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There is the idea of setting up a code or convention or dictionary between atomic propositions and capital letters.

There are compound propositions, each of which has a main connective which connects its components.

There are five propositional logical connectives:

'∼' which translates back to 'it is not the case that...'

'∧' which translates back to '... and ...'

'∨' which translates back to '... or ...'

'³' which translates back to 'if... then ...'

'≡' which translates back to '... if and only if ...'

9/1/12

Skills to be acquired in this tutorial:

Symbolizing compound propositions. Learning about logical connectives, and the notion of the main connective. Recognizing different constructions in English which have the same underlying logical form. Paraphrasing the English into a standard form.

Why this is useful:

It is the next step in learning how to symbolize. Main connectives are very important-- they are central to symbolization, they are central to the semantics, and they are central to derivations.

8/25/12

Not all propositions are atomic propositions. Consider the proposition asserted by 'It is not the case that in 2011 the United States had a female President'. This is a true proposition, yet it is not an atomic one. It is made up of the atomic proposition 'in 2011 the United States had a female President' (which is false) and negation (expressed by 'It is not the case that...'), and the resulting compound proposition, which is the negation of a false proposition, is true.

There are several types of compound proposition.

8/26/12

[The core of this is from Hugues Leblanc and William A. Wisdom [1972] Deductive Logic p.111 and f.]

Symbolization using the Universal Quantifier (of non relational English)

8/26/2012

Thus far we have considered only 'monadic' predicates-- our atomic formulas consist of a predicate followed by only one term-- for example, Fx. But in English we regularly encounter dyadic predicates or relations. For example, 'Arthur is taller than Bert' cannot be symbolized with the tools we have used so far; what is needed is a relation to represent '...is taller than ...' Txy, say, and then the proposition would be symbolized Tab.

8/26/2012

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).

8/26/2012

In predicate logic, many different styles of expression in English get cast into the same 'property-is-had-by-entity' form. For example,

8/25/12

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).

8/26/2012

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.

8/25/12

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).