Primer on Logic

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E. M. Segal:

Argument: Second concept called argument (the first being a set of premises leading to a conclusion). This refers to a component of a proposition or sentence in a predicate logic. It represents an individual thing. It is something like a noun or a pronoun. Arguments tend to be represented by lower case letters in formal logics.
Predicate: something like a verb or an adjective. It often represents a property, relation or action. When a predicate is combined with the correct number of arguments it is a simple sentence or proposition. Often a predicate is represented symbolically by a capital letter. A given predicate must have a predetermined number of arguments associated with it. eg. (1) Red, tall, smart are one argument predicates. "Tb" might mean "The boy is tall" or "The tall boy" (2) larger than, hit, kissed, are two argument predicates Kbg might mean "The boy kissed the girl." In multiple argument predicates, the order of the arguments determines their role in the predicate relation. Lab might mean "A is larger than B." It could not also mean "B is larger than A." (3) there are a few three and four place predicates as well, e.g. give, put. "John gave the book to Mary."
Quantifiers and variables:
One might say something like "Some people are tall." This might be represented as (($ x) (Px&Tx). ($ x), the existential operator, means that something represented by the variable x exists. By itself it is an incomplete expression. Px and Tx, which are said to be in the scope of  the quantifier, say 'x is P' (x is a person), and 'x is T' (x is tall). Neither Px nor Tx are complete propositions because without being quantified they have no truth value. You cannot decide whether x is a person until you know what x refers to. However "There exists an x such that x is a person" is true if at least one person exists. "There exists an x such that x is a person and x is tall" is true if there is at least one tall person. Thus the quantified expression is a proposition.
The sentence "All men are mortal" is represented using the universal operator ("x) or often (x). Logicians identified this proposition as a conditional: (x)(M_an x É M_ortalx ). This is usually expressed as "For all x, if x is a man then x is mortal."
The two propositional representations of the sentences "Some people are tall" and "All men are mortal" may not look like the English sentences, but if you think about it you may see that they would be true in exactly the same circumstances, thus the English sentences and the formal notation represent the same propositions.

For your information, there are other logics, some called modal logics, have usually been about necessity and possibility; but may be about time or space or beliefs, etc. These logics often are not strictly truth functional. That is the truth of a whole proposition may not be simple function of the truth of its parts. Think about this: If a proposition is possible it may either be true or false John is tall. If it is necessary, it must be true Either Spot is a dog or Spot is not a dog. If it is not possible, it must be false, The cat is both alive and not alive. And if it is not necessary, it may be true or false; Sosa will hit 70 homeruns in 2000. Formal Logic requires a formal representational system, a set of axioms, and rules of derivation. It does not require strict truth functionality.

A widespread AI, Cognitive Psychology, and Cognitive Science principle is that there are procedures which can be implemented on computers which represent the way that people think. These involve applying logic type rules to a formal representation base. The psychological study of reasoning in this case is the discovery of the form of the representation and the rules of inference that people use when they reason.

a.   Incoming information is transformed into a symbolic representation of the implied proposition. There are certain formal ways in which propositions are interrelated and conclusions are derived. These are implemented by the application of logic-like rules.  In information processing systems they are the application of efficient procedures which represent algorithms or heuristics.
b.   Logic (and other) problems are solved by the use of formally valid derivations on symbolic representations; apply logical rules directly to 'sentences' which represent propositions. Rules of inference include such as modus ponens, modus tollens, and hypothetical syllogism.
c.  Errors occur when a) there is an error in the representation, b) the derivation is not completed, or c) the problem solver applies the wrong rule.

A problem of application is: it is not always clear how to get from premises to representations, nor for some problems what the derivation rules are. The basic ideas for applying rules ties this process into some of those in computation. Three relevant concepts are:
Effective procedure--a procedure which transforms a form (e.g. proposition) into another one in a well specified way. The concept of 'effective procedure' is one of the more important concepts in the symbolic sciences, and one which is needed at least on an informal basis in order to work within any cognitive science. "An effective procedure is a finite, unambiguous description of a finite set of operations. The operations must be effective in the sense that there is a strictly mechanical procedure for completing them"
Algorithm--a sequence of effective procedures that is guaranteed to solve a problem. or produce a valid outcome.
Heuristic--an effective procedure that is likely to solve a problem relatively quickly.

VII. Wason's Selection task

Causal relations and mechanisms, expectancies,
Permission view. Deontic reasoning moral obligation, what one ought to do.