors in the U.S.A.' is an indicative sentence (which happens to be false). Such sentences express statements or propositions. Not all pieces of language express propositions. For example, the question 'What day is it today?' is not either true or false (although reasonable answers to it will be either true or false); again, the greeting 'Have a nice day!' is not either true or false.
Welcome!
These web pages provide an introduction to logic to the level of Propositional and Predicate Calculus.
The focus of the program is on arguments and the question of whether they are valid. Arguments have the form <list of premises> ∴<conclusion>. An argument is valid if and only if it is not possible for all its premises to be true and its conclusion false at one and the same time; an argument which is not valid is invalid.