This covers both the syntactic and the semantic dimensions.
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Syntactic has to do with the correct or formal structure of a string of symbols in a formal language, as, for example, sending a string from a formal language to a compiler program to write it as sequence of machine instructions. Semantic has to do with interpretation or use of a string of symbols, as, for example, constructing specific models or whole sets of them, in model theory.
This study of mathematics from the outside is known as metamathematics. At its core, mathematical logic deals with mathematical concepts expressed using formal logical systems. The system of first-order logic is the most widely studied because of its applicability to foundations of mathematics and because of its desirable properties. Stronger classical logics such as second-order logic or infinitary logic are also studied, along with nonclassical logics such as intuitionistic logic.
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The mathematical field of category theory uses many formal axiomatic methods resembling those used in mathematical logic, but category theory is not ordinarily considered a subfield of mathematical logic. There are many connections between mathematical logic and computer science. Many early pioneers in computer science, such as Alan Turing , were also mathematicians and logicians. The study of computability theory in computer science is closely related to the study of computability in mathematical logic.
There is a difference of emphasis, however.
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Computer scientists often focus on concrete programming languages and feasible computability, while researchers in mathematical logic often focus on computability as a theoretical concept and on noncomputability. The study of programming language semantics is related to model theory, as is program verification in particular, model checking.
Later we will explore how things fail to be logical: because of emotions, or because there is too much data for us to process, or because too much data is missing, or because there is an element of randomness. So in order to study anything logically we have to forget the pesky details that prevent things from behaving logically. In the case of the child and the cookies, if they are allowed to eat the cookies, then the situation will not behave entirely logically. So we impose the condition that they are not allowed to eat the cookies, in which case those objects might as well not be cookies, but anything inedible as long as it is separated into discrete chunks.
This move has taken us from the real world of objects to the abstract world of ideas.
What does this gain us? The advantage of making the move into the abstract world is that we are now in a place where everything behaves logically. If I add one and one under exactly the same conditions in the abstract world repeatedly, I will always get 2. They say that insanity is doing the same thing over and over again and expecting something different to happen.
I say that logic or at least part of it is doing the same thing over and over again and expecting the same thing to happen. Where my computer is concerned, it is this that causes me some insanity.
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My computer is not logical. A powerful aspect of abstraction is that many different situations become the same when you forget some details. For example, I have found a relationship between a Bach prelude for the piano and the way we might braid our hair. Finding relationships between different situations helps us understand them from different points of view, but it is also fundamentally a unifying act.eneratrep.ml
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We can emphasize differences, or we can emphasize similarities. When we look for similarities between things we often have to discard more and more layers of outer details, until we get to the deep structures that are holding things together. Shedding outer layers, or boiling an argument down to its essence, can help us understand what we think and in particular can help us understand why we disagree with other people. A particularly helpful feature of the abstract world is that everything exists as soon as you think of it.
If you have an idea and you want to play with it, you can play with it immediately. I wish my dinner would exist as soon as I think of it. More seriously, this means that we can do thought experiments with our ideas about the world, following the logical implications through to see what will happen, without having to do real and possibly impractical experiments to get those ideas. A new system was recently introduced on the London Underground, where green markings were painted onto the platforms indicating where the doors would open. Passengers waiting for the train were instructed to stand outside the green areas, so that those disembarking the arriving train would have space to do so, instead of being faced with a wall of people trying to get on.
This sounds like a good idea to me, but it was met with outcry from some regular commuters. This is not an explicitly mathematical example but this way of making analogies is the essence of mathematical thinking, where we focus on important features of a situation to clarify it, and to make connections with other situations.
It also includes producing new propositions using existing ones. Propositions constructed using one or more propositions are called compound propositions. The propositions are combined together using Logical Connectives or Logical Operators. Since we need to know the truth value of a proposition in all possible scenarios, we consider all the possible combinations of the propositions which are joined together by Logical Connectives to form the given compound proposition.
This compilation of all possible scenarios in a tabular format is called a truth table. The truth value of is the opposite of the truth value of. The truth table of is-. The conjuction is True when both and are True, otherwise False. This proposition is true only on rainy Fridays and is false on any other rainy day or on Fridays when it does not rain.
The disjuction is True when either or is True, otherwise False. This proposition is true on any day that is a Friday or a rainy day including rainy Fridays and is false on any day other than Friday when it also does not rain. The exclusive or is True when either or is True, and False when both are true or both are false.
This proposition is true on any day that is a Friday or a rainy day not including rainy Fridays and is false on any day other than Friday when it does not rain or rainy Fridays. In the implication , is called the hypothesis or antecedent or premise and is called the conclusion or consequence. The implication is is also called a conditional statement.
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The implication is false when is true and is false otherwise it is true. You might wonder that why is true when is false.
This is because the implication guarantees that when and are true then the implication is true. But the implication does not guarantee anything when the premise is false. There is no way of knowing whether or not the implication is false since did not happen.
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